# Specify the reference state for potential energy?

## Homework Statement

A spacecraft is in circular orbit around the earth. with respect to a point below it on earth, it is travelling towards the east with speed v.

Specify the reference state for potential energy.

## The Attempt at a Solution

the total energy is

E=K+U

= 1/2mv2-GmM/r

but the reference state for potential energy i don't understand?

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Reference state/level is the one relative to which you measure the potential energy. So in this case it would be _______?

Reference state/level is the one relative to which you measure the potential energy. So in this case it would be _______?
well potential energy is really the distance from the centre of the earth to the space ship, r in this case.

so the reference level is on earth which is r- the distance from the spaceship to surface of the earth?

well potential energy is really the distance from the centre of the earth to the space ship, r in this case.

so the reference level is on earth which is r- the distance from the spaceship to surface of the earth?
Um, no. Let me be more clear. The reference state, in other words, is where the potential energy is taken to be 0. So, where is the potential energy 0?

Um, no. Let me be more clear. The reference state, in other words, is where the potential energy is taken to be 0. So, where is the potential energy 0?
at the centre of the earth assuming that all the mass in the earth is evenly distributed.?

at the centre of the earth assuming that all the mass in the earth is evenly distributed.?
True. The potential is zero there. But are you calculating potential with respect to the center?

The negative of work done against the gravitational force in bringing a mass in, from infinity, to a given point in the gravitational field, is called the gravitational potential energy. Does this give you an idea?

Doc Al
Mentor
at the centre of the earth assuming that all the mass in the earth is evenly distributed.?
Not if you're using this formula for PE:
the total energy is

E=K+U

= 1/2mv2-GmM/r
Using that expression for PE, where is the zero point?

Not if you're using this formula for PE:

Using that expression for PE, where is the zero point?
r→∞

but it's seems strange to me. does that mean that all matter can never have 0 potential energy?

r→∞

but it's seems strange to me. does that mean that all matter can never have 0 potential energy?
Yep.

Visibly, as r→∞, the body will have 0 potential energy. But keep in mind that potential energy is defined relative to a reference state, which usually is infinity, and this is crucial to derive the formula that you were using, i.e $-GMm/r.$

This doesn't stop you from assuming the reference state is the ground level, in which case potential energy at ground is 0, and at a height h, it is $\int F dh$, which for small heights comes out as $mgh$

Doc Al
Mentor
r→∞
Right. Using that formula implies that you are taking PE = 0 as r → ∞.
but it's seems strange to me. does that mean that all matter can never have 0 potential energy?
The place you choose to call PE = 0 is arbitrary. (But using that formula is very convenient for some purposes.) What physically matters is the change in PE as you go from one point to another, which is independent of the arbitrary PE = 0 reference point.

Right. Using that formula implies that you are taking PE = 0 as r → ∞.

The place you choose to call PE = 0 is arbitrary. (But using that formula is very convenient for some purposes.) What physically matters is the change in PE as you go from one point to another, which is independent of the arbitrary PE = 0 reference point.
ok, but doesn't P.E only →0 since r can only ever→∞

from PE=-GmM/r

PE→0 as r→∞

Since r can never = ∞

how can it be that PE can =0

??

Doc Al
Mentor
ok, but doesn't P.E only →0 since r can only ever→∞

from PE=-GmM/r

PE→0 as r→∞

Since r can never = ∞

how can it be that PE can =0

??
Right. The PE never quite equals zero for finite r. PE goes to zero as r goes to infinity. But it's perfectly OK as a reference point.

Right. The PE never quite equals zero for finite r. PE goes to zero as r goes to infinity. But it's perfectly OK as a reference point.
Ok thanks, but if the earth is perfectly spherical with the mass evenly distributed, what would be the PE of a particle in the centre of the earth? I thought it should be zero, but are you saying that it does have a discrete amount of PE?

Doc Al
Mentor
Ok thanks, but if the earth is perfectly spherical with the mass evenly distributed, what would be the PE of a particle in the centre of the earth? I thought it should be zero, but are you saying that it does have a discrete amount of PE?
On what basis would you say that a particle at the center of the earth would have zero PE? What are you using as your reference point? Just because the gravitational force is zero doesn't mean that the PE is zero.

On what basis would you say that a particle at the center of the earth would have zero PE? What are you using as your reference point? Just because the gravitational force is zero doesn't mean that the PE is zero.
the centre of the earth. using -GmM/r when r→0 i.e. as a particle gets closer to the centre of the earth. what does this mean its PE→?
is it undefined...or should PE→∞..for a point particle can it not be exactly in the centre of the earth so that PE≠-GmM/0..ie PE≠∞?

the centre of the earth. using -GmM/r when r→0 i.e. as a particle gets closer to the centre of the earth. what does this mean its PE→?
is it undefined...or should PE→∞..for a point particle can it not be exactly in the centre of the earth so that PE≠-GmM/0..ie PE≠∞?
But the formula (-GMm/r) changes as the particle crosses the ground level, PE will not be infinite. It changes according to a corollary of the shell theorem, which gives that the gravitational force varies linearly from the earth's surface to its center, and becomes zero at the center. Hence, the potential energy will also vary in accordance to this rule.

Doc Al
Mentor
the centre of the earth. using -GmM/r when r→0 i.e. as a particle gets closer to the centre of the earth. what does this mean its PE→?
is it undefined...or should PE→∞..for a point particle can it not be exactly in the centre of the earth so that PE≠-GmM/0..ie PE≠∞?
You cannot apply the formula PE = -GmM/r for points within the surface of the earth. Instead, think in general principles: How much work against gravity would be required to raise a particle from the center of the earth to an arbitrary point?

You cannot apply the formula PE = -GmM/r for points within the surface of the earth. Instead, think in general principles: How much work against gravity would be required to raise a particle from the center of the earth to an arbitrary point?
Oh right, of course. Thanks Doc al and Infinitum for the explenations.