A simple pendulum of length $l$ has maximum angular displacement $\theta$. The maximum kinetic energy of the bob of mass $m$ is
($g =$ acceleration due to gravity)
- $mgl (1 + cos\,?)$
- $mgl (1 + cos^2\, ?)$
- $mgl (1 - cos \,?)$
- $mgl (cos\,? - 1)$
The Correct Option is C
Solution and Explanation
To solve this problem, we determine the expression for the maximum kinetic energy of a simple pendulum's bob when it oscillates with maximum angular displacement \(\theta\).
Step 1: Understanding Energy Conservation
The total mechanical energy is conserved and consists of potential energy (PE) and kinetic energy (KE). At maximum displacement, KE is zero and energy is purely potential.
Step 2: Calculate Maximum Potential Energy
The height above the lowest point is: \(h = l(1 - \cos\theta)\)
Thus, maximum potential energy is: \(\text{PE}_{\text{max}} = mgl(1 - \cos\theta)\)
Step 3: Relate to Kinetic Energy
At the lowest point, all potential energy converts into kinetic energy: \(\text{KE}_{\text{max}} = mgl(1 - \cos\theta)\)
Step 4: Conclusion
Therefore, the maximum kinetic energy is: \(mgl(1 - \cos\theta)\)
Thus, the correct answer is option (3).
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Concepts Used:
Kinetic energy
Kinetic energy of an object is the measure of the work it does as a result of its motion. Kinetic energy is the type of energy that an object or particle has as a result of its movement. When an object is subjected to a net force, it accelerates and gains kinetic energy as a result. Kinetic energy is a property of a moving object or particle defined by both its mass and its velocity. Any combination of motions is possible, including translation (moving along a route from one spot to another), rotation around an axis, vibration, and any combination of motions.
