Question

For a pendulum in damped oscillation, with a bob of mass \(\mathrm{m}\) and radius \(\mathrm{r}\), with a string of length \(\ell\) What is the time period? (A) \(\mathrm{T}=2 \pi \sqrt{(\ell / \mathrm{g})}\) (B) T depends on \(\mathrm{m}\) (C) \(\mathrm{T}>2 \pi \sqrt{(\ell / \mathrm{g})}\) (D) \(\mathrm{T}<2 \pi \sqrt{(\ell / \mathrm{g})}\)

Short answer

The correct answer is option (C), which states that for a pendulum in damped oscillation, the time period is greater than that of a simple pendulum: \[\mathrm{T}>2 \pi \sqrt{\frac{\ell}{\mathrm{g}}}\]

Step-by-step solution

1. Time period of a simple pendulum formula

The time period (T) of a simple pendulum (no damping) is given by the formula: \[\mathrm{T} = 2\pi\sqrt{\frac{\ell}{\mathrm{g}}}\] where \(\ell\) is the length of the string and \(\mathrm{g}\) is the acceleration due to gravity.

2. Adding damping to the system

Damping in a pendulum means that after each oscillation, the amplitude of the oscillations would dampen down due to the air resistance, friction, and other factors. Damping doesn't affect the mass and length of the pendulum, but it will have an effect on the time period.

3. Analyzing Option (A)

Option (A) gives the formula for the time period of a simple pendulum (T) without any damping, which is not suitable for the given problem. Therefore, option (A) is not correct.

4. Analyzing Option (B)

Option (B) states that the time period (T) depends on the mass (m). However, the time period of a simple pendulum doesn't depend on the mass, and although damping may add additional factors into the equation, mass still won't directly affect the time period. Hence, option (B) is not correct.

5. Analyzing Option (C) and (D)

Now, we are left with options (C) and (D). Option (C) states that the time period (T) of the damped pendulum is greater than that of a simple pendulum, and option (D) states that the time period (T) is less than that of a simple pendulum. When a pendulum's oscillation is damped, the bob takes more time to complete one oscillation as it has to overcome the energy loss due to damping. Thus, the time period of a damped pendulum should be greater than that of a simple pendulum. So, the correct answer is option (C), which states that \[\mathrm{T}>2 \pi \sqrt{\frac{\ell}{\mathrm{g}}}\]