Question

A point object is kept in front of a plane mirror. The plane mirror is doing SHM of amplitude $2cm$. The plane mirror moves along the x-axis which is normal to the mirror. The amplitude of the mirror is such that the object is always in front of the mirror. The amplitude of SHM of the image is
(A). $0$
(B). $2cm$
(C). $4cm$
(D). $1cm$

Answer 1

The plane mirror performs simple harmonic motion. SHM is a special type of periodic motion where the force is directly proportional to the negative of displacement. Let us assume that the plane mirror is kept at origin. Find the position of image for mean and extreme positions of the mirror and calculate the distance to calculate the amplitude.

Complete answer:
Given that the mirror is kept such that the x-axis is perpendicular to the plane of the mirror. The amplitude is $2cm$.
Let us assume that the mirror is kept at the origin and performs SHM therefore it moves between $-2cm$ and $2cm$.
Simple Harmonic motion or SHM is a periodic motion in which the force is directly proportional to the negative of displacement of the body.

Now let us assume that an object is kept at a point on the x-axis at $(2+x)$, then its image will be formed at the coordinate $-(2+x)$.
When the mirror is at coordinate $2cm$, then distance between object and mirror is x and the image is formed at coordinate $-(2+x)+2=-x$
Similarly, when the mirror is at coordinate $-2cm$, then distance between the object and mirror is 4+x and the image is formed at coordinate $-(2+x)-2=-(4+x)$. Therefore, the net difference between the position of image for the two coordinates is-
$\left| -(4+x)-(-x) \right|=4cm$
The amplitude will be half of the distance between the positions of images therefore,
$A=\dfrac{4cm}{2}=2cm$
Therefore, the amplitude of the SHM of the image is $2cm$.

So, the correct answer is “Option B”.

Note: Periodic motions are those motions which repeat itself after a fixed interval of time. Amplitude is the greatest distance from the mean position of a body in periodic motion or the distance between mean position and one extreme position. Some examples of simple harmonic motion are motion of a swing, motion of a spring, motion of a pendulum etc.