Calculating Freefall Acceleration on the Moon

[Ockonal]

Homework Statement

Mass of the Moon is nearly less in 100 times than Earth's, and the diametr is less in 4 times.
Find the acceleration of free falling at the Moon.

Homework Equations

F = G×((M×m)/R²)
g = 9.8N

The Attempt at a Solution

M_moon = M_earth / 100
D_moon = D_earth / 4
R_moon = D_moon / 2 = D_earth / 8

g_earth = G×(M_earth/R_earth²)
g_moon = G×(M_moon / R_moon²)
=>
M_earth = (g×R_earth²) / G
M_moon = (g×R_earth²) / G / 100
...
g_moon = g_earth / 25
But the answer is wrong. I think my mistake is in diameter passing and converting it into radius. Help me, please/

 

[D H]

Your mistake is that you dropped something. You were fine up to here:

g_earth = G×(M_earth/R_earth²)
g_moon = G×(M_moon / R_moon²)

Given that, what is g_moon / g_earth?

 

[Ockonal]

so:
g_moon / g_earth =

M_moon × R_earth²
-------------------
M_earth × R_moon²

Now I'm passing data instead of *_moon:

(M_earth/100) × R_earth²
----------------------
M_earth × (D_earth / 4)

How to make it easier?

 

[Ockonal]

so:

g_moon / g_earth =
 
M_moon × R_earth²
-------------------
M_earth × R_moon²

Now I'm passing data instead of *_moon:

(M_earth/100) × R_earth²
----------------------
M_earth × (D_earth / 4)

How to make it easier?

-----
I did this:

  M_earth × R_earth²
----------------------
          100
----------------------
M_earth × 2 × R_earth
----------------------
              8

=>
R_earth / 25

But R_earth shouldn't be presented.

 

[D H]

If the diameter of the Moon is roughly 1/4 of the diameter of the Earth, what can you say about the radius of the Moon in terms of the radius of the Earth?

 

[Ockonal]

If the diameter of the Moon is roughly 1/4 of the diameter of the Earth, what can you say about the radius of the Moon in terms of the radius of the Earth?

D_moon = 1/4 D_Earth
R_moon = (1/4 D_Earth)/2

I'm confused :(

 

[D H]

You just expressed the Moon's radius in terms of the Earth's diameter. I asked you to express the Moon's radius in terms of the Earth's radius. So, try again, please.

Once you have that figured out, go back to computing g_moon / g_earth.

 

[cupid.callin]

you know that g=Gm/r2

now for earth

g = GM/R²

for moon, g' = GM'/R'²

you know that M' = M/100 and D' = D/4

so R' = ?

 

[Ockonal]

You just expressed the Moon's radius in terms of the Earth's diameter. I asked you to express the Moon's radius in terms of the Earth's radius. So, try again, please.

Once you have that figured out, go back to computing g_moon / g_earth.

As I understand:
D/4 = Moon's diametr, so:
R/4 + R/4 = D/4
So radius is smaller in 4 times?

 

[cupid.callin]

yes!

now just substitute the values

 

[Ockonal]

now just substitute the values

Didn't understand right, maybe:

g_moon = G × ( M_earth / 25 × R_earth² )
g_earth = G × ( M_earth / R_earth² )

g_moon / g_earth = 
    M_eath × R_earth²
-----------------------   =
M_earth × 25 × R_earth²

1/25

 

[cupid.callin]

the ' ones are for moon


g = GM/R²

for moon, g' = GM'/R'²

you know that M' = M/100 and D' = D/4

so R' = R/4

g' = G(M/100)
      _______
        R/4

so ... g' = <something> GM/R[sup]2[/sup]
        g' = <something> g

 

[Ockonal]

the ' ones are for moon

g' = G(M/100)
      _______
        R/4

so ... g' = <something> GM/R[sup]2[/sup]
        g' = <something> g

Sorry, I can't understand from where did you get GM/R² in

g' = G(M/100)
      _______
        R/4

 

[cupid.callin]

sorry i can't give answer ,,, got 2 warnings for it

i just rearranged the eqn and dividing the numbers i can

separate GM/R² ... and as you know from eqn of Earth GM/R² = g

eqn of Earth ... sounds fascinating!