# Homework Help: Gravitional potential energy

1. May 9, 2010

### jwu

1. The problem statement, all variables and given/known data
A satellite of mass M is in a circular orbit of radius R around the earth.

(a) what is its total mechanical energy (where Ugrav is considered zero as R approaches infinity)?

(b) How much work wouldbe required to move the satellite into a new orbit, with radius 2R?

2. Relevant equations
(a)
mv²/R=GMm/R² →→ mv²=GMm/R →→ K=1/2mv²=GMm/(2R),
therefore, E=K+U=GMm/(2R)+(-GMm/R)=-GMm/(2R)

(b)
Here's where I got stuck :
This is the correct answer on the book:
From the equation Ki+Ui+W=Kf+Uf,
W=(Kf+Uf)-(Ki+Ui)
=Ef-Ei
=-GMm/(2(2R))-(-GMm/(2R))
=GMm/(4R)

Here's what I did, instead of using the equation above, Ki+Ui+W=Kf+Uf, I used the WORK-ENERGY THEOREM. But it came out the different answer.

W=Kf-Ki=GMm/(4R)-GMm/(2R)=-GMm/(4R) , the same magnitude but different sign.

What's wrong with using WORK-ENERGY THEOREM?

3. The attempt at a solution
As above.

2. May 9, 2010

### tiny-tim

Hi jwu!

Have I understood this correctly …

instead of using W = Kf - Ki + Uf - Ui,

you just used the "work-energy theorem", W = Kf - Ki ?​

ok, for the work-energy theorem, you have to include the work done by all the forces, and that includes the force of gravity, so you would have Wrocket + Wgravity = Kf - Ki … the same as the book's answer.

The only trick is that the book has replaced Wgravity by -PE.

You see, PE is just another name for (minus) work done by a conservative force (such as gravity) … you can either use work done, or you can use (minus) PE.

(btw, you have to be careful about what you regard as "energy" …

from the PF Library on potential energy …

Is potential energy energy?

There is confusion over whether "energy" includes "potential energy".

On the one hand, in the work-energy equation, potential energy is part of the work done.

On the other hand, in the conservation-of-energy equation (and conservation of course only applies to conservative forces), potential energy is part of the energy.)​

3. May 9, 2010

### bjd40@hotmail.com

think of it:
if the distance increases how the P.E., K.E., and T.E varies?

4. May 9, 2010

### jwu

So basically you mean the work-energy theorem and the conservation of energy equation are interchangable at some point?

5. May 9, 2010

### tiny-tim

For conserved forces (such as gravity), yes.

But for most applied forces (such as rockets, bits of string, etc), no … conservation of energy can't apply to them because, with them, energy isn't conserved.