Question

A rod is to move at constant speed $\mathit{v}$ along the $\mathit{x}$axis of reference frame $\mathit{S}$, with the rod’s length parallel to that axis. An observer in frame $\mathit{S}$is to measure the length $\mathit{L}$of the rod. Figure 37-23 given length $\mathit{L}$versus speed parameter $\mathit{\beta }$for a range of values for $\mathit{\beta }$. The vertical axis scale is set by ${\mathit{L}}_{\mathbf{a}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{00}\mathit{m}$. What is $\mathit{L}$ if $\mathit{v}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{95}\mathit{c}$?

Short answer

The value of L is 0.25 .

Step-by-step solution

The relation between lengths in different frames

The formula to find the length of the object in a reference frame is given by

$L=L_0\sqrt{1-{\left(\frac{v}{c}\right)}^2}$.

The length of the rod

From the graph, it is clear that when $\beta =0$,$L_0=0.8$ , and the value of $v$ is $0.95c$.

Substitute the known values and solve for $L$ as follows:

$$\begin{gathered} L=L_0\sqrt{1-{\left(\frac{v}{c}\right)}^2} \\ L=0.8\sqrt{1-{\left(\frac{0.95}{c}\right)}^2} \\ L=0.8\sqrt{1-{\left(0.95\right)}^2} \\ L=0.25 \end{gathered}$$

Thus, the value of L is 0.25m .