A torsional pendulum consists of a solid disc connected to a thin wire (coefficient of linear expansion α = 2.4 × 10^-5 /°C) at it's centre. Find the percentage change in the time period between peak winter (5°C) and peak summer (45°C).

Coefficient of linear expansion of the wire, α = 2.4 × 10–5 °C–1
Let I0 be the moment of inertia of the torsional pendulum at 0 °C.
If K is the torsional constant of the wire, then time period of torsional pendulum

T:

T=2πIK   …1Here, I = moment of inertia after change in temperature
When the temperature is changed by

∆θ, moment of inertia

I,
I = I0(1+2

α ∆θ)
On substituting the value of I  in equation(1), we get:

T=2πI01+2α∆θKIn winter,

∆θ= 5 °C

∴Time period

T1

=2πI01+2α×5KIn summer,

∆θ= 45 °C
Time period

T2

=2πI01+2α×45K

So,T2T1=1+90α1+10α    =1+90×2.4×10-51+10×2.4×10-5⇒T2T1=1.002161.00024%change =T2T1-1×100              =0.0959%⇒%change in time period≈9.6×10-2%Therefore, the percentage change in time period of a torsional pendulum between peak winters and peak summers is 9.6 × 10–2 % .