Question

A student says that he had applied a force $F=-k\sqrt{x}$ on a particle and the particle moved in simple harmonic motion. He refuses to tell whether k is a constant or not. Assume that he was worked only with positive x and no other force acted on the particle.
(a) As x increases k increases.
(b) As x increases k decreases.
(c) As x increases k remains constant.
(d) The motion cannot be simple harmonic.

Answer 1

(a) As x increases k increases.

A body is said to be in simple harmonic motion only when,
F = $-$ kx ...(1)
where F is force,
k is force constant, and
x is displacement of the body from the mean position.

Given:
F = -k$\sqrt{x}$ ...(2)

On comparing the equations (1) and (2), it can be said that in order to execute simple harmonic motion, k should be proportional to $\sqrt{x}$ . Thus, as x increases k increases.