Answer the following questions:

$(a)$ Time period of a particle in $SHM$ depends on the force constant $k$ and mass $m$ of the particle:

$T=2 \pi \sqrt{\frac{m}{k}}$. A stmple pendulum executes $SHM$ approximately. Why then is the time pertodof.anondwers period of a pendulum independent of the mass of the pendulum?

$(b)$ The motion of a simple pendulum is approximately stmple harmonte for small angle oscillations. For larger angles of oscillation, a more involved analysis shows that $T$ is greater than $2 \pi \sqrt{\frac{l}{g}} .$ Think of a qualitative argument to appreciate this result.

$(c)$ A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall?

$(d)$ What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely failing under gravity?

The time period of a simple pendulum, $T=2 \pi \sqrt{\frac{m}{k}}$

For a simple pendulum, $k$ is expressed in terms of mass $m,$ as:

$k{\propto} m$

$\frac{m}{k}=$ Constant

Hence, the time period $T$, of a simple pendulum is independent of the mass of the bob.

In the case of a simple pendulum, the restoring force acting on the bob of the pendulum is given as:

$F=-m g \sin \theta$

Where,

$F=$ Restoring force

$m=$ Mass of the bob

$g=$ Acceleration due to gravity

$\theta=$ Angle of displacement

For small $\theta, \sin \theta=\theta$

For large $\theta, \sin \theta$ is greater than $\theta$

This decreases the effective value of $g$.

Hence, the time period increases as:

$T=2 \pi \sqrt{\frac{l}{g}}$

Where, $l$ is the length of the simple pendulum

The time shown by the wristwatch of a man falling from the top of a tower is not affected by the fall. since a wristwatch does not work on the principle of a simple pendulum, it is not affected by the acceleration due to gravity during free fall. Its working depends on spring action.

When a simple pendulum mounted in a cabin falls freely under gravity, its acceleration is zero. Hence the frequency of oscillation of this simple pendulum is zero.