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 i can find the velocity of image with respect to object?????

To tackle the problem of finding the velocity of the image formed by a plane mirror when it is rotated, we can indeed use the approach you've outlined. Let's break it down step by step to clarify how the rotation of the mirror affects the image and how we can calculate the velocity of the image relative to the object.

Understanding the Setup

Imagine you have a plane mirror positioned vertically, and an object is placed in front of it. The image of the object is formed behind the mirror, and it appears to be the same distance behind the mirror as the object is in front of it. When the mirror is rotated about a point at its bottom with an angular velocity $\omega$, the image will also change its position due to this rotation.

Analyzing the Rotation

When the mirror rotates, the angle between the line joining the object to the point of rotation and the line joining the image to the same point changes. Initially, let's denote:

• Point O as the object
• Point I as the image
• P as the point of rotation at the bottom of the mirror

When the mirror rotates by an angle $\theta$, the line joining the object (O) to the point of rotation (P) rotates by $\theta$ in one direction, while the line joining the image (I) to the point of rotation (P) rotates by $\theta$ in the opposite direction. This means that the total change in angle between the lines OP and IP is $2\theta$.

Calculating the Velocity of the Image

Now, if we consider the angular velocity $\omega$ of the mirror, the angular displacement after a small time $t$ can be expressed as:

• For the object: $\theta =\omega t$
• For the image: The angle changes by $2\theta =2\omega t$

To find the linear velocity of the image relative to the object, we can use the relationship between linear velocity $v$, angular velocity $\omega$, and radius $r$ (the distance from the point of rotation to the object or image). The linear velocity $v$ can be calculated as:

Velocity of the Image

The velocity of the image $v_I$ can be expressed as:

• For the object: $v_O=r\cdot \omega$
• For the image: $v_I=r\cdot 2\omega$

Thus, the velocity of the image with respect to the object becomes:

Relative Velocity: $v_{relative}=v_I-v_O=r\cdot 2\omega -r\cdot \omega =r\cdot \omega$

Conclusion

In summary, by analyzing the rotation of the mirror and the corresponding changes in the angles of the lines connecting the object and image to the point of rotation, we can effectively determine the velocity of the image. The key takeaway is that the image moves with a velocity that is twice that of the object due to the nature of reflection and the geometry involved in the rotation. This method of visualizing the problem and breaking it down into manageable parts allows for a clearer understanding of the dynamics at play.