Gravitational Potential Energy questions near the surface of a planet

[shk]

Homework Statement

The change in gravitational potential energy of a mass m as it moves from the surface to a height h above the surface of a planet of mass M and radius R is given by:

ΔPE= GMmh/R(R+h)

a) show that when h is very small compared to R , this approximates to the more familiar expression for the change in gravitational potential energy near the surface.

b) Potential energy increases as h increases. To what value does it tend as h approaches infinity?

c) Explain the difference between potential energy and potential.

d) sketch a graph of the variation in gravitational potential with height near the surface of the planet . What is the significant of the slope of this graph?

Homework Equations

v=-GM/r
g=Gm/r^2
Ep=-GMm/r

The Attempt at a Solution

i think for part a when h is very small in compared with R, the equation would be the same equation we have for g. g=Gm/r^2. but still not sure and don't even know how to show this. I can do it Mathematically but not through physics. plus I'm not even sure if this is correct.

 

[gneill]

i think for part a when h is very small in compared with R, the equation would be the same equation we have for g. g=Gm/r^2. but still not sure and don't even know how to show this. I can do it Mathematically but not through physics.

Show us your math. The physics is described by the math.

What is the "more familiar expression for the change in gravitational potential energy near the surface"?

 

[shk]

Thanks for the reply.
I think
If h is very small then R+h≈R
Therefore the equation will be GMmh/R^2 and as g=GM/R^2 , the equation changes to mgh which is the change in gravitational potential energy near the surface. But I'm not sure about all these. And I'm not even sure about the answer to the other parts of the question

 

[gneill]

Thanks for the reply.
I think
If h is very small then R+h≈R
Therefore the equation will be GMmh/R^2 and as g=GM/R^2 , the equation changes to mgh which is the change in gravitational potential energy near the surface. But I'm not sure about all these. And I'm not even sure about the answer to the other parts of the question

Well, your math looks good, so that's part (a) done.

What have you tried for the other parts?

 

[shk]

Well, your math looks good, so that's part (a) done.

What have you tried for the other parts?

I think:
for part b)
Potential energy increases as h increases because it is inversely proportional to h as Ep=-GMm/r . So it tends to zero as h approaches infinity.

for part c)
I think potential energy depends on the the mass of the object that is causing the gravitational field and the object which is in the field but potential only depends on the mass of the object that is causing the gravitational field.
in summary: The gravitational potential is the potential energy per unit mas.

and for part d)
it's an decreasing exponential curve which shows the gravitational potential(V) decreases as the height increases. So V=0 is the asymptote.
The significant is g which I can get from the differentiating the V

 

[gneill]

Okay! You've done well.

 

[shk]

Okay! You've done well.

thanks for checking my working . It helped a lot

 

[SammyS]

Homework Statement

The change in gravitational potential energy of a mass m as it moves from the surface to a height h above the surface of a planet of mass M and radius R is given by:

ΔPE= GMmh/R(R+h)

a) show that when h is very small compared to R , this approximates to the more familiar expression for the change in gravitational potential energy near the surface.

b) Potential energy increases as h increases. To what value does it tend as h approaches infinity?

It looks to me like part b) above is an extension of part a). Otherwise it makes little sense to refer to 'h' .
So you should use $\displaystyle \ \Delta PE = \frac{GMmh}{R(R+h)} \,.\$
(By the way, when you write this expression on a single line, the entire denominator needs to be enclosed in parentheses.) :ΔPE= GMmh/(R(R+h)) .

To find the limit as h → ∞ , multiply the numerator & denominator by $\displaystyle \ \frac 1 h \,.$

 

[shk]

It looks to me like part b) above is an extension of part a). Otherwise it makes little sense to refer to 'h' .
So you should use $\displaystyle \ \Delta PE = \frac{GMmh}{R(R+h)} \,.\$
(By the way, when you write this expression on a single line, the entire denominator needs to be enclosed in parentheses.) :ΔPE= GMmh/(R(R+h)) .

To find the limit as h → ∞ , multiply the numerator & denominator by $\displaystyle \ \frac 1 h \,.$

Dear Sammys
Thanks for reminding about the parentheses. t b I should say that I have actually used part a as I had derived the equation of Ep=-GMm/r from part a. what do you think now?