Planets and satellites, law of periods circular orbit.

[J-dizzal]

Homework Statement

A 15 kg satellite has a circular orbit with a period of 4.4 h and a radius of 3.5 × 106 m around a planet of unknown mass. If the magnitude of the gravitational acceleration on the surface of the planet is 1.2 m/s2, what is the radius of the planet?

Homework Equations

The Attempt at a Solution

I think I am on the right track, but I am stuck finding the radius of the planet. in my equation at the bottom for radius what would i use for M and m?

 

[Nathanael]

In your equation $r=\sqrt{\frac{GMm}{F_g}}$ r is the distance between the two objects, not the radius of the planet.

Try to use the centripetal acceleration to find the mass of the planet (only 1 mass will cause this orbit) and then try to find the radius (with the mass known, only 1 radius will cause that value of surface gravity).

 

[TSny]

Considering Newton's 2nd law, what does F_g /m represent?

 

[J-dizzal]

In your equation r=GMmFg−−−−√r=\sqrt{\frac{GMm}{F_g}} r is the distance between the two objects, not the radius of the planet. That equation is really of no use.

would it work if M is the mass of the planet and m is a mass on the surface of the planet? the distance between them would be about the radius of the planet?

Considering Newton's 2nd law, what does Fg /m represent?

acceleration. But in my equation for r; F_g is the force of acceleration on the surface of the planet. would that work with the given a=1.2m/s²

 

[Nathanael]

would it work if M is the mass of the planet and m is a mass on the surface of the planet? the distance between them would be about the radius of the planet?

Oh my bad, yes then it would work. You just need to find the mass of the planet first.

 

[J-dizzal]

Oh my bad, yes then it would work. You just need to find the mass of the planet first.

Does my equation for M look ok, toward the top of my page M=946496.4kg?
If its ok, then would any value for m work in my equation for r?

 

[Nathanael]

Does my equation for M look ok, toward the top of my page M=946496.4kg?
If its ok, then would any value for m work in my equation for r?

Sorry for ignoring your initial work, apparently I'm not so good at skimming pictures

The equation is right but you didn't cube the radius when you solved it.

 

[J-dizzal]

Sorry for ignoring your initial work, apparently I'm not so good at skimming pictures

The equation is right but you didn't cube the radius when you solved it.

In my equation for r, I am letting F_g=a. where a=1.2m/s/s is a given in the problem statement. I am guessing this is wrong because its the force on the satellite at distance r from the surface.

 

[Nathanael]

In my equation for r, I am letting F_g=a. where a=1.2m/s/s is a given in the problem statement. I am guessing this is wrong because its the force on the satellite at distance r from the surface.

a is 1.2 m/s/s, not Newtons. Consider an object on the surface of mass m. If it accelerates at 1.2m/s/s, then what must the force on that object (of mass m) be?

 

[TSny]

acceleration. Yes, But in my equation for r; F_g is the force of acceleration on the surface of the planet. would that work with the given a=1.2m/s²

F_g = GMm/r² gives the force of gravity on m as a function of r. Using this, you can derive an expression for F_G /m as a function of r. Thus, you have an expression for the acceleration due to gravity as a function of r. Apply this for a point at the surface of the planet.

 

[J-dizzal]

a is 1.2 m/s/s, not Newtons. Consider an object on the surface of mass m. If it accelerates at 1.2m/s/s, then what must the force on that object (of mass m) be?

F=(15kg)(1.2m/s/s)=18N

Fg = GMm/r2 gives the force of gravity on m as a function of r. Using this, you can derive an expression for FG /m as a function of r. Thus, you have an expression for the acceleration due to gravity as a function of r. Apply this for a point at the surface of the planet.

F_g/M = Gm/r² so this would be the acceleration of M?

 

[TSny]

F_g/M = Gm/r² so this would be the acceleration of M?

Well, yes that would give the acceleration of the planet (M) due to the attraction of the satellite (m). But that would be extremely small and it is not of relevance to this problem.

However, F_g /m gives the acceleration of the satellite due to the attraction of the planet. This is what you would call the "acceleration due to gravity of the planet". You are given the acceleration due to gravity of the planet at the surface of the planet.

So you want to find an expression for F_g /m rather than F_g /M. Then apply the expression to a point near the surface of the planet.

 

[J-dizzal]

Well, yes that would give the acceleration of the planet (M) due to the attraction of the satellite (m). But that would be extremely small and it is not of relevance to this problem.

However, F_g /m gives the acceleration of the satellite due to the attraction of the planet. This is what you would call the "acceleration due to gravity of the planet". You are given the acceleration due to gravity of the planet at the surface of the planet.

So you want to find an expression for F_g /m rather than F_g /M. Then apply the expression to a point near the surface of the planet.

im getting R= sqrt(GM/a) =1.337x10⁶ same answer i got before...

 

[TSny]

R = sqrt(GM/a) is correct. What did you get for the mass of the planet, M, after correcting for cubing r as pointed out by Nathanael in post #7?

 

[J-dizzal]

R = sqrt(GM/a) is correct. What did you get for the mass of the planet, M, after correcting for cubing r as pointed out by Nathanael in post #7?

I got 3.219x10²²kg

 

[J-dizzal]

R = sqrt(GM/a) is correct. What did you get for the mass of the planet, M, after correcting for cubing r as pointed out by Nathanael in post #7?

ok i got it correct now thanks again.