Question

In the experiment of simple pendulum we keep \(\theta<5^{\circ}\), so as ensure \(\ldots \ldots \ldots\) (A) mass \(\mathrm{m}\) does not interfere in the time period (B) \(\mathrm{g}\) remains constant (C) the air drag is not too much (D) \(\sin \theta \cong \theta\) where by motion becomes simple harmonic.

Short answer

In the experiment of a simple pendulum, we keep the angle θ less than 5° to ensure that the motion becomes simple harmonic. This is achieved by making a small-angle approximation, which states that \(\sin \theta \cong \theta\), allowing the equation governing the motion of the pendulum to be simplified and treated as simple harmonic motion (Option D).

Step-by-step solution

Mass does not interfere in the time period

We know that the time period of a simple pendulum is given by: T = 2π√(L/g) Here, T is the time period, L is the length of the pendulum, and g is the acceleration due to gravity. Mass (m) is not a factor in this formula. Hence, keeping θ < 5° does not affect the mass interfering in the time period. So, option (A) is not the correct answer. #Step 2: Analyze Option (B)#

Acceleration due to gravity (g) remains constant

The acceleration due to gravity (g) is a constant and is not affected by the angle θ. It might change with altitude, but that is not relevant here. Therefore, option (B) is not the correct answer. #Step 3: Analyze Option (C)#

Air drag is not too much

Air drag does have some effect on the motion of the pendulum. However, the angle θ being less than 5° doesn't significantly affect the air drag, and hence the air drag doesn't really contribute to simple harmonic motion directly. So, option (C) is not the correct answer. #Step 4: Analyze Option (D)#

\(\sin \theta \cong \theta\), and motion becomes simple harmonic

A simple pendulum only exhibits simple harmonic motion when the oscillations are small. In this case, when θ < 5°, we can make a small-angle approximation, which states that: \(\sin \theta \cong \theta\) This approximation allows us to simplify the equation governing the motion of the pendulum, allowing it to be treated as simple harmonic motion. Therefore, option (D) is the correct answer. Thus, the angle must be kept less than 5° in the experiment of a simple pendulum to ensure that the motion becomes simple harmonic (Option D).