If an object is kept in front if a plane mirror and its image is getting formed. Now the mirror is rotated by angular velocity w about a point on the bottom of mirror(take mirror as a straight line for this with w in upward direction). And we have to find the velocity of image. Can i do this by stopping the mirror and by taking that the line joining object and the point of rotation is moving by w and the line joining image and the point of rotation is also rotating by w. Both in opposite direction. After a that i stop the line joining object and point of rotation . Hence now the line joining image and point of rotation rotates by 2w.hence now this velocity of image is actually velocity of image with respect to object.is this true????. As i am getting right answer always but someone told me that we cannot take relative omega(angular velocity).

To tackle your question about the behavior of an image in a rotating plane mirror, let's break it down step by step. The scenario involves a plane mirror that is rotating about a point at its bottom, and we want to analyze how the image of an object changes as the mirror rotates. Your approach of considering the angular velocities and the relative motion is insightful, but let's clarify some concepts to ensure we have a solid understanding.

Understanding the Setup

When an object is placed in front of a plane mirror, the image formed is virtual and appears behind the mirror at the same distance as the object is in front of it. If the mirror is rotated, the image will also change its position due to the properties of reflection.

Analyzing the Rotation

When the mirror rotates with an angular velocity $\omega$, the angle of incidence equals the angle of reflection. This means that the image will also rotate, but in a way that depends on the angle of rotation of the mirror. The key point here is that the image's position relative to the object will change as the mirror rotates.

Using Angular Velocity

Your method of considering the angular velocities is a valid approach. When you say that the line joining the object and the point of rotation rotates by $\omega$ in one direction, and the line joining the image and the point of rotation rotates by $\omega$ in the opposite direction, you are essentially using the principle of relative motion. This is a common technique in physics to simplify the analysis of rotating systems.

Calculating the Velocity of the Image

To find the velocity of the image with respect to the object, we can indeed consider the total angular displacement. If the mirror rotates by an angle $\theta$, the image will appear to rotate by $2\theta$ due to the reflection property. Therefore, if you stop the line joining the object to the point of rotation, the line joining the image to the point of rotation will indeed rotate by $2\omega$. This means that the angular velocity of the image relative to the object is $2\omega$.

Addressing the Concern

The concern raised about not being able to take relative angular velocities likely stems from a misunderstanding of how angular motion works in this context. In many cases, especially in rigid body dynamics, relative angular velocities can be used effectively to analyze motion. As long as you are consistent with your reference points and the directions of rotation, your approach is sound.

Conclusion

In summary, your reasoning about the velocities and the use of relative angular velocities is correct. The image's angular velocity relative to the object indeed doubles due to the nature of reflection in a rotating mirror. This understanding is crucial in problems involving rotating systems and can be applied to various scenarios in physics. Keep practicing these concepts, and you'll continue to strengthen your grasp of rotational dynamics!