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 \[\mathrm{T}>2 \pi \sqrt{\frac{\ell}{\mathrm{g}}}\]

Key concepts

Time Period of Pendulum

The time period of a pendulum is a crucial aspect in understanding its motion. For a simple pendulum, this period is the time it takes for the pendulum to swing back and forth once. The standard formula for the time period (T) of a simple, non-damped pendulum is given by:

• \[\mathrm{T} = 2\pi\sqrt{\frac{\ell}{\mathrm{g}}}\]

Here, \( \ell \) represents the length of the pendulum, and \( \mathrm{g} \) stands for the acceleration due to gravity.
This equation reveals that the time period is proportional to the square root of the pendulum's length, but it is unaffected by the mass of the bob. Whether the pendulum is in a vacuum or has a bob of varying mass, the absence of air resistance and external influences will ensure this formula holds true.

Effect of Damping

When considering the effect of damping on a pendulum, it's key to note how energy is dissipated as the pendulum swings. Damping primarily arises due to factors like air resistance and friction at the pivot point.
As damping increases, the amplitude of oscillations decreases over time, affecting the time period in a subtle yet significant way.

Damping doesn't directly change the mass or length of the pendulum. However, it does cause energy loss during each oscillation cycle. This energy loss slows down the oscillation, resulting in a longer time period. So, while damping doesn't alter the basic formula to an obvious extent, it effectively increases the period as the pendulum has to work against this energy loss.

Damped Pendulum Analysis

In analyzing a damped pendulum, it is essential to compare it with its simple non-damped counterpart. When damping occurs, it affects how long it takes for the pendulum to complete an oscillation.
The time period of the damped pendulum becomes longer than the simple pendulum due to additional forces acting on it. These forces include energy loss from air resistance or friction.

Through this analysis, one can conclude that the damped pendulum's time period can be denoted as:

• \[\mathrm{T} > 2 \pi \sqrt{\frac{\ell}{\mathrm{g}}}\]

In practical scenarios, this means the pendulum swings more sluggishly over time. It underscores the importance of damping in real-world systems where perfect oscillations are hard to maintain over repeated cycles.