A uniform metre ruler is balanced horizontally on a knife edge placed at 60 cm mark when a mass m is suspended from 75 cm mark. Draw the diagram of the arrangement. State with reason (through mathematical steps) whether the mass of the scale is greater than, less than or equal to the mass m?

The diagram of the arrangement is shown below :

Given,

• Distance of the knife edge from zero mark = 60 cm
• Distance of mass m from zero mark = 75 cm

Let,

M = mass of the scale

As,

Distance of centre of gravity of the scale from zero mark = 50 cm

Then,

Distance of centre of gravity of the scale from the knife edge (r) = 60 - 50 = 10 cm

Distance of mass m from the knife edge (R) = 75 - 60 = 15 cm

Now,

Anticlockwise moment (due to weight of the scale) = r x Mg = 10 × Mg

Clockwise moment (due to m) = R x mg = 15 × mg

For equilibrium,

Anticlockwise moment of force about knife edge = Clockwise moment of force about knife edge

$⇒ 10 \times \text {Mg} = 15 \times \text {mg}\\[1em] ⇒ 10 \text M = 15 \text m \\[1em] ⇒ \dfrac {\text M}{\text m} = \dfrac{15}{10} = 1.5$

As,

$\dfrac {\text M}{\text m}$ > 1

⇒ M > m

So, mass of the scale is greater than the mass m.