A torsional pendulum consists of a solid disc connected to a thin wire `(alpha=2.4 xx 10^(-5)C^(-1))` at its center. Find the percentage change in the time period between peak winter `(5^@C)` and peak summer `(45^@C)`

Explanation

Let the initial length at `0^@C` be `l_0`
` Time period at theta^@C, T= 2pi sqrt(l/K)`
` l=l_0(1+2 alpha Delta theta)`
` at 5^@C, `
`T_1= (2pi sqrt(l_0(1+2 alpha Delta theta)))/ (sqrt(K))`
`=(2pi sqrt(l_0(1+2 alpha xx 45)))/ (sqrt(K)),`
` =(2pi sqrt(l_0(1+ 10 alpha )))/ (sqrt(K))`
` at 45^@C, T_2=(2pi sqrt(l_0(1+2 alpha xx 45)))/ (sqrt(K))`
` = (2pi sqrt(l_0(1+90alpha)))/ (sqrt(K))`
` T_2/T_1 = (sqrt(1+90alpha))/(sqrt(1+10alpha))`
`= sqrt(1+90 xx 2.4 xx 10^-5)/ sqrt(1+10 xx 2.4 xx 10^-5)`
`= sqrt(1.00216)/ sqrt(1.00024)`
` % change = ((T_2/T_1)-1) xx 100 `
`=0.0959% ~~9.6 xx 10^-2%`