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Question

A plane mirror hinged at O is free to rotate in a vertical plane. The point O is at a distance x from a long screen placed in front of the mirror as shown in figure. A laser beam of light incident vertically downward is reflected by he mirror at O so that a bright spot is formed at the screen.

At the instant shown, the angle of inclidence is θ and the mirror is rotating clockwise with constant angular velocity w. Find the speed of the spot at this instant.

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Solution

The correct option is **A** 2x w sec2ϕ

Let P be the bright spot, shown on the screen, Let the distance of point P from O' be y at this instant shown in figure

dydt=?

From the figure,

θ+θ+ϕ=90∘ → (1)θ+ϕ+α=90∘(2) ⇒ θ=α⇒ϕ+2θ=90∘ ⇒ ‘ϕ+2α=90∘⇒dϕdt+2dαdt=0 ⇒dϕdt=2dαdt

So, the angular spped of the reflected ray is double the angular speed of the mirror since,

y=x tanϕ⇒ dydt=s sec2ϕdϕdt=2xwsec2ϕ

Let P be the bright spot, shown on the screen, Let the distance of point P from O' be y at this instant shown in figure

dydt=?

From the figure,

θ+θ+ϕ=90∘ → (1)θ+ϕ+α=90∘(2) ⇒ θ=α⇒ϕ+2θ=90∘ ⇒ ‘ϕ+2α=90∘⇒dϕdt+2dαdt=0 ⇒dϕdt=2dαdt

So, the angular spped of the reflected ray is double the angular speed of the mirror since,

y=x tanϕ⇒ dydt=s sec2ϕdϕdt=2xwsec2ϕ

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