Units & Measurements — Practice & Self Evaluation

Which of the following is not the unit of distance?

1. Light year
2. Astronomical unit
3. Parsec
4. Millisecond

Answer

Millisecond

Reason — The light year, the astronomical unit (AU) and the parsec are all units of length used for measuring large astronomical distances. One light year is the distance travelled by light in vacuum in one year, one astronomical unit is the average distance of the sun from the earth, and one parsec is the distance at which the average radius of the earth's orbit around the sun subtends an angle of 1". The millisecond, on the other hand, is a sub-multiple of the second and is therefore a unit of time, not of distance.

Which of the following is the correct unit of electric current in the International System of Units (SI)?

1. Volt
2. Coulomb
3. Ampere
4. Ohm.

Answer

Ampere

Reason — Electric current is one of the seven base quantities of the SI system, and its base unit is the ampere (A). The other options are derived units of different quantities : the volt is the unit of potential difference, the coulomb is the unit of electric charge, and the ohm is the unit of resistance.

The SI unit of luminous intensity is:

1. candela
2. lumen
3. lux
4. watt

Answer

candela

Reason — Luminous intensity is one of the seven base quantities in the SI system, and its SI base unit is the candela (cd). The other options are different photometric units : the lumen is the unit of luminous flux, the lux is the unit of illuminance, and the watt is the unit of power.

In the International System of Units (SI), the unit of pressure is:

1. pascal
2. bar
3. atmosphere
4. torr

Answer

pascal

Reason — Pressure is defined as the thrust acting per unit area, so its SI unit is the newton per square metre, which is given the special name pascal (Pa). Thus 1 Pa = 1 N m^-2. The bar, the atmosphere and the torr are also units of pressure, but they are practical units and not SI units.

The SI unit of temperature is:

1. celsius
2. kelvin
3. fahrenheit
4. rankine

Answer

kelvin

Reason — Thermodynamic temperature is one of the seven base quantities of the SI system, and its base unit is the kelvin (K). The kelvin is defined in terms of the Boltzmann constant, which is a fundamental constant of nature, so the unit is invariant and easily reproducible. The celsius, fahrenheit and rankine are other temperature scales and are not SI base units.

Which of the following units is used to measure solid angle in the SI system?

1. Radian
2. Steradian
3. Degree
4. Revolution

Answer

Steradian

Reason — The SI system has two supplementary units : the radian (rad) for plane angle and the steradian (sr) for solid angle. Hence the solid angle is measured in steradian. The degree and the revolution are practical units of plane angle and not of solid angle.

Which of the following units is used to measure the activity of a radioactive substance in the SI system?

1. Gray
2. Sievert
3. Becquerel
4. Curie

Answer

Becquerel

Reason — The activity of a radioactive substance is the number of disintegrations taking place per unit time, and its SI unit is the becquerel (Bq), where 1 Bq = 1 disintegration per second. The gray is the SI unit of absorbed dose, the sievert is the SI unit of equivalent dose, and the curie is an older practical unit of activity.

In the SI system, the unit of inductance is:

1. farad
2. henry
3. weber
4. ohm

Answer

henry

Reason — The SI unit of inductance is the henry (H). The other options are the SI units of different electrical quantities : the farad is the unit of capacitance, the weber is the unit of magnetic flux, and the ohm is the unit of resistance.

The dimensional formula for power is given by:

1. [ML²T^-1]
2. [ML²T^-3]
3. [ML²T^-2]
4. [ML²T]

Answer

[ML²T^-3]

Reason — Power is defined as the work done per unit time,

$\text{Power} = \dfrac{\text{Work}}{\text{Time}}$

The dimensional formula of work is [ML²T^-2] and that of time is [T]. Therefore,

$[\text{Power}] = \dfrac{[\text{ML}^2\text{T}^{-2}]}{[\text T]} \\[1em] = [\text{ML}^2\text{T}^{-3}]$

The Reynolds number used in fluid mechanics is:

1. a dimensional quantity representing flow velocity.
2. a dimensionless quantity indicating the type of flow.
3. a dimensional quantity representing fluid viscosity.
4. a dimensionless quantity representing fluid density.

Answer

a dimensionless quantity indicating the type of flow.

Reason — The Reynolds number is the ratio of the inertial force to the viscous force acting on a flowing fluid. Since it is a ratio of two quantities of the same kind, it is a pure number having neither units nor dimensions. Its value indicates the nature of the flow — a small value corresponds to streamline (laminar) flow and a large value corresponds to turbulent flow.

Friction, air resistance, tension and thrust are forces S.I. unit of tension is:

1. joule
2. newton
3. watt
4. henry

Answer

newton

Reason — Friction, air resistance, tension and thrust are all forces, and the SI unit of force is the newton (N), where 1 N = 1 kg m s^-2. Since tension is a force, it is also measured in newton. The other options are the SI units of different quantities : the joule is the unit of work (or energy), the watt is the unit of power, and the henry is the unit of inductance.

Which of the following physical quantities is measured in units of joules in the SI system?

1. Force
2. Energy
3. Power
4. Momentum

Answer

Energy

Reason — The SI unit of energy is the joule (J), where 1 J = 1 kg m² s^-2. Since the SI is a rational system, the same unit joule is used for all forms of energy — mechanical, heat and electrical. The other options have different SI units : force is measured in newton, power in watt, and momentum in kg m s^-1.

Which of the following is not a base unit in the International System of Units (SI)?

1. Mole
2. Second
3. Newton
4. Kilogram

Answer

Newton

Reason — The seven base units of the SI system are the metre, the kilogram, the second, the ampere, the kelvin, the mole and the candela. The mole, the second and the kilogram are therefore base units. The newton is not a base unit; it is a derived unit obtained from the base units as 1 N = 1 kg m s^-2.

The unit 'Tesla' in the SI system is used to measure:

1. magnetic flux
2. magnetic field strength
3. magnetic moment
4. magnetic permeability

Answer

magnetic field strength

Reason — The tesla (T) is the SI unit of magnetic field (magnetic flux density), and 1 T = 1 Wb m^-2. The other options are measured in different units : magnetic flux is measured in weber (Wb), magnetic moment in A m² (or J T^-1), and magnetic permeability in H m^-1 (or N A^-2).

The principle of dimensional homogeneity states that:

1. the dimensions of all physical quantities must be the same.
2. in a physically meaningful equation, all terms must have the same dimensions.
3. physical quantities with different dimensions can be equated.
4. an equation is dimensionally homogeneous if it involves only dimensionless quantities.

Answer

in a physically meaningful equation, all terms must have the same dimensions.

Reason — According to the principle of homogeneity of dimensions, every term on both sides of a physically meaningful equation must have the same dimensions. This is because only quantities of the same kind can be added to, subtracted from or equated with one another. The principle is used to check the dimensional correctness of a physical relation.

Which of the following quantities does not have dimensions?

1. Gravitational potential
2. Strain
3. Velocity
4. Work

Answer

Strain

Reason — Strain is defined as the ratio of the change in dimension to the original dimension. Since it is the ratio of two quantities of the same kind, it is a pure number and therefore has neither units nor dimensions. The remaining quantities have definite dimensional formulae : gravitational potential [M⁰L²T^-2], velocity [LT^-1] and work [ML²T^-2].

The dimensional method is particularly useful for:

1. determining the units of any physical quantity.
2. deriving relationships between physical quantities.
3. verifying the numerical coefficients in equations.
4. both 1 and 2.

Answer

both determining the units of any physical quantities and deriving relationships between them.

Reason — Once the dimensional formula of a physical quantity is known, the units of that quantity in any system can be written down directly from it. The dimensional method is also used to derive the relation between physical quantities, by writing the required quantity as a product of powers of the quantities on which it depends and then equating the dimensions on both sides. However, the dimensional method cannot verify the numerical coefficients appearing in an equation, since pure numbers are dimensionless.

Which of the following statements is correct regarding dimensions?

1. Dimensions depend on the units chosen.
2. Dimensions are independent of the system of units.
3. The dimensions of a quantity change when the units are converted.
4. Dimensions are applicable only to quantities with units.

Answer

Dimensions are independent of the system of units.

Reason — The dimensions of a physical quantity are the powers to which the base quantities must be raised in order to represent that quantity. They depend only on the nature of the quantity and not on the units in which it is measured. For example, the dimensional formula of force is [MLT^-2] whether the force is expressed in newton, dyne or pound-force. Hence the dimensions remain the same in every system of units.

Dimensional analysis can help to:

1. determine whether a physical equation is dimensionally correct.
2. predict the exact form of a physical law, including constants.
3. identify whether a quantity is a scalar or vector.
4. determine the numerical value of physical quantities.

Answer

determine whether a physical equation is dimensionally correct.

Reason — Dimensional analysis is used to check the dimensional consistency of a physical relation, that is, to verify that every term of the relation has the same dimensions. For example, in the equation of motion

$\text s = \text{ut} + \dfrac{1}{2}\text{at}^2$

the dimensions of each term are

$[\text s] = [\text L],\quad [\text{ut}] = [\text{LT}^{-1}][\text T] = [\text L],\quad [\text{at}^2] = [\text{LT}^{-2}][\text T^2] = [\text L]$

Since all the terms have the dimension of length, the equation is dimensionally correct.

However, dimensional analysis cannot determine the dimensionless constant $\dfrac{1}{2}$ appearing in the second term, nor can it tell whether a quantity is a scalar or a vector, nor give the numerical value of a physical quantity. Dimensional correctness is therefore only a necessary, and not a sufficient, condition for a relation to be physically correct.

Which of the following statements about dimensional analysis is true?

1. It can determine the exact magnitude of physical quantities.
2. It is applicable only to mechanical quantities.
3. It can be used to check the plausibility of derived equations.
4. It requires a specific unit system to be applied.

Answer

It can be used to check the plausibility of derived equations.

Reason — Dimensional analysis is used to test whether a derived equation is dimensionally consistent, and hence whether it is plausible. It cannot give the exact magnitude of a physical quantity, since dimensionless constants do not appear in a dimensional equation. It is not restricted to mechanical quantities, as electrical, thermal and other quantities also have dimensional formulae. Further, dimensions are independent of the system of units, so no particular unit system is needed to apply the method.

Which statement correctly describes a dimensionless quantity?

1. It has different values in different unit systems.
2. It possesses units but no dimensions.
3. It has neither units nor dimensions.
4. It has dimensions but no units.

Answer

It has neither units nor dimensions.

Reason — A dimensionless quantity is one whose dimensional formula is [M⁰L⁰T⁰]. Such a quantity is a pure number and has neither units nor dimensions, so its numerical value is the same in every system of units. Strain, relative density, refractive index, the Reynolds number and mathematical constants such as π are examples of dimensionless quantities.

The dimensional formula for the modulus of elasticity is the same as that for:

1. pressure
2. energy
3. force
4. momentum

Answer

pressure

Reason — The modulus of elasticity is defined as the ratio of stress to strain. Strain is dimensionless, so the modulus of elasticity has the same dimensions as stress, which is force per unit area. Pressure is also defined as force per unit area. Hence

$[\text{Modulus of elasticity}] = [\text{Stress}] = \dfrac{[\text{MLT}^{-2}]}{[\text L^2]} = [\text{ML}^{-1}\text{T}^{-2}]$

which is the same as the dimensional formula of pressure.

Out of 4.0 and 4.00 which is more accurate?

1. 4.0
2. 4.00
3. Both 4.0 and 4.00
4. Nothing can be said

Answer

4.00

Reason — In a measurement, the digits measured accurately together with the first doubtful digit are called the significant figures. The measurement having the maximum number of significant figures is the most accurate.

• 4.0 has 2 significant figures, so it is measured up to an accuracy of 0.1
• 4.00 has 3 significant figures, so it is measured up to an accuracy of 0.01

Since 4.00 has more significant figures, it is the more accurate measurement.

Which of the following pairs of physical quantities is correctly matched with their SI units?

