Solveeit Logo

Question

Question: A spot of light S rotates in a horizontal plane with a constant angular velocity of 0.1rad/s. The sp...

A spot of light S rotates in a horizontal plane with a constant angular velocity of 0.1rad/s. The spot of light P moves along the wall at a distance of 3m from S. The velocity of spot P, where θ=45\theta = 45^\circ , is?
A. 0.5m/s0.5m/s
B. 0.6m/s0.6m/s
C. 0.7m/s0.7m/s
D. 0.8m/s0.8m/s

Explanation

Solution

Hint:- The formula for angular velocity is given by ω=dθdt\omega = \dfrac{{d\theta }}{{dt}} where θ\theta is the angle between the perpendicular from the source to the wall and the line joining the source and the spot. For your better understanding a diagram is given for the situation.

Complete step-by-step solution:-
The situation in the question is given in the diagram for better understanding
As given in the question that the perpendicular distance between the source and the wall, SA=r=3mSA = r = 3m
Let angle between SA and SP be θ\theta and the distance PA be xx
Now, the velocity of the spot of light P, vv is the rate at which xx is changing i.e. v=dxdtv = \dfrac{{dx}}{{dt}}

As from figure,
x=rtanθx = r\tan \theta
So, velocity of the point P
v=dxdt=rsec2θdθdtv = \dfrac{{dx}}{{dt}} = r{\sec ^2}\theta \dfrac{{d\theta }}{{dt}}
Now as we know that angular velocity is given by ω=dθdt\omega = \dfrac{{d\theta }}{{dt}} where θ\theta is the angle between the perpendicular from the source to the wall and the line joining the source and the spot.
So, the equation becomes v=rωsec2θv = r\omega {\sec ^2}\theta
Now according to the question θ=45\theta = 45^\circ , r=3mr = 3m and angular velocity ω=0.1rad/s\omega = 0.1rad/s
So, substituting these value in the above equation, we have
v=3×0.1×sec245=3×0.1×2v = 3 \times 0.1 \times {\sec ^2}45^\circ = 3 \times 0.1 \times 2
So, v=0.6m/sv = 0.6m/s

Hence option B is correct.

Note:- Angular velocity is the time rate of change of angular displacement at which an object or a particle is rotating around a center or a specific point. It is also known as rotational velocity. The unit of Angular velocity is angle per unit time or radians per second (rad/s). The rate of change of angular velocity is angular acceleration.
Angular velocity plays an important role in the rotational motion of an object. The linear velocity of every particle of a body in circular motion is directly related to the angular velocity of the whole object.