A glass tube is attached to the bottom of a thin iron rod of length ℓ 0 as shown in figure. To what height must the glass tube be filled with mercury so that the centre of mass of the pendulum forme

Question

A glass tube is attached to the bottom of a thin iron rod of length $ℓ_{0}$ as shown in figure. To what height must the glass tube be filled with mercury so that the centre of mass of the pendulum formed will not rise or fall with changes in temperature? (Linear expansivity of glass $= α_{1}$ , that of iron $= α_{2}$ and cubical expansivity of mercury $= γ$ )

TestHub · Accepted Answer

Let m 1 ​ and m 2 ​ be the masses of the iron rod and mercury; lengths ℓ 0 ​ and L 0 ​ at 0°C, with suffix t at t°C. For the centre of mass to remain stationary: m 1 ​ 2 ℓ 0 ​ ​ + m 2 ​ ( ℓ 0 ​ − 2 L 0 ​ ​ ) = m 1 ​ 2 ℓ t ​ ​ + m 2 ​ ( ℓ t ​ − 2 L t ​ ​ ) Since the rod is thin, m 1 ​ ≈ 0 ⇒ L t ​ − L 0 ​ = 2 ℓ 0 ​ α 2 ​ t Also A 0 ​ L 0 ​ D 0 ​ ( 1 + γ t ) = A 0 ​ ( 1 + 2 α 1 ​ t ) L t ​ D 0 ​ ⇒ L t ​ − L 0 ​ = L 0 ​ ( γ − 2 α 1 ​ ) t ∴ L 0 ​ ( γ − 2 α 1 ​ ) t = 2 ℓ 0 ​ α 2 ​ t ⇒ L 0 ​ = γ − 2 α 1 ​ 2 ℓ 0 ​ α 2 ​ ​ hence (A).

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