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Author Topic: Gravitational Potential  (Read 1009 times)  Share 

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/0

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Gravitational Potential
« on: May 08, 2009, 02:38:53 am »
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Energy can be described as the area under an Force-distance graph... But what do you do when F is not constant?

a) Using calculus, derive an expression for the change in gravitational potential energy when is large.

b) Show that this is equivalent to for small .

It is by definition that the gravitational potential energy at infinity is zero, and it becomes more negative as you move closer to the attracting mass.

c) Find the escape velocity of Earth. (Minimum velocity required to escape earth's gravity and travel to infinity)

d) Find the total mechanical energy of a satellite in terms of m, the mass of the satellite, M, the mass of the planet it is orbiting, G, and R, the distance between the centre of masses of the satellite and the planet. ()

e) Hence, find the energy required to send a satellite of mass into geosynchrous orbit around the earth.

Data:
, ,
« Last Edit: May 08, 2009, 02:41:45 am by /0 »

kamil9876

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Re: Gravitational Potential
« Reply #1 on: May 08, 2009, 11:59:48 am »
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You know that work done is Hence if the force is not constant you have to split the interval up into many pieces. Basically, the work done by moving the object from 1m to 2m is equal to the work done by moving it from 1m to 1.1 plus work done from 1.1 to 1.2. etc. And for that interval of 0.1m u can assume the force is constant (the numbers are an unrealistic example, but i hope u get the idea). Basically you want to take the limit as the interval approaches zero, and you can see that this is analogous to finding an integral(which i know that you are aware of judging from a post u made when u presented that definition).

Consider this question as an example:

What is the work done when u stretch a spring from x=1 to x=2. (where x is the extension beyond natural length).

You know that initially the force is is , then it is then in the next interval and finally on the last intervral it is.

Hence to get the total work we need to multiply each force by each distance, (i have factorised the distance and put at the end because I am lazy and forgot to put it in earlier):



Which you can solve using the formula for an arithmetic sequence, or just recognize that this is the integral:



(if lower terminal was 0 you can see that it is simply the formula for elastic potential energy).

This should give you a clue as to how to approach the gravity problem.

This may be beyond the course, but I think it's a pretty good way to deepen your understanding of integration and appreciate how Newton saw it as necessary to calculate stuff.
« Last Edit: May 08, 2009, 12:06:24 pm by kamil9876 »
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

kamil9876

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Re: Gravitational Potential
« Reply #2 on: May 08, 2009, 04:33:50 pm »
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lol ok so after seeing the replies to your other thread, is this a hw question or another challenge?
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

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Re: Gravitational Potential
« Reply #3 on: May 08, 2009, 04:37:57 pm »
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lol sorry I should have made it more clear, it is a challenge problem :/

But I still liked your explanation of turning it into an integral :)

It's amazing how easy it is to evaluate complicated things by turning them into integrals

kamil9876

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Re: Gravitational Potential
« Reply #4 on: May 08, 2009, 04:47:30 pm »
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I think that is partly because integrals are ready made for you. Before newton people weren't aware of the fundamental theorem of calculus and had to find integrals not by antidifferentiating, but by calculating very wierd infinite sums and so simple things like integral of x^n took ages to prove and a lot of ingenuity.

http://www.math.wpi.edu/IQP/BVCalcHist/calc1.html
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."