Q16E - University Physics with Modern Physics

Question

An astronaut’s pack weighs 17.5 N when she is on the earth but only 3.24 N when she is at the surface of a moon.

(a) What is the acceleration due to gravity on this moon?

(b) What is the mass of the pack on this moon?

Step-by-Step Solution

Answer

(a) The acceleration due to gravity on the surface of the moon where the astronaut's pack weighs 3.24 N is $\text{1.81} \mathrm{m}/{\mathrm{s}}^2$.

(b) The mass of the astronaut's pack on the moon is 1.79 kg.

Step 1: Given data

The weight of the astronaut's pack on Earth is

${\text{W}}_{\text{E}}=17.5 \text{N}$

The weight of the astronaut's pack on the moon is

${\text{W}}_{\text{m}}=3.24 \text{N}$

The acceleration due to gravity on Earth is

$\text{g}=9.8\mathrm{m}/{\mathrm{s}}^2$

Step 2: Gravitational force

The downward gravitational force on a body of mass  $m$ is

${\mathbf{F}}_{\mathbf{g}}\mathbf{=}\mathbf{mg}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(1\right)$

Step 3: The acceleration due to gravity on the moon

From equation (1), the ratio of weights of the astronaut's pack on the moon and on Earth is

$$\begin{gathered} \frac{{\text{W}}_{\text{m}}}{{\text{W}}_{\text{E}}}=\frac{{\text{mg}}_{\text{m}}}{\text{mg}} \\ {\text{g}}_{\text{m}}=\frac{{\text{W}}_{\text{m}}}{{\text{W}}_{\text{E}}}\times \text{g} \\ {\text{g}}_{\text{m}}=\frac{3.24 \text{N}}{17.5 \text{N}}\times 9.8\mathrm{m}/{\mathrm{s}}^2 \\ {\text{g}}_{\text{m}}=1.81\mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$

Thus, the acceleration due to gravity on the moon is $1.81\mathrm{m}/{\mathrm{s}}^2$

Step 4: Mass of the astronaut's pack

From equation (1), the mass of the astronaut's pack is

$\begin{array}{rcl}\mathrm{m}& =& \frac{{\text{W}}_{\text{m}}}{{\text{g}}_{\text{m}}}\\ & =& \frac{3.24 \text{N}}{1.81\mathrm{m}/{\mathrm{s}}^2}\\ & =& 1.79·\left(\frac{1}{{\text{m/s}}^{\text{2}}}\right)·\left(1 \text{N}\times \frac{1 \text{kg}·{\text{m/s}}^2}{1 \text{N}}\right)\\ & =& 1.79 \text{kg}\\ & & \end{array}$

Thus, the mass of the pack is 1.79 kg.