Question

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

Answer 1

For motion to be simple harmonic,acceleration should be proportional to the displacement and should be directed in a direction opposite to the displacement.
When the block is moving towards the wall, the acceleration is along displacement.
So, the block does not undergo SHM.
Time taken to reach the wall is given by
$$\begin{gathered} d=ut+\frac{1}{2}a{t}^2 \\ \Rightarrow t=\sqrt{\frac{2md}{qE}}(\text{U}\mathrm{sing}u=0,\mathrm{and}a=\frac{qE}{m}) \end{gathered}$$
Since it is an elastic collision, the time taken by the block to move towards the wall is the time taken to move away from it till the velocity is zero.
Total time, T = 2t
$\Rightarrow T=2\sqrt{\frac{2dm}{qE}}=\sqrt{\frac{8dm}{qE}}$