1. Force - Joule
2. Work - Newton
3. Power - Watt
4. Energy - Pascal

Answer

Power - Watt

Reason — The SI unit of power is the watt (W), where 1 W = 1 J s^-1. Hence this pair is correctly matched. The remaining pairs are wrongly matched : the SI unit of force is the newton (not the joule), the SI unit of work is the joule (not the newton), and the SI unit of energy is the joule (the pascal is the unit of pressure).

Which of the following equations is dimensionally homogeneous?

1. $\text K = \dfrac{1}{2} \text {mv}^2$
2. $\text v = \text u + \text {at}^2$
3. $\text s = \text {ut} + \dfrac{1}{2}\text {at}^2$
4. $\text F = \text {ma + bt}$

Answer

$\text s = \text {ut} + \dfrac{1}{2}\text {at}^2$

Reason — By the principle of homogeneity of dimensions, an equation is dimensionally homogeneous when every term of the equation has the same dimensions. Checking each option :

Option 3 : $\text s = \text{ut} + \dfrac{1}{2}\text{at}^2$

$[\text s] = [\text L] \\[1em] [\text{ut}] = [\text{LT}^{-1}][\text T] = [\text L] \\[1em] [\text{at}^2] = [\text{LT}^{-2}][\text T^2] = [\text L]$

All the terms have the dimension [L], so the equation is dimensionally homogeneous.

Option 2 : v = u + at²

$[\text v] = [\text{LT}^{-1}],\quad [\text u] = [\text{LT}^{-1}],\quad [\text{at}^2] = [\text{LT}^{-2}][\text T^2] = [\text L]$

The term at² has the dimension [L] while the other terms have [LT^-1], so the equation is not dimensionally homogeneous.

Option 4 : F = ma + bt

$[\text F] = [\text{MLT}^{-2}],\quad [\text{ma}] = [\text M][\text{LT}^{-2}] = [\text{MLT}^{-2}]$

The dimensions of the constant b are not specified, so the term bt cannot be assumed to have the dimensions of force. Hence the equation is not dimensionally homogeneous.

Option 1 : $\text K = \dfrac{1}{2}\text{mv}^2$

$[\text K] = [\text{ML}^2\text{T}^{-2}],\quad \left[\dfrac{1}{2}\text{mv}^2\right] = [\text M][\text{LT}^{-1}]^2 = [\text{ML}^2\text{T}^{-2}]$

Both the terms have the dimension [ML²T^-2], so this equation is also dimensionally homogeneous.

Note: Both option 1 and option 3 are dimensionally homogeneous, which means that the dimensions on both sides of each equation are the same. Therefore, both options are correct, although the question is intended to have only one correct answer. The question or its options may contain a printing error.

If two quantities have the same dimensions, they:

1. must have the same units.
2. can be added or subtracted directly.
3. must have the same physical meaning.
4. can have different physical interpretations.

Answer

can be added or subtracted directly.

Reason — By the principle of homogeneity of dimensions, only quantities having the same dimensions can be added to or subtracted from one another. However, quantities having the same dimensions need not have the same units or the same physical meaning. For example, work and torque both have the dimensional formula [ML²T^-2], but work is a scalar quantity measured in joule while torque is a vector quantity measured in N m.

Which of the following is NOT an application of dimensional analysis?

1. Checking the dimensional correctness of equations.
2. Deriving relationships between physical quantities.
3. Determining the numerical value of dimensionless constants.
4. Converting units from one system to another.

Answer

Determining the numerical value of dimensionless constants.

Reason — The three chief applications of dimensional analysis are : to check the dimensional correctness of a physical relation, to derive the relation between physical quantities, and to convert the value of a physical quantity from one system of units to another. Dimensional analysis cannot determine the numerical value of a dimensionless constant such as π or $\dfrac{1}{2}$, because pure numbers have the dimensional formula [M⁰L⁰T⁰] and therefore do not appear in a dimensional equation.

Which of the following pairs have both the same units and the same dimensions?

1. Work and power
2. Torque and energy
3. Pressure and energy
4. Momentum and impulse

Answer

Momentum and impulse

Reason — Momentum is the product of mass and velocity, while impulse is the product of force and time. Their dimensional formulae are

$[\text{Momentum}] = [\text M][\text{LT}^{-1}] = [\text{MLT}^{-1}] \\[1em] [\text{Impulse}] = [\text{MLT}^{-2}][\text T] = [\text{MLT}^{-1}]$

and their SI units are kg m s^-1 and N s, which are the same unit since 1 N s = 1 kg m s^-1. Hence this pair has both the same units and the same dimensions.

The other pairs do not satisfy both conditions :

• Work and power — [ML²T^-2] and [ML²T^-3], with units joule and watt. Different dimensions and different units.
• Torque and energy — both [ML²T^-2], but the units are N m and joule respectively, and torque is a vector while energy is a scalar. Same dimensions but different units.
• Pressure and energy — [ML^-1T^-2] and [ML²T^-2], with units pascal and joule. Different dimensions and different units.

If percentage error in the measurement of mass and volume of an object ate 2% and 3% respectively, then the percentage error in the measurement of density of the object is:

1. 1%
2. 0.66%
3. 5%
4. 6%

Answer

5%

Reason —

Given,

• Percentage error in the measurement of mass = 2%
• Percentage error in the measurement of volume = 3%

The density of the object is

$\rho = \dfrac{\text m}{\text V}$

When a quantity is expressed as a product or quotient of measured quantities, the maximum percentage error in it is obtained by adding the percentage errors of all the quantities, each multiplied by the magnitude of its power. Here the powers of m and V are each 1. Therefore,

$\dfrac{\Delta \rho}{\rho} \times 100 = \dfrac{\Delta \text m}{\text m} \times 100 + \dfrac{\Delta \text V}{\text V} \times 100 \\[1em] = 2 + 3 \\[1em] = 5\$

The number of significant figures in 30.00 m is:

1. 1
2. 2
3. 3
4. 4

Answer

4

Reason — The rules for determining significant figures give :

• The non-zero digit 3 is significant.
• The zero lying to the right of a non-zero digit but on the left of the decimal point is significant, so the zero in 30 is significant.
• All the zeros to the right of the decimal point after a non-zero digit are significant, so both the zeros after the decimal point are significant.

Hence in 30.00 the significant figures are 3, 0, 0 and 0, that is, 4 significant figures.

The order of magnitude of 11 is:

1. 0
2. 1
3. 2
4. −1

Answer

1

Reason — To find the order of magnitude, the quantity is written in the form N × 10^x, where N is a number between 1 and 10 and x is a positive or negative integer. If N is equal to or smaller than $\sqrt{10}$ = 3.16, the order of magnitude is 10^x; if N is greater than 3.16, the order of magnitude is 10^x+1.

Here,

$11 = 1.1 \times 10^{1}$

Since N = 1.1 is smaller than 3.16, the order of magnitude is 10¹, that is, 1.

When using a screw gauge, the pitch of the screw is 0.5 mm, and the circular scale has 100 divisions. What is the least count of the screw gauge?

1. 0.01 mm
2. 0.005 mm
3. 0.1 mm
4. 0.02 mm

Answer

0.005 mm

Reason —

Given,

• Pitch of the screw = 0.5 mm
• Number of divisions on the circular scale = 100

The least count of a screw gauge is given by

$\text{Least count} = \dfrac{\text{Pitch}}{\text{Number of divisions on the circular scale}}$

Substituting the values,

$\text{Least count} = \dfrac{0.5\ \text{mm}}{100} \\[1em] = 0.005\ \text{mm}$

The significant figures in the measurement 0.004500 kg are:

1. 3
2. 4
3. 5
4. 6

Answer

4

Reason — The rules for determining significant figures give :

• All the initial zeros on the right of the decimal point but on the left of the first non-zero digit are not significant, since they only fix the position of the decimal point. Hence the zeros in 0.00 are not significant.
• All the zeros to the right of the decimal point after a non-zero digit are significant. Hence the two trailing zeros are significant.

Hence in 0.004500 the significant figures are 4, 5, 0 and 0, that is, 4 significant figures.

In a vernier callipers, if 10 divisions on the vernier scale coincide with 9 divisions on the main scale, what is the least count of the vernier callipers?

1. 0.1 mm
2. 0.01 mm
3. 0.02 mm
4. 0.05 mm

Answer

0.1 mm

Reason —

Given,

• 10 vernier scale divisions coincide with 9 main scale divisions
• Value of one main scale division = 1 mm

The least count of a vernier callipers is the difference between the value of one main scale division and one vernier scale division,

$\text{Least count} = 1\ \text{M.S.D.} - 1\ \text{V.S.D.}$

Since 10 V.S.D. = 9 M.S.D., one vernier scale division is

$1\ \text{V.S.D.} = \dfrac{9}{10}\ \text{M.S.D.} = 0.9\ \text{mm}$

Therefore,

$\text{Least count} = 1\ \text{mm} - 0.9\ \text{mm} \\[1em] = 0.1\ \text{mm}$

A spherometer is used to measure the radius of curvature of a spherical surface. If the measured height difference is 0.5 cm and the distance between the legs is 3 cm, what is the approximate radius of curvature?

1. 18 cm
2. 12 cm
3. 9 cm
4. 3.25 cm

Answer

3.25 cm

Reason —

Given,

• Height difference (sagitta), h = 0.5 cm
• Distance between the legs, l = 3 cm

The radius of curvature measured by a spherometer is given by

$\text R = \dfrac{\text l^2}{6\text h} + \dfrac{\text h}{2}$

Substituting the values,

$\text R = \dfrac{(3)^2}{6 \times 0.5} + \dfrac{0.5}{2} \\[1em] = \dfrac{9}{3} + 0.25 \\[1em] = 3 + 0.25 \\[1em] = 3.25 \text { cm}$

The reading on the main scale of a screw gauge is 2.5 mm, and the circular scale reading is 30 divisions. If the least count of the screw gauge is 0.01 mm, what is the total measurement?

1. 2.53 mm
2. 2.55 mm
3. 2.80 mm
4. 2.85 mm

Answer

2.53 mm

Reason —

Given,

• Main scale reading = 2.5 mm
• Circular scale reading = 30 divisions
• Least count = 0.01 mm

The total reading of a screw gauge is given by

$\text{Total reading} = \text{Main scale reading} + (\text{Circular scale reading} \times \text{Least count})$

Substituting the values,

$\text{Total reading} = 2.5 + (30 \times 0.01) \\[1em] = 2.5 + 0.30 \\[1em] = 2.53\ \text{mm}$

The dimensional formula of a physical quantity is defined as:

1. the numerical value assigned to the quantity in a specific unit system.
2. the expression showing how and which of the base quantities represent the dimensions of a physical quantity.
3. the process of converting one unit to another.
4. the scalar multiple of the unit of the quantity.

Answer

the expression showing how and which of the base quantities represent the dimensions of a physical quantity.

Reason — The dimensional formula of a physical quantity is the expression which shows which of the base quantities, and with what powers, are contained in that quantity. It is written in terms of the symbols of the base quantities, such as [M] for mass, [L] for length and [T] for time. For example, the dimensional formula of force is [MLT^-2], which shows that force contains mass to the power 1, length to the power 1 and time to the power −2.

Assertion (A): The SI unit of time is the second.

Reason (R): The second is defined based on the frequency of radiation corresponding to the transition between two energy levels of the caesium-133 atom.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The second is the SI base unit of time.

Reason (R) is also correct: The second is defined in terms of the frequency of the radiation emitted in the transition between two hyperfine energy levels of the ground state of the caesium-133 atom. One second is equal to 9,192,631,770 periods of this radiation. This definition fixes the exact standard by which the second is measured, so it explains what the SI unit of time is. Hence the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The SI unit of electric current is the ampere.

Reason (R): The ampere is defined based on the force between two parallel conductors carrying current.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: Electric current is one of the seven base quantities of the SI system, and the ampere (A) is its base unit.

