Satellite Orbiting Homework: Calculating Period of Planet X

[Peach]

Homework Statement

Planet X rotates in the same manner as the earth, around an axis through its north and south poles, and is perfectly spherical. An astronaut who weighs w on the Earth weighs w_1 at the north pole of Planet X and only w_2 at its equator. The distance from the north pole to the equator is L, measured along the surface of Planet X.

If a m satellite is placed in a circular orbit h above the surface of Planet X, what will be its orbital period?

Homework Equations

T = 2(pi)(r)/v

The Attempt at a Solution

I found the first part of the soln, which is the period of the planet. But I'm stuck on the second question because there's so many unknowns. I know angular velocity has to be the same but I got stuck with the calculation by using omega = v/r. Is this even correct though?

I know there's probably a simple soln but I'm so stuck, I can't think straight.

R' = R + h is the only thing I'm sure about...so pls help, I'm going crazy over this problem.

Many thanks in advance.

 

[HallsofIvy]

You know that a person who weighs w₁ at the north pole weighs w_2 at the equator. The difference, w₁- w₂ is due to the "centrifugal force" of the rotation of the planet so you can figure out the period as a function of w₁- w₂.

 

[Peach]

Thanks for helping. I found that already, but I'm stuck on finding the satellite at height h above planet X. I can't use v = (GM/R)^(1/2) because I don't have the mass of the planet and I'm not sure about using omega = v/r because the answer didn't make sense. The angular velocity must be the same, that much I got.

 

[morbidpotato]

I've been trying to solve for the period, but I've been getting a case sensitive error when using T=(2piL^(3/2))/(sqroot(G(w/g)). Am I miscalculating something?

 

[Peach]

For the satellite or the planet..?

 

[cycam]

I am having trouble with the first portion of the problem too. Do you think you could lead me in the right direction, Peach? Are you supposed to change L into some form of r, or is L = r?

I'm not quite sure what to do. Any help would be great. Thanks in advance.

 

[morbidpotato]

I thought L= r So far I've been using the T= 2(pi)(r)^(3/2) all over the square root of Gm-sub-e. I don't know what Gm-sub-e is so I tried solving for it using the weight equation but that isn't working. What am I missing?

 

[Peach]

Yeah, you have to convert L to the radius, because L is not the radius, it's only the distance from the north pole to the equator. :)

GM_e? M_e is the mass of the earth...which doesn't apply here because you're not calculating the mass of earth. Look up apparent weight in the text guys, there's an equation and explanation you can use there.

 

[cycam]

Okay, that is what I was stuck on; L. I know L is measured on the surface of the planet, like arclength, and s = rtheta. But I don't know if I can get the r from that since we don't have theta?

for the apparent weight, do we use w=Gm_Em/r^2 ?

 

[Peach]

Have you drawn a diagram yet? If you did, then you can see what theta is. The equator is around half of the sphere and the north pole is at the top. Radius is from the center to the top so...

No, because you're not on earth. You're on planet X. Remember that mass stays the same, no matter which planet you're on. The only thing that changed is g. You have the weight on earth, weight at the north pole, weight at the equator. Use weight on Earth to find your mass and then find the new g on planet X.

Big hint: It's all on page 460 of the text. Also, download the slide from mar. 9th, slide number 4.

 

[cycam]

OH sorry! Thanks for the info; I solved part A already.

 

[Peach]

Well, when you solve part B, can you give me a hint? I have an idea about it, but since I only have one try left, I don't want to get it wrong. :x

 

[cycam]

Oh man! I know, I've been stuck on this problem for hours! I think it is lame. I'll give you a heads up if I get anything.

 

[denverdoc]

anyone else wondering why we have such an interest in this problem?

 

[Peach]

Thanks. I have an answer already, but again, not sure if it's correct.

I don't know about anyone else, but I've spent more than 6 hours on this problem already so I'm very interested. :x

 

[morbidpotato]

Thank you Peach, I was able to solve part A, still having trouble with part B though :(

 

[denverdoc]

well I kno for one you have crossed boundaries by PMing asking for solns. I bet I'm not the only one. This is an inappropriate use of the forum.

 

[Peach]

morbid - I am too, that's why I'm waiting for someone to help us with this problem. So far, I have two eqns. T and g_planet.

T = 2(pi)(r^(3/2)) / sqr(Gm_planet)
g_planet = (Gm_planet) / (r_p)^2
g_planet = from part A

Solving for that, I get the mass of the planet and with that, I plugged it into the period eqn with the new radius. I'm not sure if this is entirely correct, maybe I misused something. Someone pls help?

denver - Okay.

 

[Bowser]

Your T equation is right, but to find the mass of the planet you use the equation:

w_1 = (G*m_p*m_astronaut)/(r_p)^2

You find the m_p and then plug that into the T equation, and that's it!

 

[Peach]

Bower: Somehow I got the wrong answer from that, maybe I entered it wrong. It's too late and I'm too tired from this problem so that's probably the case. Anyway, thanks for helping, appreciate it much. :)