For the following questions, statement as well as the...

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement $-1$ is true, statement $-2$ is true; statement $-2$ is the correct explanation of statement $-1$ (b) Statement $-1$ is true, statement $-2$ is true but statement $-2$ is not the correct explanation of statement $-1$ (c) Statement $-1$ is true, statement $-2$ is false (d) Statement $-1$ is false, statement $-2$ is true (A) a (B) $b$ (C) $\mathrm{c}$ (D) $\mathrm{d}$ Statement $-1:$ Periodic time of a simple pendulum is independent of the mass of the bob. Statement $-2:$ The restoring force does not depend on the mass of the bob. (A) a (B) $b$ (C) $\mathrm{c}$ (D) $\mathrm{d}$

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option.
(a) Statement $-1$ is true, statement $-2$ is true; statement $-2$ is the correct explanation of statement $-1$
(b) Statement $-1$ is true, statement $-2$ is true but statement $-2$ is not the correct explanation of statement $-1$
(c) Statement $-1$ is true, statement $-2$ is false
(d) Statement $-1$ is false, statement $-2$ is true
(A) a
(B) $b$
(C) $\mathrm{c}$
(D) $\mathrm{d}$
Statement $-1:$ Periodic time of a simple pendulum is independent of the mass of the bob. Statement $-2:$ The restoring force does not depend on the mass of the bob.
(A) a
(B) $b$
(C) $\mathrm{c}$
(D) $\mathrm{d}$

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Key Concepts

Restoring Force versus Acceleration

The restoring force in a pendulum is given by the component of the gravitational force acting along the arc of the pendulum’s swing, typically expressed as -mg sin?. Although the magnitude of this force depends on the mass of the bob, the resultant acceleration (which determines the periodic motion) does not, due to the cancellation of mass when applying Newton’s Second Law.

Simple Pendulum Dynamics

This concept involves understanding that the time period of a simple pendulum, given by the formula T = 2??(L/g), depends only on the length of the pendulum and the acceleration due to gravity. This derivation traditionally uses the small-angle approximation to linearize the motion, showing that the mass of the pendulum bob cancels out of the equation.

Newton’s Second Law and Mass Cancellation

When applying Newton’s Second Law (F = ma) to the pendulum, the gravitational force acting on the bob (which is proportional to its mass) results in an acceleration that is independent of the mass, as the mass cancels from both sides of the equation. This concept explains why the pendulum’s period does not depend on the mass of the bob despite the force itself being mass-dependent.

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