A block of mass m containing a net positive charge q is placed on a smooth horizontal table which terminates in a vertical wall as shown in figure. The distance of the block from the wall is d. A horizontal electric field E towards right is switched on. Assuming elastic collisions (if any) find the time period of the resulting oscillatory motion. Is it a simple harmonic motion?

Question

Philoid · Accepted Answer

Given:
Mass of the block : m
Charge on the block: q
Distance between block and the wall : d
Horizontal electric field: E
Initial velocity: u=0

Formula used:

It is not Simple Harmonic Motion. In SHM the acceleration is directly proportional to the displacement of the body and in opposite direction of the displacement. In this case, acceleration is proportional to the displacement but not in the opposite direction.
We know that,

Here, F is the horizontal Electric force , q is the charge of the body and E is the electric force (horizontal), a is the acceleration and m is the mass of the body
To find time required by the block to collide into the wall can be calculated by laws of motion.
Here s is the displacement; s=d. u is the initial velocity , a is the acceleration of the block and t is the time required to travel the displacement.

This is the time required by the block to travel d and hit the wall.
Since it is assumed to be elastic collision , after collision with the wall the block will take same time ‘t’ to come back till it’s velocity is zero.
Thus Total time take:T

Time period of the resulting oscillatory motion is