Answer to Question #180087 in Physics for Elessia

Question

The period of a simple mathematical pendulum is 1500 s. The pendulum is made of wire at the end of which a body is hung. The wire is made of a material whose linear coefficient of thermal expansion is 0.19 × 10^-4 (°C)^-1. By how much does the wire temperature need to increase for the pendulum period to be 1506 s? (Solution: t₂ = 422 ° C)

Answer 1

Equation for period for the pendulum in normal conditions:

$T=2\pi\sqrt{\frac lg}.$

Express length:

$l=\frac{T^2g}{4\pi^2}.$

Equation for period of 'heated' pendulum:

$T_h=2\pi\sqrt{\frac{l(1+\alpha\Delta t)}{g}},\\\space\\ \Delta t=\frac{T_h^2g}{4\alpha\pi^2l}-\frac1\alpha.$

Substitute length found above:

$\Delta t=\frac1\alpha\bigg(\frac{T_h^2}{T^2}-1\bigg)=\frac{1}{0.19·10^{-4}}\bigg(\frac{1506^2}{1500^2}-1\bigg)=422º\text C.$