Question

An experimenter measures the length of a rod. Initially the experimenter and the rod are at rest with respect to the lab. Consider the following statements.
(A) If the rod starts moving parallel to its length but the observer stays at rest, the measured length will be reduced.
(B) If the rod stays at rest but the observer starts moving parallel to the measured length of the rod, the length will be reduced.

(a) A is true but B is false
(b) B is true but A is false
(c) Both A and B are true
(d) Both A and B are false

Answer 1

(c) Both A and B are true.

If a rod is moving with speed v parallel to its length l​_o and the observer is at rest, its new length will be given as
$l=l_o\sqrt{1-\frac{v^2}{c^2}}$
$$\begin{gathered} \mathrm{As},v<c \\ \therefore \frac{v^2}{c^2}<1 \\ \Rightarrow \sqrt{1-\frac{v^2}{c^2}}<1 \\ \mathrm{Therefore},l<l_o \end{gathered}$$
If the rod is at rest and the observer is moving with speed v parallel to measured length of the rod, the rod's length will be given as
$$\begin{gathered} l=l_o\sqrt{1-\frac{(-v{)}^2}{c^2}} \\ \Rightarrow l=l_o\sqrt{1-\frac{v^2}{c^2}} \\ \mathrm{As},v<c \\ \therefore {\left(\frac{-v}{c}\right)}^2<1 \\ \Rightarrow \sqrt{1-\frac{v^2}{c^2}}<1 \\ \Rightarrow l<l_o \end{gathered}$$
Therefore, the length will be reduced in both the cases.