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Question

A particle which can move along x-axis is released from rest at the position $x = x_{0} .$ The potential energy (U) of the block is described below :
$U = {\begin{matrix} - a x & ; x < 0 b x^{2} & ; x \geq 0 \end{matrix}}$
Which of the following statements is/are correct :

The subsequent motion is simple harmonic
The subsequent motion is periodic
The speed is a continuous function of time
The magnitude of acceleration is a continuous function of time

A

The subsequent motion is simple harmonic

B

The subsequent motion is periodic

C

The speed is a continuous function of time

D

The magnitude of acceleration is a continuous function of time

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Solution

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The potential is not quadratic so the motion is not sinusoidal or simple harmonic. The motion is periodic since the particle is bounded by the potential. The energy is constant and so the particle will move back and forth between a maximum negative position and the position $x = x_{0} .$
The potential energy is continuous and hence $K . E . = T E - U$ is continuous. Therefore, speed is continuous.
The derivative of the potential energy is not continuous at x = 0. Since, force is equal to the negative derivative of the potential energy, the force is not continuous at $x = 0$ . Therefore, acceleration $(\frac{F}{m})$ is not continuous at $x = 0$ .

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