Question

The time period of a second’s pendulum on the surface of the moon is found to be 5s. The acceleration due to gravity on moon is:
\eqalign{ & {\text{A}}.\;3.2m{s^{ - 2}} \cr & {\text{B}}.\;1.6m{s^{ - 2}} \cr & {\text{C}}.\;0.8m{s^{ - 2}} \cr & {\text{D}}.\;0.6m{s^{ - 2}} \cr}

Answer 1

Second’s pendulum means it measures the time in seconds by completing half oscillation in 1 second. So, the total time period of the pendulum is 2 seconds. To the pendulum, the necessary restoring torque is provided by gravity, hence gravity has an effect on the time period of the pendulum. If gravity is more the restoring torque is more and hence pendulum will oscillate faster and hence time period will become less and if torque is less, it will oscillate slower and the time period will be more or it will take more time to complete the oscillation.

Formula used:
$T = 2\pi \sqrt {\dfrac{l}{g}}$
where ‘l’ is the length of string to which bob is attached
and ‘g’ is the acceleration due to gravity.

Complete step-by-step answer:
Second’s pendulum means its time period on earth is two seconds and ‘g’ on earth is $9.8m{s^{ - 2}}$.
Putting values in the equation and solving for ‘l’ :
\eqalign{ & \Rightarrow 2 = 2\pi \sqrt {\dfrac{l}{g}} \cr & \Rightarrow l = \dfrac{g}{{{\pi ^2}}} \cr & \Rightarrow l = \dfrac{{9.8}}{{{\pi ^2}}} \cr}
Now, solving for the time period with moon conditions.
Given, T=5s
Putting in formula:
\eqalign{ & 5 = 2\pi \sqrt {{{\dfrac{l}{g}}_m}} \cr & \Rightarrow {\left( {\dfrac{5}{{2\pi }}} \right)^2} \times {g_m} = l \cr}
Also, $l = \dfrac{{9.8}}{{{\pi ^2}}}$
Hence, equating both will give;
\eqalign{ & {\left( {\dfrac{5}{{2\pi }}} \right)^2} \times {g_m} = \dfrac{{9.8}}{{{\pi ^2}}} \cr & \Rightarrow \left( {\dfrac{{25}}{{4{\pi ^2}}}} \right) \times {g_m} = \dfrac{{9.8}}{{{\pi ^2}}} \cr & \Rightarrow {g_m} = \dfrac{{9.8 \times 4}}{{25}} \cr & \Rightarrow {g_m} = 1.568 \cr & \therefore {g_m} \simeq 1.6m{s^{ - 2}} \cr}

So, the correct answer is “Option B”.

Note: Whenever studying new concepts in physics, we must be curious about different terms appearing in formula which states that particular quantity depends upon which parameters, for example Time period depends upon acceleration due to gravity but not on the mass. This is an important result.
Chance of mistake in pendulum oscillation questions is that being master means lesser time period i.e. if speed of oscillation increases, it’s time period will decrease because both these physical quantities are inversely proportional to each other.