Reason (R) is also correct: The ampere is defined as that constant current which, when maintained in two straight parallel conductors of infinite length and negligible cross-section placed 1 m apart in vacuum, produces a force of 2 × 10^-7 N per metre of length between them. This gives the standard by which the ampere is fixed, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Note: The SI definition of the ampere was revised on 20 May 2019. Under the revised definition, the ampere is fixed by taking the numerical value of the elementary charge to be e = 1.602176634 × 10^-19 C. If the revised definition is followed, the Reason is false and the answer becomes option 3. The answer given above follows the definition used in the prescribed textbook.

Assertion (A): The SI unit of temperature is kelvin.

Reason (R): Kelvin is defined as 1/273.16 of the thermodynamic temperature of the triple point of water.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: Thermodynamic temperature is one of the seven base quantities of the SI system, and the kelvin (K) is its base unit.

Reason (R) is also correct: The kelvin is defined as $\dfrac{1}{273.16}$ of the thermodynamic temperature of the triple point of water. This fixes the standard by which the kelvin is measured, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Note: The SI definition of the kelvin was revised on 20 May 2019, and it is now defined by fixing the numerical value of the Boltzmann constant. The answer given above follows the definition used in the prescribed textbook.

Assertion (A): The SI unit of luminous intensity is the candela.

Reason (R): Candela is defined based on the intensity of light, emitted in a particular direction by a source of a specific frequency.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: Luminous intensity is one of the seven base quantities of the SI system, and the candela (cd) is its base unit.

Reason (R) is also correct: The candela is defined as the luminous intensity, in a given direction, of a source emitting monochromatic radiation of a specified frequency and of a specified radiant intensity in that direction. This gives the standard by which the candela is fixed, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The SI unit of mass is the kilogram.

Reason (R): The kilogram is defined by the mass of the international prototype, a platinum-iridium cylinder stored in France.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: Mass is one of the seven base quantities of the SI system, and the kilogram (kg) is its base unit.

Reason (R) is false: The kilogram was formerly defined as the mass of the international prototype, a platinum-iridium cylinder kept at Sèvres in France. Under the revised SI, the kilogram is no longer defined in this way; it is defined by fixing the numerical value of the Planck constant as h = 6.62607015 × 10^-34 J s. The base units of the SI are now defined in terms of fundamental constants of nature so that they are invariant and easily reproducible, and are not tied to any physical object.

Therefore, assertion is true but reason is false.

Note: Some editions of the prescribed text still give the older definition of the kilogram in terms of the platinum-iridium prototype. If that definition is followed, the Reason is true and the answer becomes option 1.

Assertion (A): The number of significant figures in the measurement 0.00450 m is three.

Reason (R): Leading zeros are not considered significant figures.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: In 0.00450, the significant figures are 4, 5 and the trailing zero. The trailing zero lies to the right of the decimal point after a non-zero digit, so it is significant since it arises due to measurement. Hence there are three significant figures.

Reason (R) is also correct: All the initial zeros on the right of the decimal point but on the left of the first non-zero digit are not significant, since they only fix the position of the decimal point. Leaving out these leading zeros in 0.00450 leaves exactly three significant digits, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The least count of a vernier calliper is the smallest measurement it can accurately measure.

Reason (R): The least count of a vernier calliper is determined by the difference between the values of one main scale division and one vernier scale division.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The least count of a measuring instrument is the smallest quantity that the instrument is capable of measuring accurately.

Reason (R) is also correct: For a vernier callipers, the least count is given by

$\text{Least count} = 1\ \text{M.S.D.} - 1\ \text{V.S.D.}$

that is, the difference between the values of one main scale division and one vernier scale division. This difference fixes the smallest measurable value, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The relative error in a measurement can be reduced by taking measurements with higher precision instruments.

Reason (R): Relative error is the absolute error divided by the true value of the measured quantity.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: An instrument of smaller least count gives a smaller absolute error, and since the relative error depends on the absolute error, the relative error is also reduced.

Reason (R) is also correct: The relative error is defined as

$\text{Relative error} = \dfrac{\text{Absolute error}}{\text{True value}}$

However, the Reason only states the definition of relative error. It does not by itself explain why the use of a more precise instrument reduces the error; that follows from the fact that a smaller least count gives a smaller absolute error. Hence the Reason is not the correct explanation of the Assertion.

Therefore, both assertion and reason are true but reason is not the correct explanation of assertion.

Assertion (A): Random errors in measurements are due to unpredictable variations in experimental conditions.

Reason (R): Random errors can be minimized by increasing the number of observations.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: Random errors arise from small changes in the conditions of the experiment and from incorrect judgement of the observer in taking readings. Such causes are unknown and uncontrollable, so the exact cause of a random error cannot be traced.

Reason (R) is also correct: Since random errors occur irregularly and are equally likely to be positive or negative, they are minimised by taking a large number of readings of the same quantity and then taking their arithmetic mean, the positive and negative errors tending to cancel one another. This follows directly from the unpredictable nature of these errors, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The error in the sum of two measurements is the sum of the absolute errors in the individual measurements.

Reason (R): Errors propagate linearly when adding quantities.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: If z = x + y, where the limiting errors in x and y are ±Δx and ±Δy, then

$\text z \pm \Delta \text z = (\text x \pm \Delta \text x) + (\text y \pm \Delta \text y)$

so that the maximum possible error in z is

$\Delta \text z = \Delta \text x + \Delta \text y$

Reason (R) is also correct: In addition (and in subtraction), the absolute errors combine directly, that is, the limiting error in the final result is the sum of the absolute errors in the quantities involved. This linear propagation is exactly what leads to the result stated in the Assertion, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The absolute error in a measurement is the difference between the measured value and the true value.

Reason (R): Absolute error can be positive, negative, or zero, depending on the measured value relative to the true value.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: The absolute error in a measurement is the difference between the measured value of the quantity and its true value.

Reason (R) is false: The absolute error is taken as the magnitude of this difference, the sign being ignored. Hence the absolute error is always positive or zero, and can never be negative.

Therefore, assertion is true but reason is false.

Assertion (A): The metre was originally defined based on the Earth's meridian but is now defined in terms of the speed of light.

Reason (R): The speed of light in vacuum is a universal constant, making it an ideal basis for defining units of length.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The metre was originally defined as a fraction of the length of the earth's meridian. It is now defined as the length of the path travelled by light in vacuum in $\dfrac{1}{299,792,458}$ part of a second.

Reason (R) is also correct: The speed of light in vacuum is a fundamental constant of nature and has the same value at all places and at all times. A unit defined in terms of such a constant is invariant and easily reproducible, which is why the metre is now defined in this way. Hence the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The precision of a measuring instrument is reflected in the number of significant figures it can record.

Reason (R): More significant figures indicate higher precision because they show finer resolution in the measurement.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The precision of an instrument is determined by its least count — the smaller the least count, the greater is the precision. An instrument of smaller least count records a greater number of significant figures, so the precision is reflected in the number of significant figures recorded.

Reason (R) is also correct: A greater number of significant figures means that the measurement has been made with finer divisions, that is, with a smaller least count. This is precisely why more significant figures indicate higher precision, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): In a series of measurements, the mean value gives the most accurate estimate of the true value.

Reason (R): The mean value reduces the impact of random errors on the final measurement.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: When a quantity is measured a large number of times, the arithmetic mean of all the readings is taken as the most accurate value of the quantity, that is, as the true value.

Reason (R) is also correct: Random errors are equally likely to be positive or negative, so on taking the arithmetic mean the positive and negative errors tend to cancel one another and the mean comes very close to the correct value. This is exactly why the mean is the best estimate, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The vernier scale of a vernier calliper provides more precise measurements than the main scale alone.

Reason (R): The vernier scale has a finer resolution, allowing smaller divisions of the main scale to be measured.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: With the main scale alone, a length can be read only up to one main scale division. The vernier scale reduces the least count of the instrument, so a length can be read to a fraction of a main scale division, giving a more precise measurement.

Reason (R) is also correct: The divisions of the vernier scale are slightly smaller than those of the main scale, so that

$\text{Least count} = 1\ \text{M.S.D.} - 1\ \text{V.S.D.}$

which is a fraction of one main scale division. This finer resolution is the reason for the greater precision, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The standard deviation is a measure of the spread of data in a set of measurements.

Reason (R): A smaller standard deviation indicates that the data points are closer to the mean value.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The standard deviation measures how much the individual readings deviate from the arithmetic mean, and therefore indicates the spread or dispersion of the set of measurements.

Reason (R) is also correct: A small standard deviation means that the readings are closely clustered about the mean, that is, the spread is small; a large standard deviation means the readings are widely scattered. This is exactly what makes the standard deviation a measure of spread, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): Surface tension and surface energy have the same dimensions.

Reason (R): Because both have the same SI unit.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: Surface tension is the force acting per unit length and surface energy is the energy per unit area. Writing their dimensional formulae,

$[\text{Surface tension}] = \dfrac{[\text{MLT}^{-2}]}{[\text L]} = [\text{ML}^0\text T^{-2}] \\[1em] [\text{Surface energy}] = \dfrac{[\text{ML}^2\text T^{-2}]}{[\text L^2]} = [\text{ML}^0\text T^{-2}]$

Hence both have the same dimensional formula [ML⁰T^-2].

Reason (R) is also correct: The SI unit of surface tension is N m^-1 and that of surface energy is J m^-2. Since 1 J = 1 N m,

$\text{J m}^{-2} = \text{N m} \times \text m^{-2} = \text{N m}^{-1}$

so both have the same SI unit. Two quantities having the same unit must have the same dimensions, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The dimensional formula for force is [MLT^-2].

Reason (R): Force is the product of mass and acceleration.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The dimensional formula of force is [MLT^-2].

Reason (R) is also correct: From Newton's second law, F = m × a. Writing the dimensional formulae,

$[\text F] = [\text M][\text{LT}^{-2}] = [\text{MLT}^{-2}]$

The dimensional formula of force is obtained directly from this definition, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The unit of power, the watt, is equivalent to one joule per second.

Reason (R): Power is the rate at which work is done or energy is transferred.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The SI unit of power is the watt (W), and 1 W = 1 J s^-1.

Reason (R) is also correct: Power is defined as the work done per unit time,

$\text{Power} = \dfrac{\text{Work}}{\text{Time}}$

Since the SI is a coherent system, the unit of power follows directly from this definition as the joule per second, without any numerical factor. Hence the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): Systematic errors can be eliminated by calibrating the measuring instrument.

Reason (R): Calibration helps to adjust the instrument to give correct readings by comparing it with a standard reference.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: Systematic errors occur in one direction and follow a definite rule, for example the calibration error or the zero error of an instrument. Since the rule governing them can be identified, they can be removed by applying proper corrections, in particular by re-calibrating the instrument.

Reason (R) is also correct: Calibration consists in comparing the readings of the instrument with a standard reference and adjusting the instrument so that it gives correct readings. This removes the fixed bias of the instrument, which is exactly the source of the systematic error. Hence the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): In y = A sin (ωt - kx), (ωt - kx) is dimensionless.

Reason (R): Because dimension of ω = [M⁰L⁰T].

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: The argument of a trigonometric function is an angle and must therefore be dimensionless. Checking the two terms,

$[\omega \text t] = [\text T^{-1}][\text T] = [\text M^0\text L^0\text T^0] \\[1em] [\text{kx}] = [\text L^{-1}][\text L] = [\text M^0\text L^0\text T^0]$

Both terms are dimensionless, and hence (ωt − kx) is dimensionless.

Reason (R) is false: The dimensional formula of the angular frequency ω is [M⁰L⁰T^-1], and not [M⁰L⁰T] as stated.

Therefore, assertion is true but reason is false.

Assertion (A): Dimensional constants are quantities whose values are constant.

Reason (R): Dimensional constants are dimensionless.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: Dimensional constants are quantities which have a fixed value and also possess dimensions, for example the universal gravitational constant G and Planck's constant h.

Reason (R) is false: Dimensional constants have dimensions — this is why they are called dimensional constants. For example, the dimensional formula of G is [M^-1L³T^-2] and that of h is [ML²T^-1]. It is the dimensionless constants, such as π and e, which have no dimensions.

Therefore, assertion is true but reason is false.

