A clock pendulum made of invar has a period of $0.5 \mathrm{~s}$, at $20^{\circ} \mathrm{C}$. If the clock…
A clock pendulum made of invar has a period of $0.5 \mathrm{~s}$, at $20^{\circ} \mathrm{C}$. If the clock is used in a climate where the temperature averages to $30^{\circ} \mathrm{C}$, how much time does the clock lose in each oscillation? (For invar, $\alpha=9 \times 10^{-7} /{ }^{\circ} \mathrm{C}$, $g=$ constant $)$
$2.25 \times 10^{-6} \mathrm{~s}$
$2.5 \times 10^{-7} \mathrm{~s}$
$5 \times 10^{-7} \mathrm{~s}$
$1.125 \times 10^{-6} \mathrm{~s}$
Solution
Time period of oscillation,
$\begin{aligned}
T & =2 \pi \sqrt{\frac{l}{g}} \\
\Rightarrow \quad \frac{d T}{T} & =\frac{1}{2} \frac{d l}{l} \\
\text { As, } \quad \frac{d l}{l} & =\alpha d t \\
\Rightarrow \quad \frac{d T}{T} & =\frac{1}{2} \alpha d t \\
& =\frac{1}{2} \times 9 \times 10^{-7} \times(30-20) \\
& =4.5 \times 10^{-6} \\
\therefore \quad \text { Loss in time }= & 4.5 \times 10^{-6} \times 0.5 \\
= & 2.25 \times 10^{-6} \mathrm{~s}
\end{aligned}$