Question

Uniform metal rod is used as a bar pendulum if the room temperature rises by 10 degree celsius and the coefficient of linear expansion of metal of the rod is 2 into 10 raise to power minus 6 degree celsius the period of the pendulum will have percentage increase of

Answer 1

we know , Time period is directly proportional to square root of length of pendulum.
so, T = k√l , where k is proportionality constant.

from thermal expansion,
$l=l_0(1+\alpha\Delta T)$

where $l_0$ is Initial length of pendulum, $\alpha$ is coefficient of linear expansion and ∆T is temperature change.

now, T = k√l
differentiate both sides,
dT = k/2√l × dl

or, 2√l dT/l = k dl/l

or, 2dT/√l = k dl/l

or, 2dT/T = dl/l

or, 2 ∆T/T = ∆l/l

now, $\Delta l=l\alpha\Delta T$

so, ∆l = l × 2 × 10^-6 × 10 = 2 × 10^-5 m

now, 2 ∆T/T = 2 × 10^-5

∆T/T = 1 × 10^-5

∆T/T × 100 = 10^-3 = 0.001

% change in Time period = 0.001 %