Gravitational Potential Energy questions near the surface of a planet

In summary: I just don't know how to show this in physics.In summary, h is very small compared to radius and the equation for change in potential energy near the surface is approximated by GMmh/R. The limit as h approaches infinity is found by multiplying the numerator and denominator by ##\displaystyle \ \frac 1 h \,.##
  • #1
shk
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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.
 
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  • #2
shk said:
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"?
 
  • #3
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
 
  • #4
shk said:
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?
 
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  • #5
gneill said:
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
 
  • #6
Okay! You've done well.
 
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  • #7
gneill said:
Okay! You've done well.
thanks for checking my working . It helped a lot
 
  • #8
shk said:

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 \,. ##
 
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  • #9
SammyS said:
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?
 

Related to Gravitational Potential Energy questions near the surface of a planet

1. What is gravitational potential energy near the surface of a planet?

Gravitational potential energy near the surface of a planet is the energy that an object possesses due to its position in a gravitational field. It is the potential for an object to move towards the center of the planet under the influence of gravity.

2. How is gravitational potential energy calculated near the surface of a planet?

The formula for calculating gravitational potential energy near the surface of a planet is PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above the surface of the planet.

3. How does gravitational potential energy change as an object moves closer or further from the surface of a planet?

As an object moves closer to the surface of a planet, its gravitational potential energy decreases. This is because the object is moving towards the center of the planet, where the gravitational force is strongest. Conversely, as an object moves further from the surface of a planet, its gravitational potential energy increases because the object is moving away from the center of the planet, where the gravitational force is weaker.

4. What is the relationship between gravitational potential energy and mass?

The gravitational potential energy near the surface of a planet is directly proportional to the mass of the object. This means that as the mass of the object increases, its gravitational potential energy also increases. This can be seen in the formula PE = mgh, where m is the mass of the object.

5. How does the strength of gravity affect gravitational potential energy near the surface of a planet?

The strength of gravity directly affects gravitational potential energy near the surface of a planet. The stronger the gravitational force, the higher the potential energy of an object near the surface of the planet. This is because a stronger gravitational force means that the object would require more energy to move away from the surface of the planet.

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