Question

A rod with a proper length of 3 m moves along x-axis, making an angle of 30° with respect to the x-axis. If its speed is $\frac{c}2m/s$, where c is the speed of light, the change in length due to Lorentz contraction is _________ m (Round off to 2 decimal places). [Use $c = 3 x 10^8 m/s$]

Answer 1

Correct Answer: 0.29

The problem requires calculating the Lorentz contraction of a rod moving at $\frac{c}{2}$ m/s, where $c=3 \times 10^8$ m/s. The rod's proper length is 3 m, and its angle with the x-axis is 30°.

To calculate Lorentz contraction, use the formula for the length contraction:
$$ L = L_0\sqrt{1-\frac{v^2}{c^2}} $$
where \( L_0 \) is the proper length and \( v \) is the velocity.

First, find the component of the velocity along the rod's axis. Given the speed is \( \frac{c}{2} \) and the angle is 30°, only the parallel component affects contraction:
$$ v_{\parallel} = v\cos(30°) = \frac{c}{2}\cos(30°) = \frac{c\sqrt{3}}{4} $$

Plug this component into the length contraction formula:
$$ L = 3\sqrt{1-\left(\frac{(\frac{c\sqrt{3}}{4})^2}{c^2}\right)} = 3\sqrt{1-\frac{3c^2}{16c^2}} = 3\sqrt{1-\frac{3}{16}} $$

Simplifying further:
$$ 3\sqrt{\frac{16-3}{16}} = 3\sqrt{\frac{13}{16}} = 3 \times \frac{\sqrt{13}}{4} $$

Calculating \( \frac{\sqrt{13}}{4} \approx 0.901 \):
Thus, contracted length \( L \) is:
$$ 3 \times 0.901 \approx 2.703 \ m $$

The change in length due to contraction: \( 3 - 2.703 = 0.297 \ m \).