The Relativity of Length — Question 4

Question 4

A moving box changes volume through one dimension only

A rectangular box has rest dimensions $a=3.0\,\mathrm m$ along x, $b=2.0\,\mathrm m$ along y and $d=1.0\,\mathrm m$ along z. It moves at $0.80c$ along x. Use $\gamma=(1-v^2/c^2)^{-1/2}$ and $c=3.00\times10^8\,\mathrm{m/s}$.

1. Find its simultaneous laboratory dimensions and volume.

2. Compare the area of a face normal to x with the area of a face parallel to the xy plane.

Question 4 - Solution

A moving box changes volume through one dimension only

(a) Method and calculation

$$a_{\mathrm{lab}}=a/\gamma=1.80\,\mathrm m,\quad b_{\mathrm{lab}}=b,\quad d_{\mathrm{lab}}=d.$$$$\boxed{V=abd/\gamma=3.60\,\mathrm m^3}.$$

(b) Interpretation and further calculation

$$\boxed{A_{yz}=bd=2.00\,\mathrm m^2}.$$$$\boxed{A_{xy}=ab/\gamma=3.60\,\mathrm m^2}.$$

Physical check

Only the dimension parallel to the boost contracts; applying the same factor to all three dimensions would be incorrect.