A problem on angles from different inertial frames

[Debdutta]

Homework Statement

A rod lies at an angle α with the x'-axis of an inertial frame moving at a speed v along the x-axis(x and x' are parallel) of another inertial frame. The rod makes angle β with the x-axis of this frame. Find the relation between α and β.

Variables: α,β,v
and define
γ=1/√1-(v/c)2;
x'=projection of rod along x.-axis with respect to first inertial frame;
z'=projection of rod along z'-axis(perpendicular to x') with respect to first inertial frame;
x=projection of rod along x-axis with respect to second inertial frame;
z=projection of rod along z-axis with respect to second inertial frame;

Homework Equations

Since lengths perpendicular to the relative motion remains unchanged, z'=z;
According to the second inertial frame, the length of the rod along the direction of motion is contracted by a factor of γ. Thus, x=(1/γ)x'.

tanα=z'/x' and tanβ=z/x.

The Attempt at a Solution

Thus, tanβ=γtanα.

The problem is, the answer given in the book is, tanα=γtanβ. I am confused, please help.

 

[mark726]

Thank you for your post. It seems that you have made a small mistake in your solution. The correct relation between α and β should be:

tanβ = tanα/γ

This can be derived by using the fact that the length of the rod along the direction of motion is contracted by a factor of γ, and the fact that the lengths perpendicular to the relative motion remain unchanged. Therefore, the correct solution should be:

tanβ = tanα/γ

I hope this helps clarify your confusion. If you have any further questions, please feel free to ask.
Scientist