Question

Find the proper length of a rod if in the laboratory frame of reference its velocity is $v = \dfrac{c}{2}$ , the length $l = 1.00\,m$ , and the angle between the rod and its direction of motion is $\theta = {45^ \circ }$.

Answer 1

We can find the proper length of the rod by using the concept of length contraction of the body moving at an angle with respect to the X-axis which is the length of the body in motion that appears to be contracted. Also, the concept of rest frame and laboratory is used.

Formula used:
The contracted length of the rod by length contraction is given by
$l = {l_0}\sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} = \dfrac{{{l_0}}}{\alpha }$
Where, ${l_0}$ - proper length of the rod.
$\alpha = \dfrac{1}{{\sqrt {1 - {\beta ^2}} }}$ and $\beta = \dfrac{v}{c}$

Complete step by step answer:
Consider the rest frame in which the ends of the rod are expressed in the terms of the proper length ${l_0}$. $A(0,0)$ at origin, $B({l_0}\cos \theta ,{l_0}\cos \theta )$ at an angle $\theta = {45^ \circ }$ at time $t$. And in the laboratory frame which is moving with the velocity $v = \dfrac{c}{2}$ at time $t'$ are $A'(vt',0)$ and $B(l\cos \theta \sqrt {1 - {\beta ^2}} + vt',l\sin \theta )$ as the rod is moving along the X-axis, there is no length contraction along Y- axis. So, we can write,
${l_0}\cos \theta = l\cos \theta \sqrt {1 - {\beta ^2}} - - - - - (1)$ and
$\Rightarrow {l_0}\sin \theta = l\sin \theta - - - - - - - (2)$

Now, squaring and adding equations $(1)$ and $(2)$ , we get
$l_0^2 = {l^2}\left[ {{{\cos }^2}\theta (1 - {\beta ^2}) + {{\sin }^2}\theta } \right]$
Substituting the given values, $l = 1.00m$ and $\theta = {45^ \circ }$ , we get
$l_0^2 = {1^2}\left[ {{{\cos }^2}{{45}^ \circ }\left( {1 - {{\left( {\dfrac{v}{c}} \right)}^2}} \right) + {{\sin }^2}{{45}^ \circ }} \right]$
Also, $v = \dfrac{c}{2}$ , then
$\Rightarrow l_0^2 = \left[ {\dfrac{1}{2}\left( {1 - \dfrac{1}{4}} \right) + \dfrac{1}{2}} \right]$
$\Rightarrow l_0^2 = \left[ {\dfrac{3}{8} + \dfrac{1}{2}} \right]$
$\therefore {l_0} = 1.08m$

Hence, the proper length of the rod is ${l_0} = 1.08\,m$.

Note: Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame, always note that proper length is greater than the contracted length in moving frame.