Assertion (A): The given equation $\text x = \text x_\text o + \text u_\text o \text t + \dfrac{1}{2}\text {at}^2$ is dimensionally correct, where $\text x$ is distance travelled by a particle in time $\text t$, initial position $\text x_\text o$, initial velocity $\text u_\text o$ and uniform acceleration '$\text a$' is along the direction of motion.

Reason (R): Dimensional analysis can be used for checking dimensional consistency or homogeneity of the equation.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: Writing the dimensional formula of each term,

$[\text x] = [\text L],\quad [\text x_\text o] = [\text L] \\[1em] [\text u_\text o \text t] = [\text{LT}^{-1}][\text T] = [\text L] \\[1em] \left[\dfrac{1}{2}\text{at}^2\right] = [\text{LT}^{-2}][\text T^2] = [\text L]$

Since every term has the dimension [L], the equation is dimensionally homogeneous and hence dimensionally correct.

Reason (R) is also correct: By the principle of homogeneity of dimensions, every term of a physically meaningful equation must have the same dimensions, so dimensional analysis is used to check the dimensional consistency of an equation. It is by applying this very method that the equation in the Assertion is found to be correct, so the Reason correctly explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): Systematic errors in measurement can be minimized by taking repeated measurements.

Reason (R): Systematic errors arise from predictable and consistent factors.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are false.

Explanation

Assertion (A) is false: Repeated measurements minimise random errors, not systematic errors. Systematic errors occur in one direction and remain the same in every measurement, so taking the arithmetic mean of a large number of readings does not remove them. They are removed by applying proper corrections, such as correcting the zero error or re-calibrating the instrument.

Reason (R): Systematic errors do arise from definite and consistent causes such as a defective instrument, a zero error, an imperfect experimental technique or the personal bias of the observer, and they follow a definite rule. This statement, as it stands, is correct.

Since the assertion is false while the reason states the nature of systematic errors correctly, the correct combination is "assertion false, reason true", which is not listed among the four options. Of the given choices, option 4 is marked as the answer, on the understanding that the assertion is false.

Note: The given options do not include the case “Assertion is false, but Reason is true.” The Assertion is false because repeated measurements help to reduce random errors, not systematic errors. The Reason is true because it correctly describes systematic errors. Therefore, the required correct option is missing from the question.

Assertion (A): The significant figures in the product of 2.5 x 10² and 4.56 is three.

Reason (R): When multiplying two numbers, the number of significant figures in the result is equal to the number with the fewest significant figures.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are false.

Explanation

Assertion (A) is false: In the product of 2.5 × 10² and 4.56, the number 2.5 × 10² has 2 significant figures and 4.56 has 3 significant figures. In multiplication, the result carries the least number of significant figures among the given numbers, that is, 2. Hence the product has 2 significant figures, not three, so the assertion is false.

Reason (R): When two numbers are multiplied, the number of significant figures in the result is equal to that of the number having the fewest significant figures. This rule, as stated, is correct.

Since the assertion is false while the reason states the rule correctly, the correct combination is "assertion false, reason true", which is not listed among the four options. Of the given choices, option 4 is marked as the answer, on the understanding that the assertion is false.

Note: The given options do not include the case “Assertion is false, but Reason is true.” The Assertion is false because the product has 2 significant figures, not 3. The Reason is true because it correctly states the rule for determining significant figures. Therefore, the required correct option is missing from the question.

Assertion (A): Force can be added to pressure.

Reason (R): Force and pressure have same dimensions.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are false.

Explanation

Assertion (A) is false: By the principle of homogeneity of dimensions, only quantities having the same dimensions can be added to one another. Force and pressure are quantities of different kinds, so force cannot be added to pressure.

Reason (R) is also false: The dimensional formulae of the two quantities are

$[\text{Force}] = [\text{MLT}^{-2}] \\[1em] [\text{Pressure}] = \dfrac{[\text{MLT}^{-2}]}{[\text L^2]} = [\text{ML}^{-1}\text T^{-2}]$

which are not the same.

Therefore, both assertion and reason are false.

Why length, mass and time are chosen as base quantities in mechanics?

Answer

Length, mass and time are chosen as base quantities in mechanics because : They are independent of one another, that is, none of them can be expressed in terms of the other two.

Are all constants dimensionless?

Answer

No, all constants are not dimensionless. Only pure numbers such as π and e are dimensionless constants. Physical constants such as the gravitational constant G, whose dimensional formula is [M^-1L³T^-2], and Planck's constant h, whose dimensional formula is [ML²T^-1], have definite dimensions.

How is S.I. a coherent system of units?

Answer

The S.I. is called a coherent system of units because all the derived units in this system can be obtained from the base units by simple multiplication or division, without introducing any numerical factor (other than unity).

For example, the S.I. unit of force (newton) is obtained from the base units as 1 N = 1 kg m s^-2, and the S.I. unit of work (joule) is obtained as 1 J = 1 kg m² s^-2. In each case the derived unit follows directly from the base units with no extra numerical factor.

Hence, the S.I. is a coherent system of units.

Does the magnitude of a physical quantity change with change in system of unit?

Answer

No, the magnitude of a physical quantity does not change with a change in the system of units. Only the numerical value and the size of the unit change, in such a way that their product remains the same, that is,

$\text n_1\text u_1 = \text n_2\text u_2$

If the size of the unit is small, the numerical value is large, and vice versa.

Are light year (ly) and parsec (pc) units of time?

Answer

No, the light year and the parsec are units of distance and not of time. One light year is the distance travelled by light in vacuum in one year, and one parsec is the distance at which the average radius of the earth's orbit around the sun subtends an angle of 1" (second of arc).

Which one is larger, light year or parsec? How much?

Answer

The parsec is larger. Since 1 ly = 9.46 × 10¹⁵ m and 1 parsec = 3.08 × 10¹⁶ m,

$\dfrac{1\ \text{parsec}}{1\ \text{ly}} = \dfrac{3.08 \times 10^{16}\ \text m}{9.46 \times 10^{15}\ \text m} = 3.26$

Hence, 1 parsec = 3.26 light years.

How many times larger is kg than mg?

Answer

Since 1 kg = 10³ g and 1 g = 10³ mg,

$1\ \text{kg} = 10^3 \times 10^3\ \text{mg} \\[1em] = 10^{6}\ \text{mg}$

Hence, the kilogram is 10⁶ times larger than the milligram.

Why has second been defined in terms of period of vibration of the atom Cs-133?

Answer

The second has been defined in terms of the period of vibration of the caesium-133 atom because a good unit should be precisely defined, easily reproducible and should not change with time, place or physical conditions such as temperature and pressure. The vibrations of the caesium-133 atom satisfy all these conditions, since the fundamental properties of caesium atoms remain constant everywhere. Hence the second so defined is invariant and easily reproducible.

Are there more microseconds in a second than number of seconds in a year?

Answer

No. In one second there are 10⁶ microseconds, whereas in one year there are

$365 \times 24 \times 60 \times 60 = 3.15 \times 10^{7}\ \text s$

Since 3.15 × 10⁷ is greater than 10⁶, the number of seconds in a year is greater than the number of microseconds in a second.

How much greater is a millisecond than a microsecond?

Answer

Since 1 millisecond = 10^-3 s and 1 microsecond = 10^-6 s,

$\dfrac{1\ \text{millisecond}}{1\ \text{microsecond}} = \dfrac{10^{-3}\ \text s}{10^{-6}\ \text s} \\[1em] = 10^{3}$

Hence, a millisecond is 10³ times greater than a microsecond.

Are inertial and gravitational masses of an object different?

Answer

No, the inertial mass and the gravitational mass of an object are not different. They represent different physical concepts — the inertial mass is a measure of the inertia of the body, while the gravitational mass determines the gravitational force on it — but experimentally they are found to be equal in magnitude.

Which is the most accurate clock?

Answer

The atomic clock is the most accurate clock. The latest hydrogen maser clock is accurate to 1 s in 3 × 10⁷ years.

What is the number of significant figures in the result obtained by multiplying or dividing a measurement by a single-digit pure number?

Answer

The number of significant figures remains unchanged, that is, it is the same as in the original measurement. This is because pure numbers are not obtained by measurement and have unlimited accuracy, so they are not counted while deciding the number of significant figures in the result.

How do you take care of significant figures in calculations?

Answer

The accuracy of a result obtained by calculation can never be greater than the accuracy of the original measurements. Therefore the number of significant figures in the final result should not be more than the number of significant figures in the least accurate quantity given, and the non-significant figures are dropped by rounding off.

What is the difference between the length measurements 2.0 cm and 2.000 cm ?

Answer

The measurement 2.0 cm has two significant figures and is accurate up to the first place of decimal only, whereas the measurement 2.000 cm has four significant figures and is accurate up to the third place of decimal. Since the measurement having the maximum number of significant figures is the most accurate, 2.000 cm is the more accurate measurement of the two.

Name two physical quantities having dimensions of work.

Answer

Energy and torque (moment of force). Both have the dimensional formula [ML²T^-2], which is the same as that of work.

Does a mechanical quantity have different dimensions in different systems of units?

Answer

No, a mechanical quantity has the same dimensions in all systems of units. The dimensions depend only on the nature of the quantity and not on the units chosen; only the units in which the quantity is expressed may change.

Can a quantity has unit, but still be dimensionless?

Answer

Yes. The plane angle and the solid angle have the units radian (rad) and steradian (sr) respectively, yet both are dimensionless quantities, since each is the ratio of two quantities of the same kind.

Can a quantity has dimensions, but no unit?

Answer

No. If a quantity has dimensions, it must also have a unit. The dimensions show which base quantities are contained in the quantity, and every base quantity has a unit, so the quantity is necessarily measured in a unit derived from those base units.

Does the magnitude of a 'dimensionless' quantity depend upon the system of units used ?

Answer

No, the magnitude of a dimensionless quantity does not depend upon the system of units used. A dimensionless quantity is a pure number having no units, so its numerical value remains the same in every system of units.

What is meant by torr?

Answer

The torr is a practical unit of pressure. One torr is the pressure exerted by a mercury column of height 1 mm, that is, 1 torr = 1 mm of mercury.

What is the order of mass of our universe?

Answer

The order of the mass of our universe is 10⁵⁵ kg.

Name the balance used to measure the weight of a body.

Answer

Spring balance.

What is the order of the age of earth?

Answer

The order of the age of the earth is 10¹⁷ s.

How can random error be minimised?

Answer

Random errors occur irregularly and are equally likely to be positive or negative. They can therefore be minimised by taking a large number of readings of the same quantity and then taking their arithmetic mean, since the positive and negative errors tend to cancel one another.

What are the dimensions of angular displacement?

Answer

Angular displacement (angle) is the ratio of the arc length to the radius, that is, the ratio of two lengths. Hence it is dimensionless and its dimensional formula is [M⁰L⁰T⁰].

Give an example of a dimensionless constant.

Answer

Avogadro's number is a dimensionless constant. Other examples of dimensionless constants are the pure numbers π and e.

Express 0.000003 kg in power of 10.

Answer

$0.000003\ \text{kg} = \dfrac{3}{10^{6}}\ \text{kg} \\[1em] = 3 \times 10^{-6}\ \text{kg}$

Write a measured size corresponding to the following orders of length:

(a) 10⁷ m,

(b) 10⁴ m,

(c) 10³ m,

(d) 10² m,

(e) 10^-3 m,

(f) 10^-6 m,

(g) 10^-15 m.

Answer

(a) Radius of the earth,

(b) Height of Mount Everest,

(c) Distance travelled by sound in air in 3 s,

(d) Length of a playground,

(e) Thickness of a cardboard,

(f) Mean free path of an air molecule,

(g) Size of an atomic nucleus.

State the number of significant figures in the following:

(i) 0.050 cm,

(ii) 0.0009 m,

(iii) 0.039 m,

(iv) 6.37 x 10⁶ m,

(v) 0.009203 m²,

(vi) 1.99 x 10³⁰ kg,

(vii) 9.1 x 10^-31 kg,

(viii) 3.08 x 10⁶ s,

(ix) 2.99 x 10⁸ m s^-1

(x) 1.60 x 10^-19 J.

