A wire of uniform cross - section is suspended vertically from a fixed point , with a load attached at the lower end . Calculate the fractional change in frequency of the wire due to rise in temperature by `t^(@) C`. The coefficient of expansion of the wire is `alpha`.

Explanation

With a temperature change , the tension and mass of the wire will not change but length `(I)` and linear density `(m)` will change.
` n = (1)/(2) sqrt ((T)/(M)) and m = (M)/(I)`
Where `M` = mass of the wire
`:. N = (1)/(2) sqrt((T)/(MI)) = (k)/( sqrt(I))`
where ` k = (1)/(2) sqrt(T//M) = a ( constant)`
Differentiating both sides
` Delta n = k (-1/2) I^(-3//2) Delta I`
`:. (Delta n)/( n) (fractional change ) = ((1)/(2) KI^(-3//2) Delta I)/( KI^(-1//2))`
or `(Delta n)/( n) = (1)/( 2) ( Delta I)/( I) = (1)/(2) alpha t ( Delta I = I prop t)`