Question

A rod of length Lo = 1m, moves with speed v equivalent to 77% of the speed of light c along the y-axis direction. At rest, the rod makes an angle of 35 ° with respect to the x- axis.
a) Determine the length of the rod as measured by a stationary observer.
b) Determine the angle θ the rod makes with the x axis when it reaches its speed.

Answer 1

a). Since, initially angle made with x axis is $35^\circ$ thus

$$L_{ox}=1\cdot\cos(35^\circ)=0.82m\\ L_{oy}=1\cdot\sin(35^\circ)=0.57m\\$$

As there is only motion along y axis, and $v=\frac{77}{100}c$

$$L_y=\frac{L_{oy}}{\sqrt{1-v^2/c^2}}\implies L_y=0.89m$$

Thus,

$$L=\sqrt{L_y^2+L_{ox}^2}=1.21m$$

b)Now,

$$\theta=\tan^{-1}(L_y/L_{0x})=47.34^\circ$$