Answer

(i) Two — the initial zeros are not significant, while the digits 5 and 0 are significant.

(ii) One — the initial zeros are not significant, so only the digit 9 is significant.

(iii) Two — the initial zero is not significant, so the digits 3 and 9 are significant.

(iv) Three — the digits 6, 3 and 7 are significant; the power of ten is not counted.

(v) Four — the initial zeros are not significant, while the digits 9, 2, 0 and 3 are significant.

(vi) Three — the digits 1, 9 and 9 are significant.

(vii) Two — the digits 9 and 1 are significant.

(viii) Three — the digits 3, 0 and 8 are significant, the zero lying between two non-zero digits.

(ix) Three — the digits 2, 9 and 9 are significant.

(x) Three — the digits 1, 6 and 0 are significant, the trailing zero after the decimal point being significant.

Write the dimensional formulae of the following quantities:

(i) Velocity, Force, Momentum, Energy and Surface tension.

(ii) Impulse, Moment of force, Angle.

(iii) Pressure, Kinetic energy, potential energy, Frequency and Strain.

(iv) Work, Acceleration, Gravitational constant G.

(v) Stress, Power.

(vi) Moment of Inertia.

(vii) Young's modulus of Elasticity or Modulus of Rigidity.

(viii) Angular Momentum

(ix) Velocity gradient

(x) Force constant

(xi) Coefficient of ViscositY

(xii) Gravitational Potential

(xiii) Gravitational Potential Energy

(xiv) Latent Heat

(xv) Specific Heat

(xvi) Coefficient of Thermal Conductivity

(xvii) Boltzmann's Constant

(xviii) Gas Constant

(xix) Planck's Constant

Answer

(i) Velocity → [M⁰LT^-1]

Force → [MLT^-2]

Momentum → [MLT^-1]

Energy → [ML²T^-2]

Surface tension → [ML⁰T^-2]

(ii) Impulse → [MLT^-1]

Moment of force → [ML²T^-2]

Angle → [M⁰L⁰T⁰]

(iii) Pressure → [ML^-1T^-2]

Kinetic energy → [ML²T^-2]

Potential energy → [ML²T^-2]

Frequency → [M⁰L⁰T^-1]

Strain → [M⁰L⁰T⁰]

(iv) Work → [ML²T^-2]

Acceleration → [M⁰LT^-2]

Gravitational constant (G) → [M^-1L³T^-2]

(v) Stress → [ML^-1T^-2]

Power → [ML²T^-3]

(vi) Moment of inertia → [ML²T⁰]

(vii) Young's modulus of elasticity or Modulus of rigidity → [ML^-1T^-2]

(viii) Angular momentum → [ML²T^-1]

(ix) Velocity gradient → [M⁰L⁰T^-1]

(x) Force constant → [ML⁰T^-2]

(xi) Coefficient of viscosity → [ML^-1T^-1]

(xii) Gravitational potential → [M⁰L²T^-2]

(xiii) Gravitational potential energy → [ML²T^-2]

(xiv) Latent heat → [M⁰L²T^-2]

(xv) Specific heat → [M⁰L²T^-2Θ^-1]

(xvi) Coefficient of thermal conductivity → [MLT^-3Θ^-1]

(xvii) Boltzmann's constant → [ML²T^-2Θ^-1]

(xviii) Gas constant → [ML²T^-2Θ^-1 mol^-1]

(xix) Planck's constant → [ML²T^-1]

Round off the following to three significant figures.

(i) 1.0084,

(ii) 36.99,

(iii) 0.005135,

(iv) 6.225,

(v) 0.03828,

(vi) 3.15360 x 10⁷.

Answer

(i) 1.01 — the digit to be dropped is 8, which is more than 5, so the preceding digit 0 is increased by 1.

(ii) 37.0 — the digit to be dropped is 9, which is more than 5, so the preceding digit 9 is increased by 1, giving 37.0.

(iii) 0.00514 — the digit to be dropped is 5, and the preceding digit 3 is odd, so it is increased by 1.

(iv) 6.22 — the digit to be dropped is 5, and the preceding digit 2 is even, so it is retained unchanged.

(v) 0.0383 — the digit to be dropped is 8, which is more than 5, so the preceding digit 2 is increased by 1.

(vi) 3.15 × 10⁷ — the digit to be dropped is 3, which is less than 5, so the preceding digit 5 is retained unchanged.

An unknown quantity X multiplied by velocity equals power. Recognise X, using method of dimensions.

Answer

Given,

• X × velocity = power

Writing the dimensional formulae of the known quantities,

$[\text{Power}] = [\text{ML}^2\text{T}^{-3}],\quad [\text{Velocity}] = [\text{LT}^{-1}]$

Therefore,

$[\text X] = \dfrac{[\text{Power}]}{[\text{Velocity}]} \\[1em] = \dfrac{[\text{ML}^2\text{T}^{-3}]}{[\text{LT}^{-1}]} \\[1em] = [\text{MLT}^{-2}]$

This is the dimensional formula of force.

Hence, the unknown quantity X is force.

What is the order of size of our galaxy and that of the height of an average man and of mean free path of an air molecule.

Answer

The orders of magnitude are :

• Size of our galaxy ⟶ 10²⁰ m
• Height of an average man ⟶ 10⁰ m
• Mean free path of an air molecule ⟶ 10^-6 m

Which of the following measured lengths is most accurate and why?

(i) 3.0 cm, (ii) 3.00 cm, (iii) 3.000 cm.

Answer

3.000 cm is the most accurate measurement. The measurement having the maximum number of significant figures is the most accurate, and 3.000 cm has four significant figures whereas 3.0 cm has two and 3.00 cm has three.

The mass of a body measured by three persons was expressed as

(i) 2000 g, (ii) 2.00 kg, (iii) 2.0 x 10³ g.

Which one expresses the most accurate measurement and why?

Answer

2000 g expresses the most accurate measurement. The zeros to the right of a non-zero digit are significant when they arise due to measurement, so 2000 g has four significant figures, whereas 2.00 kg has three and 2.0 × 10³ g has only two. Since the measurement having the maximum number of significant figures is the most accurate, 2000 g is the most accurate of the three.

If f = x², then how many times is the fractional error in f than the fractional error in x?

Answer

Given,

• f = x²

When a quantity is raised to a power, its fractional error is multiplied by that power. For f = x², taking the fractional error on both sides,

$\dfrac{\Delta \text f}{\text f} = 2 \times \dfrac{\Delta \text x}{\text x}$

Hence, the fractional error in f is 2 times the fractional error in x.

Is the digit 0 in a number significant? Explain.

Answer

The digit 0 may or may not be significant, depending on its position in the number.

The zero is significant when it lies between two non-zero digits (as in 60.5 g), and also when it lies to the right of the decimal point after a non-zero digit, since it then arises due to actual measurement (as in 6.50 cm).

The zero is not significant when it only fixes the position of the decimal point, that is, when it lies to the right of the decimal point but to the left of the first non-zero digit (as in 0.065 m).

Which quantity in a given formula should be measured maximum accurate?

Answer

The quantity occurring with the highest power, say n, in the formula should be measured with the maximum accuracy. This is because the fractional error in such a quantity is multiplied by n in the final result, so any error in its measurement is magnified n times.

If all measurements in an experiment are taken up to same number of significant figures, then which measurement is responsible for maximum error?

Answer

The measurement occurring with the highest power in the formula is responsible for the maximum error, since its fractional error is multiplied by that power. If all the quantities occur with equal powers, then the measurement smallest in magnitude is responsible for the maximum error, because for the same absolute error the fractional error is greatest for the smallest measurement.

Which of the following has the same dimensions as Planck's constant ?

Torque, work, angular momentum, coefficient of viscosity.

Answer

The dimensional formula of Planck's constant is [ML²T^-1]. Comparing with the given quantities,

• Torque → [ML²T^-2]
• Work → [ML²T^-2]
• Angular momentum → [ML²T^-1]
• Coefficient of viscosity → [ML^-1T^-1]

Hence, angular momentum has the same dimensions as Planck's constant.

Can variables be dimensionless?

Answer

Yes, variables can be dimensionless. For example, angle, strain and specific gravity (relative density) are dimensionless variables, since each of them is the ratio of two quantities of the same kind.

Obtain SI unit of work in terms of fundamental units.

Answer

Work is defined as the product of force and displacement,

$\text{Work} = \text{Force} \times \text{displacement}$

The force is the product of mass and acceleration,

$\text{Force} = \text{mass} \times \text{acceleration}$

Writing the units of each quantity in terms of the fundamental units,

• Unit of mass = kg
• Unit of acceleration = m s^-2
• Unit of displacement = m

Therefore, the SI unit of work is

$\text{Unit of work} = (\text{kg} \times \text{m s}^{-2}) \times \text{m} \\[1em] = \text{kg m}^2 \text{ s}^{-2}$

This unit is called the joule (J).

Hence, the SI unit of work in terms of the fundamental units is kg m² s^-2.

How can you estimate the number of air molecules in your room at NTP?

Answer

Given,

• Volume occupied by 1 mole of air at NTP = 22.4 L = 22.4 × 10^-3 m³
• Number of molecules in 1 mole (Avogadro's number), N_A = 6.023 × 10²³

At NTP, 22.4 × 10^-3 m³ of air contains 6.023 × 10²³ molecules. Hence the number of molecules contained in 1 m³ of air is

$\dfrac{6.023 \times 10^{23}}{22.4 \times 10^{-3}}$

If the length, breadth and height of the room are measured, its volume V is obtained, and the number of air molecules in the room at NTP is

$\text N = \dfrac{6.023 \times 10^{23}}{22.4 \times 10^{-3}} \times \text V$

Hence, the number of air molecules in a room of volume V m³ at NTP is $\dfrac{6.023 \times 10^{23}}{22.4 \times 10^{-3}} \times \text V$.

Rule out, on dimensional arguments, from the following, the wrong formulae for the kinetic energy E of a body of mass m:

(i) E = m²v³

(ii) E = $\dfrac{1}{2}$ mv²

(iii) E = ma

(iv) E = $\dfrac{3}{16}$ mv²

(v) E = $\dfrac{1}{2}$ mv² + ma,

where v is velocity and a is acceleration of the body.

Answer

By the principle of homogeneity of dimensions, the dimensions of both sides of a correct formula must be the same, and only quantities of the same dimensions can be added to one another.

The dimensional formula of kinetic energy is

$[\text E] = [\text{ML}^2\text{T}^{-2}]$

Writing the dimensions of the right hand side of each formula,

(i) $[\text m^2\text v^3] = [\text M^2][\text{LT}^{-1}]^3 = [\text M^2\text L^3\text T^{-3}]$ — different from [ML²T^-2], so this formula is wrong.

(ii) $\left[\dfrac{1}{2}\text{mv}^2\right] = [\text M][\text{LT}^{-1}]^2 = [\text{ML}^2\text{T}^{-2}]$ — same as [ML²T^-2], so this formula is dimensionally correct.

(iii) $[\text{ma}] = [\text M][\text{LT}^{-2}] = [\text{MLT}^{-2}]$ — different from [ML²T^-2], so this formula is wrong.

(iv) $\left[\dfrac{3}{16}\text{mv}^2\right] = [\text M][\text{LT}^{-1}]^2 = [\text{ML}^2\text{T}^{-2}]$ — same as [ML²T^-2], so this formula is dimensionally correct.

(v) Here the term $\dfrac{1}{2}\text{mv}^2$ has the dimension [ML²T^-2] while the term ma has the dimension [MLT^-2]. Two quantities of different dimensions cannot be added, so this formula is wrong.

Hence, the formulae (i), (iii) and (v) are ruled out on dimensional grounds.

Dimensional analysis cannot decide between (ii) and (iv), since both are dimensionally correct; it cannot determine the value of the dimensionless constant. From the definition of kinetic energy, the correct formula is (ii), that is, E = $\dfrac{1}{2}$mv².

