Spherical/Curved mirrors

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Question

A plane mirror is rotating about ‘O’ with constant angular velocity 4 rad/sec in anticlockwise direction. A point ‘P’ is moving in a circle of radius 50 cm with constant speed 3 m/s in anticlockwise direction about point O. At a given instant as shown in figure, the angular velocity of image is
(in rad/s) is

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Solution

$C o n s i d e r i n g t h e \tan g e n t i a l c o m p o n e n t s o f v e l o c i t i e s, v_{I M} = - v_{O M}$
$\Rightarrow v_{I} = 2 v_{M} - v_{O}$
$\Rightarrow ω_{i m a g e} = 2 ω_{m i r r o r} - \frac{V_{o b j e c t}}{R}$
$\Rightarrow ω_{i m a g e} = 2 (4) - \frac{3}{0.5} = 2 r a d / s$

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Spherical/Curved mirrors

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Considering the tangential components of velocities, vIM=-vOM⇒vI=2vM-vO⇒ωimage=2ωmirror−VobjectR⇒ωimage=24-30.5=2 rad/s

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$C o n s i d e r i n g t h e \tan g e n t i a l c o m p o n e n t s o f v e l o c i t i e s, v_{I M} = - v_{O M}$
$\Rightarrow v_{I} = 2 v_{M} - v_{O}$
$\Rightarrow ω_{i m a g e} = 2 ω_{m i r r o r} - \frac{V_{o b j e c t}}{R}$
$\Rightarrow ω_{i m a g e} = 2 (4) - \frac{3}{0.5} = 2 r a d / s$