Calculating mass of a planet (Law of Universal Gravitation)

[Humbleness]

Homework Statement

You are on a deep space mission to search for Earth-like planets. Your crew locates a possible planet and with scanners finds the radius to be 7.5 x 10⁶ m. A team lands on the surface. There, they hang a 1.0 kg mass from a spring scale. It reads 8.5 N. Determine the mass of the planet and whether an astronaut standing on this new planet weighs more, less, or the same as on Earth. Show your work.

Homework Equations

F = G M₁M₂/r²
M₂ = F (r²) / Gm₁
F = ma
F = mg

The Attempt at a Solution

First using the equation to find the mass of this new Earth-like planet:
M₂ = 8.5 N (7.5 x 10⁶)² / (6.67 x 10^-11) (1.0 kg)
M₂ = 8.5 N (56.25 x 10¹²) / (6.67 x 10^-11) (1.0 kg)
M₂ = 7.168 x 10²⁴

Then the gravitational field strength on this new planet:
a = G x M₂ / r²
a = (6.67 x 10^-11) (7.168 x 10²⁴) / (56.25 x 10¹²)
a = 47.81 x 10¹³ / 56.25 x 10¹² = 8.5m/s²

Now let's suppose that the astronaut's mass is 50 kg. To calculate how much he weighs on this new planet:
F = ma = (50 kg) x (8.5m/s²) = 425 N

Now calculating his weight on Earth:
F = mg = (50 kg) x (9.8m/s²) = 490 N

Therefore the astronaut weighs more on planet Earth, than he does on this newly-found planet.

Did I do everything correctly? I really appreciate confirmations and guidance on pointing me in the right direction if I made mistakes anywhere. Thank you in advance to anyone who helps and/or confirms.

 

[gneill]

You did fine, but be sure to always include units when you present a result. When you calculated the mass of the planet, M₂, you didn't show the units.

After you calculated the acceleration due to gravity on the planet's surface, it would have sufficed to compare that value to the acceleration due to gravity on Earth and noting that it is smaller on the planet, hence the astronaut would weigh less there.

 

[Humbleness]

You did fine, but be sure to always include units when you present a result. When you calculated the mass of the planet, M₂, you didn't show the units.

After you calculated the acceleration due to gravity on the planet's surface, it would have sufficed to compare that value to the acceleration due to gravity on Earth and noting that it is smaller on the planet, hence the astronaut would weigh less there.

Understood. Thank you so much for the confirmation, and yes I agree, I could've simply made it easier for myself at the acceleration point, I have not thought about it though :)