Question

Find the angle by which the mirror must be rotated such that the reflected ray will be travelling along the vertical (dotted line) as shown in the figure.
[CW - clockwise, CCW - counter-clockwise]

• A. ${15}^{\circ }CCW$
• B. ${210}^{\circ }CW$
• C. ${105}^{\circ }CCW$
• D. ${75}^{\circ }CCW$

Answer 1

Answer: (A), (C)

${105}^{\circ }CCW$

When the plane mirror is not rotated, the reflected ray makes an angle of ${30}^{\circ }$ with the normal, as shown in the figure below.



So, for the reflected ray to become vertical.

Case -I:

Angle by which the reflected ray must be rotated be $\theta ={30}^{\circ }CCW$.

So,

Angle by which mirror must be rotated be $\varphi =\frac{\theta }{2}=\frac{{30}^{\circ }}{2}={15}^{\circ }CCW.$

Case -II :

Angle by which the reflected ray must be rotated be $\theta ={210}^{\circ }CCW$

So,

Angle by which mirror must be rotated be $\varphi =\frac{\theta }{2}=\frac{{210}^{\circ }}{2}={105}^{\circ }CCW.$

Case-III:

Angle by which the reflected ray must be rotated be $\theta ={150}^{\circ }CW$

So,

Angle by which mirror must be rotated be $\varphi =\frac{\theta }{2}=\frac{{150}^{\circ }}{2}={75}^{\circ }CW.$

Case -IV:

Angle by which the reflected ray must be rotated be $\theta ={330}^{\circ }CW$

So,

Angle by which mirror must be rotated be $\varphi =\frac{\theta }{2}=\frac{{300}^{\circ }}{2}={165}^{\circ }CW.$

These are the four cases by which the reflected will be along the vertical.

It is clear that, Case -I and Case -II are valid and Case- III and Case- IV are invalid for a reason that, a mirror can have only one reflecting surface and the other side is coated.

Hence, options (a) and (c) are correct.

$\begin{array}{c}\text{Why this Question}?\\ \text{This question provides student a conceptual clarity of rotation of mirror, incident \& reflected ray.}\\ \text{Note-1 :- The angle turned by the reflected ray is twice the angle turned by the mirror.}\\ \text{Note-2 :- If mirror rotates clockwise, then reflected ray will rotate clockwise}\\ \text{Extra credit :- If the mirror is kept fixed and the incident ray is rotated,then the reflected ray}\\ \text{will rotate in opposite sense by same angle.}\end{array}$