The displacement of a particle is represented by the formula s = ct³, where t is time and c is a constant. Write down the dimensional formula for c.

Answer

Given,

• s = ct³

By the principle of homogeneity of dimensions, the dimensions of both sides must be the same,

$[\text s] = [\text c][\text t^3]$

The dimensional formula of displacement is [L] and that of time is [T]. Therefore,

$[\text L] = [\text c][\text T^3] \\[1em] \Rightarrow [\text c] = \dfrac{[\text L]}{[\text T^3]} \\[1em] = [\text{LT}^{-3}]$

Hence, the dimensional formula for c is [LT^-3].

A + B = C. Dimensions of each of A and C are [M L^-1 T^-2]. Write down the dimensions of B.

Answer

Given,

• A + B = C
• [A] = [C] = [ML^-1T^-2]

By the principle of homogeneity of dimensions, only quantities having the same dimensions can be added to one another. Since B is added to A, it must have the same dimensions as A.

Hence, the dimensions of B are [M L^-1 T^-2].

The velocity of a particle is expressed by the equation $\text v = \text {at} + \dfrac{\text b}{\text t}$, where t is time. Determine the dimensional formula for a and b.

Answer

Given,

• $\text v = \text{at} + \dfrac{\text b}{\text t}$

By the principle of homogeneity of dimensions, each term on the right hand side must have the same dimensions as the velocity v,

$[\text v] = [\text{at}] = \left[\dfrac{\text b}{\text t}\right] = [\text{LT}^{-1}]$

For a :

$[\text a][\text T] = [\text{LT}^{-1}] \\[1em] \Rightarrow [\text a] = [\text{LT}^{-1}][\text T^{-1}] \\[1em] = [\text{LT}^{-2}]$

For b :

$\dfrac{[\text b]}{[\text T]} = [\text{LT}^{-1}] \\[1em] \Rightarrow [\text b] = [\text{LT}^{-1}][\text T] \\[1em] = [\text L]$

Hence, the dimensional formula for a is [LT^-2] and for b is [L].

The velocity $\text v$ of a particle depends upon the time $\text t$ according to the equation $\text v = \text a + \text {bt} + \dfrac{\text c}{\text {d + t}}$. Write the dimensions of a, b, c and d.

Answer

Given,

• $\text v = \text a + \text{bt} + \dfrac{\text c}{\text d + \text t}$

By the principle of homogeneity of dimensions, each term on the right hand side must have the same dimensions as the velocity v, and only quantities of the same dimensions can be added to one another.

For a : Since a is added to the other terms which have the dimensions of velocity,

$[\text a] = [\text v] = [\text{LT}^{-1}]$

For b :

$[\text b][\text T] = [\text{LT}^{-1}] \\[1em] \Rightarrow [\text b] = [\text{LT}^{-2}]$

For d : Since d is added to the time t,

$[\text d] = [\text t] = [\text T]$

For c :

$\dfrac{[\text c]}{[\text d + \text t]} = [\text{LT}^{-1}] \\[1em] \Rightarrow [\text c] = [\text{LT}^{-1}][\text T] \\[1em] = [\text L]$

Hence, [a] = [LT^-1], [b] = [LT^-2], [c] = [L] and [d] = [T].

From ideal gas equation PV = RT, obtain dimensional formula for R.

Answer

Given,

• PV = RT

By the principle of homogeneity of dimensions,

$[\text P][\text V] = [\text R][\text T]$

The dimensional formulae of pressure, volume and thermodynamic temperature are [ML^-1T^-2], [L³] and [θ] respectively. Therefore,

$[\text{ML}^{-1}\text{T}^{-2}][\text L^3] = [\text R][\theta] \\[1em] \Rightarrow [\text{ML}^2\text{T}^{-2}] = [\text R][\theta] \\[1em] \Rightarrow [\text R] = [\text{ML}^2\text{T}^{-2}\theta^{-1}]$

Hence, the dimensional formula for R is [M L² T^-2 θ^-1].

The velocity v of a particle depends upon time t according to the equation v = At² + Bt + C, where v is in ms^-1 and t in second. Write the units of A, B and C.

Answer

Given,

• v = At² + Bt + C, where v is in m s^-1 and t is in s

By the principle of homogeneity, only quantities of the same kind can be added, so each term on the right hand side must have the same unit as the velocity v, that is, m s^-1.

For A :

$\text{Unit of A} \times \text s^2 = \text{m s}^{-1} \\[1em] \Rightarrow \text{Unit of A} = \text{m s}^{-1} \times \text s^{-2} = \text{m s}^{-3}$

For B :

$\text{Unit of B} \times \text s = \text{m s}^{-1} \\[1em] \Rightarrow \text{Unit of B} = \text{m s}^{-1} \times \text s^{-1} = \text{m s}^{-2}$

For C : Since C is added directly to the other terms,

$\text{Unit of C} = \text{m s}^{-1}$

Hence, the unit of A is m s^-3, the unit of B is m s^-2 and the unit of C is m s^-1.

The equation $\dfrac{1}{2}$mv² = hν - W is the equation of photoelectric effect. What is the unit of W?

Answer

Given,

• $\dfrac{1}{2}\text{mv}^2 = \text h\nu - \text W$

By the principle of homogeneity, only quantities of the same kind can be added to or subtracted from one another, and both sides of the equation must represent the same physical quantity.

The left hand side, $\dfrac{1}{2}$mv², is the kinetic energy of the emitted electron, and hν is the energy of the incident photon. Hence W, which is subtracted from hν, must also be an energy — it is the work function of the metal.

Hence, the unit of W is the joule (J).

The speed of sound v in a medium depends on the modulus of volume elasticity E and density d of the medium. Establish formula for the speed of sound on the basis of dimensional analysis.

Answer

Let the speed of sound v depend upon the modulus of volume elasticity E and the density d as

$\text v \propto \text E^\text a\ \text d^\text b \\[1em] \Rightarrow \text v = \text k\ \text E^\text a\ \text d^\text b \quad \text{...............(i)}$

where k is a dimensionless constant.

Writing the dimensions of both sides,

$[\text M^0\text{LT}^{-1}] = [\text{ML}^{-1}\text{T}^{-2}]^\text a\ [\text{ML}^{-3}\text T^{0}]^\text b \\[1em] = [\text M^{\text a + \text b}\ \text L^{-\text a - 3\text b}\ \text T^{-2\text a}]$

Equating the powers of M, L and T on both sides,

• a + b = 0
• −a − 3b = 1
• −2a = −1

From the third equation, a = $\dfrac{1}{2}$, and from the first equation, b = −a = −$\dfrac{1}{2}$.

Substituting these values in equation (i),

$\text v = \text k\ \text E^{1/2}\ \text d^{-1/2} \\[1em] = \text k\sqrt{\dfrac{\text E}{\text d}}$

Experimentally, the value of k is found to be 1.

Hence, the speed of sound in a medium is $\text v = \sqrt{\dfrac{\text E}{\text d}}$.

The velocity v of a transverse wave in a stretched wire depends on the tension F in the wire, and its mass per unit length m. Establish the formula for velocity of a transverse wave with the help of dimensions.

Answer

Let the velocity v of the transverse wave depend upon the tension F and the mass per unit length m as

$\text v \propto \text F^\text a\ \text m^\text b \\[1em] \Rightarrow \text v = \text k\ \text F^\text a\ \text m^\text b \quad \text{...............(i)}$

where k is a dimensionless constant.

Writing the dimensions of both sides,

$[\text M^0\text{LT}^{-1}] = [\text{MLT}^{-2}]^\text a\ [\text{ML}^{-1}\text T^{0}]^\text b \\[1em] = [\text M^{\text a + \text b}\ \text L^{\text a - \text b}\ \text T^{-2\text a}]$

Equating the powers of M, L and T on both sides,

• a + b = 0
• a − b = 1
• −2a = −1

From the third equation, a = $\dfrac{1}{2}$, and from the first equation, b = −a = −$\dfrac{1}{2}$.

Substituting these values in equation (i),

$\text v = \text k\ \text F^{1/2}\ \text m^{-1/2} \\[1em] = \text k\sqrt{\dfrac{\text F}{\text m}}$

Experimentally, the value of k is found to be 1.

Hence, the velocity of a transverse wave in a stretched wire is $\text v = \sqrt{\dfrac{\text F}{\text m}}$.

If the time-period T of a drop due to its surface tension depends upon its density d, radius r and surface tension S, then find out the formula for the time-period by the method of dimensions.

Answer

Let the time period T of the drop depend upon the density d, the radius r and the surface tension S as

$\text T \propto \text d^\text a\ \text r^\text b\ \text S^\text c \\[1em] \Rightarrow \text T = \text k\ \text d^\text a\ \text r^\text b\ \text S^\text c \quad \text{...............(i)}$

where k is a dimensionless constant.

Writing the dimensions of both sides,

$[\text M^0\text L^0\text T] = [\text{ML}^{-3}\text T^{0}]^\text a\ [\text M^0\text{LT}^{0}]^\text b\ [\text{ML}^{0}\text T^{-2}]^\text c \\[1em] = [\text M^{\text a + \text c}\ \text L^{-3\text a + \text b}\ \text T^{-2\text c}]$

Equating the powers of M, L and T on both sides,

• a + c = 0
• −3a + b = 0
• −2c = 1

From the third equation, c = −$\dfrac{1}{2}$.

From the first equation, a = −c = $\dfrac{1}{2}$.

From the second equation, b = 3a = $\dfrac{3}{2}$.

Substituting these values in equation (i),

$\text T = \text k\ \text d^{1/2}\ \text r^{3/2}\ \text S^{-1/2} \\[1em] = \text k\sqrt{\dfrac{\text{dr}^3}{\text S}}$

Experimentally, the value of k is found to be 1.

Hence, the time period of the drop is $\text T = \sqrt{\dfrac{\text{dr}^3}{\text S}}$.

The force F acting on a ball falling with a constant velocity in a liquid depends on the radius r of the ball, its terminal velocity v, and the coefficient of viscosity η of the liquid. Find the formula for F by the method of dimensions.

Answer

Let the force F depend upon the radius r, the terminal velocity v and the coefficient of viscosity η as

$\text F \propto \text r^\text a\ \text v^\text b\ \eta^\text c \\[1em] \Rightarrow \text F = \text k\ \text r^\text a\ \text v^\text b\ \eta^\text c \quad \text{...............(i)}$

where k is a dimensionless constant.

Writing the dimensions of both sides,

$[\text{MLT}^{-2}] = [\text M^0\text{LT}^{0}]^\text a\ [\text M^0\text{LT}^{-1}]^\text b\ [\text{ML}^{-1}\text T^{-1}]^\text c \\[1em] = [\text M^{\text c}\ \text L^{\text a + \text b - \text c}\ \text T^{-\text b - \text c}]$

Equating the powers of M, L and T on both sides,

• c = 1
• a + b − c = 1
• −b − c = −2, that is, b + c = 2

From the third equation, b = 2 − c = 2 − 1 = 1.

From the second equation, a = 1 − b + c = 1 − 1 + 1 = 1.

Substituting these values in equation (i),

$\text F = \text k\ \text r\ \text v\ \eta \\[1em] = \text k\ \eta\ \text{rv}$

Experimentally, the value of k is found to be 6π.

Hence, the force acting on the ball is F = 6πηrv, which is Stokes' law.

The air bubble formed by explosion inside water performs oscillations with time-period T which is directly proportional to P^a d^b E^c, where P is pressure, d is density and E is the energy due to explosion. Find the values of a, b and c.

