Question

A particle oscillates along the \( x \)-axis according to the law, \( x(t) = x_0 \sin^2 \left( \frac{\pi t}{T} \right) \), where \( x_0 = 1 \, \text{m} \) and \( T \) is the time period of oscillation. The kinetic energy (\( K \)) of the particle as a function of \( x \) is correctly represented by the graph:

• A.
• B.
• C.
• D.

Hint:

To determine the kinetic energy as a function of displacement, start by differentiating the displacement function to find the velocity. Then square the velocity and use the formula for kinetic energy.

Answer 1

The Correct Option is A

The kinetic energy \( K \) of a particle is defined as: \[ K = \frac{1}{2} m v^2, \] where \( v \) represents the particle's velocity. Velocity is the rate of change of displacement \( x(t) \) over time: \[ v(t) = \frac{d}{dt} \left( x_0 \sin^2 \left( \frac{\pi t}{T} \right) \right) = 2x_0 \sin \left( \frac{\pi t}{T} \right) \cos \left( \frac{\pi t}{T} \right) \frac{\pi}{T}. \] Consequently, the velocity is directly proportional to \( \sin \left( \frac{\pi t}{T} \right) \). Since kinetic energy is proportional to the square of the velocity, the kinetic energy graph demonstrates an increase as the particle moves from the origin to its furthest point, followed by a symmetrical decrease.
Final Answer: Graph 1.