A cube with edge a = 10 cm is made of a metal of density rho…

A cube with edge $a = 10\ \mathrm{cm}$ is made of a metal of density $\rho = 8600\ \mathrm{kg/m^3}$. Find (a) its mass on Earth and on the Moon, and (b) its weight on the surface of Earth and of the Moon. ($g_E = 9.81\ \mathrm{m/s^2}$, $g_M = 1.62\ \mathrm{m/s^2}$.)

Explanation of Method

A metal cube on Earth and on the Moon.

Step 1: (a) Mass

\[ V = a^3 = 0.1^3 = 1.0 \cdot 10^{-3}\ \mathrm{m^3}, \]

\[ m = \rho V = 8600 \cdot 10^{-3}. \]

\[m = 8.6\ \mathrm{kg} \ \text{on Earth and on the Moon}\]

Mass measures the amount of matter (and its inertia), so it does not depend on where the body is.

Step 2: (b) Weight

\[ W_E = m g_E = 8.6 \cdot 9.81, \]

\[ W_M = m g_M = 8.6 \cdot 1.62. \]

\[W_E \approx 84.4\ \mathrm{N}, \qquad W_M \approx 13.9\ \mathrm{N}\]

$\text{Earth},\ g = 9.81\ \mathrm{m/s^2}$ $8.6\ \mathrm{kg}$ $W = 84.4\ \mathrm{N}$ $\text{Moon},\ g = 1.62\ \mathrm{m/s^2}$ $8.6\ \mathrm{kg}$ $W = 13.9\ \mathrm{N}$

The same cube (same mass, 8.6 kg) on Earth and on the Moon. The weight arrows are to scale: on the Moon the cube presses on its support about six times less.

Result

\[\boxed{\,m = 8.6\ \mathrm{kg} \ \text{on Earth and on the Moon}\,}\]

\[\boxed{\,W_E \approx 84.4\ \mathrm{N}, \qquad W_M \approx 13.9\ \mathrm{N}\,}\]

Remark

Check. The original results are correct. The ratio of the weights, $84.4/13.9 \approx 6.1$, equals $g_E/g_M$, as it must for the same mass.