Answer

Given,

• T ∝ P^a d^b E^c, that is, T = k P^a d^b E^c, where k is a dimensionless constant

Writing the dimensions of both sides,

$[\text M^0\text L^0\text T] = [\text{ML}^{-1}\text T^{-2}]^\text a\ [\text{ML}^{-3}\text T^{0}]^\text b\ [\text{ML}^{2}\text T^{-2}]^\text c \\[1em] = [\text M^{\text a + \text b + \text c}\ \text L^{-\text a - 3\text b + 2\text c}\ \text T^{-2\text a - 2\text c}]$

Equating the powers of M, L and T on both sides,

• a + b + c = 0   ...............(i)
• −a − 3b + 2c = 0   ...............(ii)
• −2a − 2c = 1   ...............(iii)

From equation (iii),

$\text a + \text c = -\dfrac{1}{2}$

Substituting this in equation (i),

$\text b = -(\text a + \text c) = \dfrac{1}{2}$

Substituting b = $\dfrac{1}{2}$ in equation (ii),

$-\text a - \dfrac{3}{2} + 2\text c = 0 \\[1em] \Rightarrow -\text a + 2\text c = \dfrac{3}{2}$

Also, from equation (iii), a = $-\dfrac{1}{2} - \text c$. Substituting this,

$\dfrac{1}{2} + \text c + 2\text c = \dfrac{3}{2} \\[1em] \Rightarrow 3\text c = 1 \\[1em] \Rightarrow \text c = \dfrac{1}{3}$

Therefore,

$\text a = -\dfrac{1}{2} - \dfrac{1}{3} = -\dfrac{5}{6}$

Hence, a = $-\dfrac{5}{6}$, b = $\dfrac{1}{2}$ and c = $\dfrac{1}{3}$.

Given, $\text Z = \dfrac{\text A^4 \text B^{1/3}}{\text {CD}^{3/2}}$ where A, B, C and D are measured quantities. What is the maximum fractional error in Z?

Answer

Given,

$\text Z = \dfrac{\text A^4 \text B^{1/3}}{\text {CD}^{3/2}}$

When a quantity is expressed as a product or quotient of powers of measured quantities, the maximum fractional error in it is obtained by adding the fractional errors of all the quantities, each multiplied by the magnitude of its power. The sign of the power is not taken into account, since the errors always add up in the worst case.

Here the powers of A, B, C and D are 4, $\dfrac{1}{3}$, 1 and $\dfrac{3}{2}$ respectively.

Therefore, the maximum fractional error in Z is

$\dfrac{\Delta \text Z}{\text Z} = 4\dfrac{\Delta \text A}{\text A} + \dfrac{1}{3}\dfrac{\Delta \text B}{\text B} + \dfrac{\Delta \text C}{\text C} + \dfrac{3}{2}\dfrac{\Delta \text D}{\text D}$

Hence, the maximum fractional error in Z is $\dfrac{\Delta \text Z}{\text Z} = 4\dfrac{\Delta \text A}{\text A} + \dfrac{1}{3}\dfrac{\Delta \text B}{\text B} + \dfrac{\Delta \text C}{\text C} + \dfrac{3}{2}\dfrac{\Delta \text D}{\text D}$.

"To call a dimensional quantity large or small is meaningless without specifying a standard for comparison." Explain this statement clearly.

Answer

A dimensional quantity has a definite unit, and its measured value depends on the unit chosen. Calling such a quantity large or small has no meaning by itself, because the same quantity may appear large when compared with one standard and small when compared with another. A quantity can be called large or small only when it is compared with a standard (or reference) quantity of the same kind.

For example : The statement "the mass of the earth is very large" is meaningless as it stands. The mass of the earth is very large compared with the mass of a ship, but it is very small compared with the mass of the sun.

The statement becomes meaningful when a standard of comparison is specified, for example : The mass of the earth is larger than the mass of the moon.

The diameter of a thin rod is being measured by a screw gauge. A set of ten measurements is expected to give a more reliable value of the diameter than a set of five measurements. Why?

Answer

The different measurements give slightly different readings because of random errors, which arise from causes such as :

(i) the rod may not be perfectly uniform in diameter at all places, and

(ii) the rod may be held between the stud and the screw with a different pressure in each measurement.

Random errors occur irregularly and are equally likely to be positive or negative, so they are minimised by taking a large number of readings of the same quantity and then taking their arithmetic mean, since the positive and negative errors tend to cancel one another.

Hence, the mean of ten measurements is closer to the actual value of the diameter than the mean of five measurements.

Mention some repetitive phenomena which could serve as standard of time. Which one is most suitable?

Answer

Any phenomenon which repeats itself regularly can be used as a standard of time. Some such repetitive phenomena are :

• the oscillations of a simple pendulum
• the oscillations of a loaded spring
• the electrically sustained vibrations of a quartz crystal
• the characteristic frequency of the radiation emitted by certain atoms
• the rotation of the earth about its own axis

The axial rotation of the earth was used as the standard of time for centuries.

The most suitable standard is the atomic clock. The caesium (atomic) clock and the hydrogen maser clock are the most precise clocks available. Two hydrogen maser clocks could run for 30,000,000 years before their readings would differ by 1 s. The second is therefore now defined in terms of the vibrations of the caesium-133 atom, since this standard is precisely defined, easily reproducible and does not change with time, place or physical conditions.

Two clocks A and B are tested against a standard clock. Daily exactly at 12 noon (according to standard clock), the time shown by each clock is noted for seven days. Clock A shows, on the average, the time 11:59 AM with a range of variation of 162 s, while clock B shows average time 10 AM with a range of variation of 31 s. Which clock will you prefer for an experiment requiring precision time-interval measurements?

Answer

Given,

• Clock A : average time 11 : 59 AM, range of variation 162 s
• Clock B : average time 10 AM, range of variation 31 s

The average reading of clock A is closer to the standard time, so clock A is the more accurate of the two. However, its readings vary over a range of 162 s, whereas the readings of clock B vary over a range of only 31 s. Hence clock B is the more precise of the two.

For measuring time-intervals precisely, what matters is the consistency of the readings and not the difference from the standard time, because the constant difference of clock B is a systematic (zero) error which follows a definite rule and can be corrected by applying a proper correction. The larger variation of clock A is a random error, which cannot be removed in this way.

Hence, clock B is to be preferred for an experiment requiring precision time-interval measurements.

Two students measure the length of a rod as 2.5 m and 2.54 m. Which measurement is more accurate and why?

Answer

The accuracy of a measurement is judged from its fractional (relative) error,

$\text{Fractional error} = \dfrac{\text{Absolute error}}{\text{Measured value}}$

For the measurement 2.5 m : the measurement is made up to one place of decimal, so the absolute error is 0.1 m,

$\dfrac{\Delta \text l}{\text l} = \dfrac{0.1}{2.5} = 0.04$

For the measurement 2.54 m : the measurement is made up to two places of decimal, so the absolute error is 0.01 m,

$\dfrac{\Delta \text l}{\text l} = \dfrac{0.01}{2.54} = 0.004$

Hence, the measurement 2.54 m is more accurate, since it has the smaller fractional error.

Which of the following length measurements is maximum accurate and why?

5.00 cm, 0.005 mm, 50.00 cm

Answer

The accuracy of a measurement is judged from its fractional (relative) error,

$\text{Fractional error} = \dfrac{\text{Absolute error}}{\text{Measured value}}$

5.00 cm — the measurement is made up to two places of decimal, so the absolute error is 0.01 cm,

$\dfrac{0.01}{5.00} = 0.002$

0.005 mm — the measurement is made up to three places of decimal, so the absolute error is 0.001 mm,

$\dfrac{0.001}{0.005} = 0.2$

50.00 cm — the measurement is made up to two places of decimal, so the absolute error is 0.01 cm,

$\dfrac{0.01}{50.00} = 0.0002$

Hence, the measurement 50.00 cm is the most accurate, since it has the smallest fractional error.

Assuming force (F), length (L) and time (T) to be the fundamental units, find out the dimensions of mass. If energy (E) is considered in place of (F), then what will be the dimensions of mass ?

Answer

(i) When F, L and T are taken as the fundamental quantities.

From Newton's second law, force is the product of mass and acceleration,

$\text F = \text m \times \text a \\[1em] \Rightarrow \text m = \dfrac{\text F}{\text a}$

The dimensional formula of acceleration is [LT^-2]. Therefore,

$[\text M] = \dfrac{[\text F]}{[\text{LT}^{-2}]} = [\text F][\text L^{-1}\text T^{2}] \\[1em] = [\text F\ \text L^{-1}\ \text T^{2}] \quad \text{...............(i)}$

Hence, the dimensions of mass are [F L^-1 T²].

(ii) When E, L and T are taken as the fundamental quantities.

Energy is the work done, which is the product of force and distance,

$\text E = \text F \times \text L \\[1em] \Rightarrow \text F = \dfrac{\text E}{\text L}$

so that

$[\text F] = [\text E][\text L^{-1}] = [\text E\ \text L^{-1}]$

Substituting this in equation (i),

$[\text M] = [\text E\ \text L^{-1}][\text L^{-1}\text T^{2}] \\[1em] = [\text E\ \text L^{-2}\ \text T^{2}]$

Hence, the dimensions of mass are [E L^-2 T²].

If dimensions of length are expressed as [G^x c^y h^z], where G, c and h are universal gravitational constant, speed of light in vacuum and Planck’s constant respectively, then what are the values of x, y and z?

Answer

Given,

• [L] = [G^x c^y h^z]

The dimensional formulae of the three constants are

• Universal gravitational constant, [G] = [M^-1L³T^-2]
• Speed of light in vacuum, [c] = [LT^-1]
• Planck's constant, [h] = [ML²T^-1]

Therefore,

$[\text G^\text x\ \text c^\text y\ \text h^\text z] = [\text M^{-1}\text L^{3}\text T^{-2}]^\text x\ [\text{LT}^{-1}]^\text y\ [\text{ML}^{2}\text T^{-1}]^\text z \\[1em] = [\text M^{-\text x + \text z}\ \text L^{3\text x + \text y + 2\text z}\ \text T^{-2\text x - \text y - \text z}]$

Since this represents the dimensions of length,

$[\text M^0\text{LT}^0] = [\text M^{-\text x + \text z}\ \text L^{3\text x + \text y + 2\text z}\ \text T^{-2\text x - \text y - \text z}]$

Equating the powers of M, L and T on both sides,

• −x + z = 0, so z = x
• 3x + y + 2z = 1
• −2x − y − z = 0

Substituting x = z in the third equation,

$-2\text z - \text y - \text z = 0 \\[1em] \Rightarrow \text y = -3\text z$

Substituting x = z and y = −3z in the second equation,

$3\text z - 3\text z + 2\text z = 1 \\[1em] \Rightarrow 2\text z = 1 \\[1em] \Rightarrow \text z = \dfrac{1}{2}$

Therefore,

$\text x = \text z = \dfrac{1}{2}, \quad \text y = -3 \times \dfrac{1}{2} = -\dfrac{3}{2}$

Hence, x = $\dfrac{1}{2}$, y = $-\dfrac{3}{2}$ and z = $\dfrac{1}{2}$.

If velocity, force and time are taken to be the fundamental quantities, then find the dimensional formula for energy.

Answer

Given,

• Velocity (v), force (F) and time (T) are taken as the fundamental quantities.

Energy is the work done, which is the product of force and distance,

$\text{Energy} = \text{Force} \times \text{distance}$

The distance is the product of velocity and time,

$\text{distance} = \text{velocity} \times \text{time}, \quad \text{that is,}\quad [\text L] = [\text{vT}]$

Therefore,

$[\text E] = [\text F][\text v][\text T] \\[1em] = [\text F\ \text v\ \text T]$

Hence, the dimensional formula for energy is [F v T].

Verify with the help of dimensional analysis that the equation of time period (T) of a simple pendulum, $\text T = \text {2π} \sqrt {\dfrac{\text l}{\text g}}$ is correct.

Answer

The given relation is

$\text T = 2\pi\sqrt{\dfrac{\text l}{\text g}}$

By the principle of homogeneity of dimensions, the dimensions of both sides of the relation must be the same.

