Question

Simple pendulum of length \(l\) has a maximum angular displacement \(\theta\). The maximum kinetic energy of the bob is

• A. \(mgl(1 - \cos\theta)\)
• B. \(0.5\ mgl\)
• C. \(mgl\)
• D. \(2mgl\)

Hint:

For small \(\theta\), \(\cos\theta \approx 1 - \frac{\theta^2}{2}\), so KE \(\approx \frac{1}{2}mgl\theta^2\).

Answer 1

The Correct Option is A

To determine the maximum kinetic energy of a bob in a simple pendulum, we need to consider the conversion of potential energy at the highest point to kinetic energy at the lowest point. Let's break down the problem:

1. The potential energy at the maximum displacement (highest point) is converted entirely into kinetic energy at the lowest point (where the speed is maximum).
2. The potential energy at the height can be expressed as: \(PE = mgh\), where \(h\) is the height of the bob.
3. From the geometry of the pendulum, the height \((h)\) can be related to the length of the pendulum \((l)\) and the angular displacement \((\theta)\).
4. Using the relation: \(h = l - l \cos \theta = l(1 - \cos \theta)\)
5. Thus, the potential energy at maximum displacement is: \(PE = mg \cdot l(1 - \cos \theta)\)
6. At the lowest point, all potential energy is converted into kinetic energy, so: \(KE_{\text{max}} = mg \cdot l(1 - \cos \theta)\)

Hence, the correct option is \(mgl(1 - \cos\theta)\), as this expression accurately represents the maximum kinetic energy of the pendulum bob.