Question

A pendulum oscillates simple harmonically and only if
(I) the sizer of the bob of pendulum is negligible in comparison with the length of the pendulum
(II) the angular amplitude is less than 10°.

• A. Both (I) and (II) are correct
• B. Both (I) and (II) are incorrect
• C. Only (I) is correct
• D. Only (II) is correct

Answer 1

The Correct Option is A

1. The size of the bob of pendulum is negligible in comparison with the length of the pendulum.

Explanation: In the ideal model of a simple pendulum, we assume that the entire mass of the pendulum is concentrated at a single point (point mass) and is suspended by a massless, inextensible string. If the size of the bob is not negligible compared to the length of the pendulum, then the pendulum is more accurately described as a physical pendulum. In a physical pendulum, the mass is distributed, and the moment of inertia plays a role.

Impact on SHM: For a simple pendulum to exhibit true Simple Harmonic Motion (SHM), the restoring force must be directly proportional to the displacement. This proportionality is derived from the assumption of a point mass bob. While a physical pendulum can approximate SHM for small oscillations, the condition for ideal SHM is best met when the bob's size is negligible, thus approximating a point mass.

Conclusion for (I): Statement (I) is correct. Negligible bob size is a key assumption in the simple pendulum model that leads to SHM.

2. The angular amplitude is less than 10°.

Explanation: The restoring force in a pendulum is given by $F = -mg \sin \theta$, where $\theta$ is the angular displacement from the vertical equilibrium position. For SHM, the restoring force must be proportional to the displacement, which in this case is the angular displacement $\theta$. The approximation $\sin \theta \approx \theta$ (for $\theta$ in radians) is used to achieve this proportionality.

Conclusion for (II): Statement (II) is also correct. Limiting the angular amplitude to less than 10° ensures the validity of the small angle approximation, which is crucial for the pendulum to exhibit SHM.

Overall Conclusion:

Both statement (I) and statement (II) describe necessary conditions for a pendulum to oscillate in Simple Harmonic Motion (within the approximations considered in introductory physics). Statement (I) relates to the idealization of the pendulum as a simple pendulum, and statement (II) relates to the small angle approximation needed for the restoring force to be linear with displacement.

Final Answer: 
Therefore, the correct answer is: (A) Both (I) and (II) are correct

Answer 2

For a pendulum to oscillate with simple harmonic motion (SHM), the restoring force must be directly proportional to the displacement and directed towards the mean position. This condition is satisfied when the following conditions are met:

(I) the size of the bob of pendulum is negligible in comparison with the length of the pendulum: This is correct. If the bob's size is comparable to the pendulum's length, the moment of inertia of the system becomes more complex, and the restoring force is no longer directly proportional to the displacement.

(II) the angular amplitude is less than 10°: This is correct. For small angles, the sine of the angle is approximately equal to the angle itself (in radians). This approximation is crucial for the simple harmonic motion equation to hold true. For larger angles, the approximation breaks down, and the motion is no longer simple harmonic.

The correct answer is (A) Both (I) and (II) are correct.