Question

A pendulum clock which keeps correct time at the bottom of mountain losses 30sec/day when it is taken to the top of mountain. If the height of the mountain is $1.1\mu $ km. Find the values of $\mu \left( {{R_e} = 6400km} \right)$

Answer 1

Hint: This could be simply solved by applying the formula of time period. Also, the concept of linear expansion is applied.

Formula used:
Here, we will use the basic formula of speed, distance and time:
$T = 2\pi \sqrt {\dfrac{l}{g}} $
Here, $T$ is the time
$l$ is the length

Complete step by step answer:
We will start by considering the formula:
$T = 2\pi \sqrt {\dfrac{l}{g}} $
As $g = {g_0}\left( {1 - \dfrac{{2h}}{R}} \right)$
So, ${T_{mountain}} = 2\pi \sqrt {\dfrac{1}{{{g_0}\left( {1 - \dfrac{{2h}}{{6400}}} \right)}}} $
Now, we consider a simple pendulum which is having a time period of 2 sec.
\[2 = 2\pi \sqrt {\dfrac{l}{g}} \]
\[ \Rightarrow 2 + \dfrac{{23}}{{86400}} = 2\pi \sqrt {\dfrac{1}{{{g_0}\left( {1 - \dfrac{{2h}}{{6400}}} \right)}}} \]
And by solving further,
$2h = \left( {6400 - 5184} \right)$
$ \Rightarrow h = 608km$
Thus, we need to find the value of $\mu = 552$

Additional Information: Simple harmonic motion, in physics, repetitive movement back and forth through an equilibrium, or central, position, so that the maximum displacement on one side of this position is equal to the maximum displacement on the other side. The time interval of each complete vibration is the same.

Note: As the temperature increases, the length of the pendulum increases and hence the time period of the pendulum increases. Due to increment in its time period, a pendulum dock becomes slow in summer and will lose time. The time period of a pendulum does not depend on the mass of bob. So, if the mass of bob doubled, the time period remains the same.
Some of the examples of SHM are Swings that we see in the park is an example of simple harmonic motion. The back and forth, repetitive movements of the swing against the restoring force is the simple harmonic motion.
The pendulum oscillating back and forth from the mean position is an example of simple harmonic motion.
The process of hearing is impossible without simple harmonic motion. The sound waves that enter our ear causes the eardrum to vibrate back and forth.