Question

A cube has a density of $2000kg/m^3$while at rest in the laboratory. What is the cube’s density as measured by an experimenter in the laboratory as the cube moves through the laboratory at $90\%$of the speed of light in a direction perpendicular to one of its faces?

Short answer

$\rho =4600kg/m^3$.

Step-by-step solution

Given Information

We have given that, a cube has a density of $2000kg/m^3$.

The cube moves through the laboratory at $90\%$of the speed of light in a direction perpendicular to one of its faces.

Simplify

At rest, the cube has a density of $2000kg/m^3$. That is, a cube with dimensions of $lm\times lm\times lm$has a mass of $2000kg$. The cube's dimension in the direction of motion is contracted as the cube moves at a speed of $0.9c$:

localid="1650133944245" $$\begin{gathered} L\text{'}=\sqrt{1-{\left(\frac{v}{c}\right)}^2}1 \\ L\text{'}=\sqrt{1-\frac{{(0.9·c)}^2}{c^2}}1 \\ L\text{'}=\sqrt{1-0.81·}\left(1m\right) \\ L\text{'}=0.436m \end{gathered}$$

Because the other dimensions are unaffected, the new density is:

localid="1650133299285" $$\begin{gathered} \rho =\frac{2000kg}{1m\times 1m\times 0.436m} \\ \rho =4600kg/m^3 \end{gathered}$$

Therefore, Cubes density is$\rho =4600kg/m^3$.