Question

The temperature of a pendulum, the time period of which is $t$ , is raised by $\Delta T$ The change in its time period is:
A. $\dfrac{1}{2}\alpha t\Delta T$
B. $2\alpha t\Delta T$
C. $\dfrac{1}{2}\alpha \Delta T$
D. $2\alpha \Delta T$

Answer 1

Hint: To answer this topic, we must first comprehend the fundamentals of a pendulum, including its bob and the material it is constructed of. We can evaluate if its volume rises with temperature and what implications this will have on its time period based on this knowledge.

Complete step by step answer:
As we know, the time period of a simple pendulum at room temperature is $1\sec $ .
And $t = 2\pi \sqrt {\dfrac{L}{g}} $ -----(1)
And we know that the actual length with the change in temperature is given by,
$L = {L_0}\left( {1 + \alpha \Delta \theta } \right)$ and we can write this as,
$\Rightarrow \dfrac{{L - {L_0}}}{L} = \dfrac{{\Delta L}}{L} \\
\Rightarrow \dfrac{{L - {L_0}}}{L} = \alpha \Delta \theta $ -----(2)
Now with the help of equation (1)
$\dfrac{{\Delta t}}{t} = \dfrac{1}{2}\alpha \Delta T \\
\Rightarrow t = 1 \\
\therefore \Delta t = \dfrac{1}{2}\alpha \Delta T \\ $
Hence the correct option is A.

Note: The period of a basic pendulum is defined as the time it takes to complete one complete cycle, which includes a left and right swing. The length of a pendulum and, to a lesser extent, the amplitude, or the width of the pendulum's swing, determine its time period. As the temperature rises, the length of the pendulum lengthens, and the time period of the pendulum lengthens. In the summer, a pendulum dock becomes slow and loses time due to an increase in its time period.