An astro-explorer finds a planet with the same radius as the Earth. After landing, she does some experiments and finds that the acceleration due to gravity on this new planet is [tex]g_x = 8.08 \, m/s^2[/tex]. What is the mass of this new planet? Please give your answer in units of Earth masses (i.e., [tex]M = f R_{\text{Earth}}[/tex], where [tex]f[/tex] is just a pure number).

[tex]\square[/tex]



Answer :

Sure! Let's go through the solution step-by-step to find the mass of the new planet in units of Earth masses.

1. Given values:
- Gravitational constant, \( G = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)
- Radius of the Earth, \( R_{\text{Earth}} = 6.371 \times 10^6 \, \text{meters} \)
- Acceleration due to gravity on Earth, \( g_{\text{Earth}} = 9.81 \, \text{m/s}^2 \)
- Acceleration due to gravity on the new planet, \( g_x = 8.08 \, \text{m/s}^2 \)

2. Formula for gravitational acceleration:
The gravitational acceleration \( g \) at the surface of a planet is given by:
[tex]\[ g = \frac{G M}{R^2} \][/tex]
where \( M \) is the mass of the planet and \( R \) is the radius of the planet.

3. Calculate the mass of the new planet:
Rearrange the formula to solve for the mass \( M \):
[tex]\[ M = \frac{g R^2}{G} \][/tex]
For the new planet:
[tex]\[ M_{\text{new planet}} = \frac{g_x R_{\text{Earth}}^2}{G} \][/tex]

4. Calculate the mass of the Earth:
Using the same formula for Earth:
[tex]\[ M_{\text{Earth}} = \frac{g_{\text{Earth}} R_{\text{Earth}}^2}{G} \][/tex]

5. Calculate the mass of the new planet in units of Earth masses:
The mass of the new planet as a fraction of the Earth’s mass (denoted as \( f \)) is:
[tex]\[ f = \frac{M_{\text{new planet}}}{M_{\text{Earth}}} \][/tex]

6. Plug in the values and solve step-by-step:
[tex]\[ M_{\text{new planet}} = \frac{8.08 \times (6.371 \times 10^6)^2}{6.67430 \times 10^{-11}} \][/tex]

[tex]\[ M_{\text{Earth}} = \frac{9.81 \times (6.371 \times 10^6)^2}{6.67430 \times 10^{-11}} \][/tex]

Then:
[tex]\[ f = \frac{M_{\text{new planet}}}{M_{\text{Earth}}} \][/tex]

7. Result:
After performing these calculations, we find:
- Mass of the new planet: \( M_{\text{new planet}} \approx 4.91383814452452 \times 10^{24} \, \text{kg} \)
- Mass of the Earth: \( M_{\text{Earth}} \approx 5.965934677943755 \times 10^{24} \, \text{kg} \)
- Fraction of Earth’s mass: \( f \approx 0.8236493374108053 \)

Therefore, the mass of the new planet is approximately [tex]\( 0.824 \)[/tex] times the mass of the Earth.