Question

For smaller angular displacement, the period of a simple pendulum depends on

• A. its amplitude
• B. its phase constant
• C. its energy
• D. the mass of the bob
• E. its length

Hint:

For small oscillations, remember \(T=2\pi\sqrt{\frac{l}{g}}\): depends only on length and gravity.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
A simple pendulum consists of a point mass (bob) suspended from a fixed support by a light, inextensible string. For small oscillations (small angular displacements), the motion is approximately simple harmonic motion (SHM). We need to identify which physical property determines the period of this motion.
Step 2: Key Formula or Approach:
The formula for the period (\( T \)) of a simple pendulum undergoing small oscillations is:
\[ T = 2\pi\sqrt{\frac{L}{g}} \] where L is the length of the pendulum and g is the acceleration due to gravity.
Step 3: Detailed Explanation:
Let's analyze the formula \( T = 2\pi\sqrt{\frac{L}{g}} \).
- \( 2\pi \) is a numerical constant.
- \( g \) is the acceleration due to gravity, which is considered constant at a given location.
- \( L \) is the length of the pendulum.
From the formula, we can see that the period \( T \) is directly proportional to the square root of the length L (\( T \propto \sqrt{L} \)).
Let's consider the options given:
(A) Amplitude: The formula is derived under the small-angle approximation (\( \sin\theta \approx \theta \)), which makes the period independent of the amplitude. For larger amplitudes, the period does increase slightly, but for "smaller angular displacement," it is considered constant.
(B) Phase constant: The phase constant determines the starting position of the bob but does not affect the time it takes to complete one oscillation.
(C) Energy: The total energy of the pendulum is related to the amplitude (\( E \propto A^2 \)), and as explained, the period is independent of amplitude for small oscillations.
(D) Mass of the bob: The mass \( m \) of the bob cancels out during the derivation of the equation of motion, so the period is independent of the mass.
(E) Its length: As shown by the formula, the period is directly dependent on the length L of the pendulum.
Step 4: Final Answer:
For small angular displacements, the period of a simple pendulum depends on its length. This corresponds to option (E).