Question

A particle of mass \( m \) is performing U.C.M. along a circle of radius \( r \). The relation between centripetal acceleration \( a \) and kinetic energy \( E \) is given by

• A. \( a = \frac{2E}{mr} \)
• B. \( a = \left( \frac{2E}{mr} \right)^2 \)
• C. \( a = \frac{E}{mr} \)
• D. \( a = 2Em \)

Hint:

To relate centripetal acceleration and kinetic energy, use the expression \( a = \frac{2E}{mr} \), derived from the relationship between kinetic energy and velocity.

Answer 1

The Correct Option is A

Step 1: Centripetal acceleration and kinetic energy.
The centripetal acceleration \( a \) is given by: \[ a = \frac{v^2}{r} \] where \( v \) is the speed of the particle. The kinetic energy \( E \) of the particle is: \[ E = \frac{1}{2} m v^2 \]
Step 2: Relating acceleration and kinetic energy.
From the kinetic energy equation, solve for \( v^2 \): \[ v^2 = \frac{2E}{m} \] Substitute this into the equation for centripetal acceleration: \[ a = \frac{\frac{2E}{m}}{r} = \frac{2E}{mr} \]
Step 3: Conclusion.
The correct relation is \( a = \frac{2E}{mr} \), so the correct answer is (A).