Left hand side. The dimensional formula of the time period is

$[\text{L.H.S.}] = [\text T]$

Right hand side. The factor 2π is a pure number and is dimensionless. The dimensional formulae of the length l and the acceleration due to gravity g are [L] and [LT^-2] respectively. Therefore,

$[\text{R.H.S.}] = \sqrt{\dfrac{[\text L]}{[\text{LT}^{-2}]}} = \sqrt{\dfrac{1}{[\text T^{-2}]}} \\[1em] = \sqrt{[\text T^{2}]} = [\text T]$

Since the dimensions of the two sides are the same,

$[\text{L.H.S.}] = [\text{R.H.S.}] = [\text T]$

Hence, the equation $\text T = 2\pi\sqrt{\dfrac{\text l}{\text g}}$ is dimensionally correct.

If velocity, force and time are taken to be fundamental quantities, then find the fundamental formula for mass.

Answer

Given,

• Velocity (v), force (F) and time (T) are taken as the fundamental quantities.

From Newton's second law, force is the product of mass and acceleration,

$\text F = \text m \times \text a$

and acceleration is the rate of change of velocity,

$\text a = \dfrac{\text v}{\text T}$

Therefore,

$\text F = \text m \times \dfrac{\text v}{\text T} \\[1em] \Rightarrow \text m = \dfrac{\text F \times \text T}{\text v}$

Writing the dimensions in terms of the new fundamental quantities,

$[\text m] = [\text F][\text T][\text v]^{-1} \\[1em] = [\text F\ \text v^{-1}\ \text T]$

Hence, the fundamental formula for mass is [F v^-1 T].

The accuracy of an instrument depends more on systematic errors (for example calibration error, zero error etc.) present in it rather than some other factors. Therefore, it can be improved by re-calibration or by applying proper corrections. Higher the accuracy, smaller is the error.

Thus, an error gives the indication of accuracy. On the other hand precision depends on the random errors. Therefore, it cannot be eradicated. Moreover, higher the precision, larger is the number of significant figures. Thus, precision gives the indication of number of significant figures in a measurement.

(i) What do you mean by accuracy?

(ii) How can you increase the accuracy of a measurement?

(iii) How can you obtain more precised measurement?

Answer

(i) The closeness of a measurement to the true value of a physical quantity is known as its accuracy. An instrument which gives repeated readings close to the true value of the quantity is called an accurate instrument.

(ii) The accuracy of a measurement depends mainly on the systematic errors present in the instrument, such as the calibration error and the zero error. Since these errors follow a definite rule, they can be identified and removed. Hence the accuracy can be increased by re-calibrating the instrument and by applying proper corrections to the observed readings.

(iii) The precision of a measurement is determined by the least count of the measuring instrument — the smaller the least count, the greater is the precision. Hence a more precise measurement can be obtained by using an instrument of smaller least count.

The rule for multiplication and division is quite different from that in addition and subtraction. We first locate the measured quantity having least number of significant figures. Now we round off all other measured quantities so that they have one more significant figure than the least accurate one. Now, we carry out the multiplication or division and finally round off the result to the same number of significant figures as are contained in the least accurate one.

(i) What do you mean by the significant figure?

(ii) The length and breadth of a rectangular lamina are 5.245 m and 1.24 m, respectively.

Find area of the lamina to appropriate significant figures.

(iii) The mass of a block is 4.50 kg and its volume is 1.204 m³. Find the density of the material of the block.

Answer

(i) In the measurement of a quantity, the digits which are measured accurately together with the first doubtful digit are called the significant figures. Since any measuring instrument can measure accurately only up to a certain limit, the last digit of an observation is always doubtful.

(ii) Given,

• Length of the lamina, l = 5.245 m (4 significant figures)
• Breadth of the lamina, b = 1.24 m (3 significant figures)

The area of the rectangular lamina is

$\text A = \text l \times \text b$

Substituting the values,

$\text A = 5.245 \times 1.24 \\[1em] = 6.5038\ \text m^2$

In multiplication, the result is rounded off to the least number of significant figures in the given data, which is 3. Hence

$\text A = 6.50\ \text m^2$

Hence, the area of the lamina is 6.50 m².

(iii) Given,

• Mass of the block, m = 4.50 kg (3 significant figures)
• Volume of the block, V = 1.204 m³ (4 significant figures)

The density of the material of the block is

$\text{Density} = \dfrac{\text{Mass}}{\text{Volume}}$

Substituting the values,

$\text{Density} = \dfrac{4.50\ \text{kg}}{1.204\ \text m^3} \\[1em] = 3.7375\ \text{kg m}^{-3}$

In division, the result is rounded off to the least number of significant figures in the given data, which is 3. Hence

$\text{Density} = 3.74\ \text{kg m}^{-3}$

Hence, the density of the material of the block is 3.74 kg m^-3.

The nature of physical quantity is described by its dimension. All the physical quantities can be expressed in terms of some combination of seven fundamental units. The dimensions of a physical quantity are thus the powers to which the base quantities are raised to represent that quantity. If a given physical quantity depends on, a^th power of mass, b^th power of length and c^th power of time etc, then its dimensions are expressed as [M^a L^b T^c].

(i) The dimensions of Planck's constant h equal to that of:

1. energy
2. momentum
3. angular momentum
4. power

(ii) If force (F) velocity (v) and time (T) are taken as fundamental units then dimensions of mass are:

1. [F V T^-2]
2. [F V^-1 T^-1]
3. [F V^-1 T]
4. [F V T^-1]

Answer

(i) angular momentum

Reason — Planck's constant is defined by the relation E = hν, where E is the energy and ν is the frequency. Therefore

$\text h = \dfrac{\text E}{\nu}$

Writing the dimensional formulae,

$[\text h] = \dfrac{[\text{ML}^2\text{T}^{-2}]}{[\text T^{-1}]} = [\text{ML}^2\text{T}^{-1}]$

The angular momentum is the product of the moment of inertia and the angular velocity, L = Iω, so

$[\text L] = [\text{ML}^2][\text T^{-1}] = [\text{ML}^2\text{T}^{-1}]$

Both have the same dimensional formula [ML²T^-1]. The other options have different dimensions : energy [ML²T^-2], momentum [MLT^-1] and power [ML²T^-3].

(ii) [F V^-1 T]

Reason —

Given,

• Force (F), velocity (v) and time (T) are taken as the fundamental quantities.

From Newton's second law, force is the product of mass and acceleration,

$\text F = \text m \times \text a$

and acceleration is the rate of change of velocity,

$\text a = \dfrac{\text v}{\text T}$

Therefore,

$\text F = \text m \times \dfrac{\text v}{\text T} \\[1em] \Rightarrow \text m = \dfrac{\text F \times \text T}{\text v}$

Writing the dimensions in terms of the new fundamental quantities,

$[\text m] = [\text F][\text v]^{-1}[\text T] = [\text F\ \text v^{-1}\ \text T]$

The principle of homogeneity states that relation A ± B = C is valid only when physical quantities A, B and C all have same dimensions. In van der Waals' equation of state, that is, in $\left(\text P + \dfrac {\text a}{\text V^2}\right)(\text V - \text b) = \text {RT}$, P is pressure, V is volume, R is universal gas constant, T is absolute temperature and a and b are dimensional constants.

(i) The dimensional formula of constant $b$ will be same as:

1. P
2. V²
3. V
4. PV²

(ii) The dimensional formula of $a$ will be same as:

1. P
2. PV²
3. RT
4. V²

(iii) The dimensional formula for RT will be same as:

1. energy
2. force
3. latent heat
4. specific heat capacity

(iv) Dimensional formula of $\dfrac{\text {ab}}{\text {RT}}$ is:

1. [M L⁵ T^-2]
2. [M⁰ L³ T⁰]
3. [M L^-1 T^-2]
4. [M⁰ L⁶ T⁰]

Answer

(i) V

Reason — In the given equation, the constant b is subtracted from the volume V in the bracket (V − b). By the principle of homogeneity of dimensions, only quantities having the same dimensions can be subtracted from one another. Hence

$[\text b] = [\text V] = [\text M^0\text L^3\text T^0]$

(ii) PV²

Reason — In the given equation, the term $\dfrac{\text a}{\text V^2}$ is added to the pressure P in the bracket. By the principle of homogeneity of dimensions, only quantities having the same dimensions can be added. Hence

$\left[\dfrac{\text a}{\text V^2}\right] = [\text P] \\[1em] \Rightarrow [\text a] = [\text P][\text V^2]$

Therefore the dimensional formula of a is the same as that of PV².

(iii) energy

Reason — Expanding the given equation,

$\left(\text P + \dfrac{\text a}{\text V^2}\right)(\text V - \text b) = \text{RT} \\[1em] \Rightarrow \text{PV} - \text{Pb} + \dfrac{\text a}{\text V} - \dfrac{\text{ab}}{\text V^2} = \text{RT}\quad \text{...............(1)}$

By the principle of homogeneity of dimensions, every term on both sides of the equation must have the same dimensions. Hence

$[\text{RT}] = [\text{PV}] = [\text{ML}^{-1}\text{T}^{-2}][\text L^3] \\[1em] = [\text{ML}^2\text{T}^{-2}]$

which is the dimensional formula of energy.

(iv) [M⁰ L⁶ T⁰]

Reason — From equation (1) obtained in part (iii), the term $\dfrac{\text{ab}}{\text V^2}$ appears along with RT, so by the principle of homogeneity of dimensions

$\left[\dfrac{\text{ab}}{\text V^2}\right] = [\text{RT}]$

Therefore,

$\left[\dfrac{\text{ab}}{\text{RT}}\right] = [\text V^2] = [\text L^3]^2 \\[1em] = [\text M^0\text L^6\text T^0]$

What are fundamental and derived units? Give three examples of derived units.

Answer

Fundamental units : The units of the fundamental (base) quantities are called fundamental units. The fundamental quantities are those which are independent of one another, that is, none of them can be expressed in terms of the others. In the S.I. system there are seven fundamental units — the metre (m), the kilogram (kg), the second (s), the ampere (A), the kelvin (K), the mole (mol) and the candela (cd).

Derived units : The units of the derived quantities are called derived units. A derived quantity is obtained from a combination of the base quantities through multiplication or division, and hence its unit is obtained from the fundamental units in the same way. Since the S.I. is a coherent system, the derived units are obtained from the fundamental units without introducing any numerical factor.

Examples of derived units :

• Unit of velocity = m s^-1
• Unit of force = newton (N), where 1 N = 1 kg m s^-2
• Unit of work (or energy) = joule (J), where 1 J = 1 kg m² s^-2

What do you understand by errors in measurement? Discuss random errors and systematic errors and their elimination.

Answer

Error in measurement : The difference between the measured value of a quantity and its true value is called the error in measurement. However carefully a measurement is made, some error is always present, so no measurement can be perfectly accurate.

The errors in measurement are mainly of two kinds : systematic errors and random errors.

Systematic errors :

The errors which tend to occur in one direction, either positive or negative, are called systematic errors. Such errors follow a definite rule, so they can be identified and corrected.

The main sources of systematic errors are :

(i) Instrumental errors — These arise due to imperfect design or zero error of the measuring instrument, for example a zero error in a screw gauge or a wrongly marked scale.

(ii) Errors due to imperfect technique (or procedure) — These arise due to the limitations of the experimental method, for example neglecting the effect of temperature, pressure or buoyancy of air.

(iii) Personal errors — These arise due to the carelessness or individual bias of the observer, for example an error due to improper setting of the eye while taking a reading (parallax error).

Elimination : Systematic errors can be minimised or removed by improving the design of the instrument, by correcting for the zero error, by using a better experimental technique, and by removing personal bias in taking the observations.

Random errors :

The errors which occur irregularly and are random in magnitude and direction are called random errors. They arise due to unknown and uncontrollable causes, such as small changes in the conditions of the experiment, so they cannot be traced to a definite source.

Because these errors occur equally in the positive and negative directions, they can be reduced by taking a large number of observations of the same quantity and then taking their arithmetic mean. The arithmetic mean is taken as the true value, since the positive and negative errors tend to cancel one another.

Elimination : Random errors can be minimised by repeating the measurement a large number of times and taking the arithmetic mean of all the readings as the most accurate value of the quantity.

What is meant by significant figures? State the rules of finding the significant figures in the sum, difference, product and quotient of two numbers. Support your answer with examples.

Answer