Why is the gradient constant in Figure 2?

[nokia8650]

Can someone please explain the answer to the question shown below:

http://img152.imageshack.us/img152/1743/21985820pa0.jpg


http://img61.imageshack.us/img61/2455/41357835ph4.th.jpg

I know that V= - GM/r , and that allows me to sketh the answer for figure 1. Why, however, is it a constant gradient for figure 2?

Thanks

 

[turin]

I know that V= - GM/r ...

This is your problem. While, strictly speaking, this is true outside of a perfect spherical mass distribution, like a planet, the problem is basically telling you not to use this for Figure 2. Do you know how to get the potential from the field strength?

If you zoom into the curve for Figure 1 so that the percent different between r_min and r_max as very small, what does that part of the curve look like?

 

[nokia8650]

Ah right, so figure two is esentially the case over a small distance. Is this due to the equation "E=-dv/dr"? Why is this not the case for figure one then, for the larger distance?

Thanks a lot!

 

[turin]

... so figure two is esentially the case over a small distance.

Yes.

Is this due to the equation "E=-dv/dr"?

No.

... "E=-dv/dr"? Why is this not the case for figure one then, for the larger distance?

It IS the case for both figures. It is just that, in Figure 1, r is interpretted as a radial distance, whereas in Figure 2, r is interpretted as a height. You can obtain V from this equation by integration. BTW, I'm assuming that you realize that this is not an electrostatics problem, but that the similarities are such that you can use the same mathematical construction. So, in your notation, E is the force per test mass, and V is the energy per test mass.

 

[nokia8650]

Thank you very much, yes, sorry, I meant "g" not "E". I am struggling to understand what the difference between radial distance and height, is it just that height is over a very small distance?

Thanks

 

[turin]

I am struggling to understand what the difference between radial distance and height, is it just that height is over a very small distance?

It is just an (largely irrelevant) interpretation. It is more phenomenological than physical. For example, if you climb to the next floor of a building, do you imagine that your radial distance from the center of the Earth has increased by 4 meters or that your height above the surface of the Earth has increased by 4 meters. These two interpretations are equivalent; however, the height interpretation is usually more convenient when you are talking about such situations. Conversely, it sounds kind of silly (to me) to say that Mars is 200 million km high today. Typically, we speak of radial distance when the change in distance is on the order of, or larger than, the size of the gravitational bodies, and we speak of height when the change in distance is much smaller than the size of the large gravitational body.

 

[nokia8650]

That makes sense, thank you ever so much, I really appreciate your help!