Answer to Question #135594 in Mechanics | Relativity for Hasnaira Hasan

Question

A meter stick at rest makes an angle of 30.0° with the x-axis. Find the length of the meter stick and the angle it makes with the x'-axis for an observer moving parallel to the x-axis at a speed of 0.650c

Answer 1

Considering length of the stick as the Hypotenuse of Right Angled Triangle, we have-

Length of stick along x-axis = 1$\times$ cos30°

= $\frac{\sqrt{3}}{2}$ m

Since motion is along x-direction, length contraction will also take place along x direction only. Therefore,

l_x = l_ox$\sqrt{1-\frac{v^2}{c^2}}$

= l_ox$\sqrt{1-0.4225}$

= l_ox (0.76)

= 0.658m

l_y = l_oy

= sin30°

= 0.5m

Length of stick in the moving frame-

= $\sqrt{0.5^2+0.658^2}$

= 0.8264 m

The 1m long stick is contracted to 0.8264m in the moving frame

Now,

Angle stick make with x-axis-

$\theta$ = tan^-1($\frac{0.5}{0.658}$ )

= 37.23^°