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).

18 days ago

To tackle your question about the behavior of an image in a rotating plane mirror, let's break down the concepts involved and clarify the reasoning behind your approach. Your intuition about using angular velocities and the relationship between the object and its image is on the right track, but we need to ensure that we apply the principles of rotational motion correctly.

Understanding the Setup

When an object is placed in front of a plane mirror, the image formed is virtual and appears to be the same distance behind the mirror as the object is in front. If the mirror is rotated about a point at its bottom with an angular velocity \( \omega \), the image will also change its position due to the rotation of the mirror.

Analyzing the Motion

To analyze the motion of the image, consider the following steps:

• Initial Configuration: Let’s denote the object as point O and its image as point I. The distance from O to the mirror is equal to the distance from the mirror to I.
• Rotation of the Mirror: When the mirror rotates, the angle between the line joining the object to the point of rotation (let's call it P) and the line joining the image to P changes. If the mirror rotates by an angle \( \theta \), the line joining O to P will rotate by \( \theta \) in one direction, and the line joining I to P will rotate by \( \theta \) in the opposite direction.
• Relative Angular Velocity: The key point here is that the angular displacement of the image relative to the object is indeed \( 2\theta \). This is because both lines (OP and IP) are rotating, but in opposite directions.

Velocity of the Image

Now, to find the velocity of the image with respect to the object, we can use the relationship between angular velocity and linear velocity. The linear velocity \( v \) of a point at a distance \( r \) from the axis of rotation is given by:

v = r * ω

In this case, if we denote the distances from P to O and P to I as \( r_O \) and \( r_I \) respectively, the velocities can be expressed as:

• The velocity of the object O with respect to P: \( v_O = r_O * \omega \)
• The velocity of the image I with respect to P: \( v_I = r_I * \omega \)

Since the image and object are moving in opposite directions due to the rotation of the mirror, the relative velocity of the image with respect to the object is:

v_{relative} = v_I + v_O = r_I * \omega + r_O * \omega

This simplifies to:

v_{relative} = (r_I + r_O) * \omega

Clarifying the Use of Relative Angular Velocity

Your method of stopping the mirror and considering the relative angular velocities is a valid approach in this context. However, it’s crucial to ensure that you are consistently applying the principles of rotational dynamics. The concept of relative angular velocity can be used effectively, but it must be applied carefully to avoid confusion. The key takeaway is that the total angular displacement of the image relative to the object is indeed \( 2\theta \), leading to the conclusion that the relative velocity of the image with respect to the object is correctly derived.

Conclusion

In summary, your reasoning about the angular velocities and the resulting velocities of the image is sound. By understanding the relationships between the object, image, and the point of rotation, you can accurately determine the velocity of the image with respect to the object. Just remember to keep track of the directions of rotation and the distances involved to ensure clarity in your calculations.