Why GMm/d^2 instead of G(M +m)/d^2?

Discussion in 'Physics & Math' started by Dinosaur, Mar 30, 2002.

  1. Dinosaur Rational Skeptic Valued Senior Member

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    A long time ago, I took a physics course which showed how to derive a formula for gravitational potential. The derivative of the potential function was said to be the gravitational force.

    When all this was worked out for a sphere with radially symmetric density, the following formulae resulted.

    Potential(At distance d) = G*Mass/d
    Force(At distance d) = G*Mass/d^2

    Where G is the gravitational constant and d is the distance from the center of the sphere.

    Somehow this got translated to G*Mass1*Mass2/d^2 for the force between two spheres whose centers are distance d apart.

    Why not G*(Mass1 + Mass2)/d^2?
     
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  3. James R Just this guy, you know? Staff Member

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    Some of your equations are wrong.

    In the Newtonian picture, the gravitational acceleration due to a body of mass M is, at distance r:

    g = GM/r<sup>2</sup>.

    The force on another body of mass m due to the first one is the mass of the that body multiplied by the acceleration, according to F=ma (Newton's second law). i.e.:

    F = GMm/r<sup>2</sup>

    The potential energy of the system of two bodies is:

    U = -GMm/r

    As to why the masses are multiplied and not added, consider the case when m=0. If the masses are multiplied (in the correct way), the resulting force of gravity is zero. But if they were added instead, the massless object would still be under a gravitational force.

    The bottom line is: observations support the masses being multiplied, not added. That's the way the universe works.
     
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  5. Dinosaur Rational Skeptic Valued Senior Member

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    James R: I know that the correct formula is GMm/d^2. I have even written a fairly slick Visual Basic application which does gravitational computations using this formula.

    I just wonder how it is derived from the derivative of the potential. As mentioned in previous post, the formulae are as follows.

    Potential(At distance d) = G*Mass/d
    Force = Derivative(Potential)
    Force(At distance d) = -G*Mass/d^2

    I have a compendium of Math & Physics formula which shows the above. This book also mentions the following as the force between two objects as.

    Force = G*Mass1*Mass2/d^2

    It gives no indication of how the latter is derived from the former.

    I do not deny the validity of the latter formula. I just wonder how it relates to the derivative of potential, which is referred to as the force due to one object.

    BTW: This all came up due to a discussion about gravity inside a hollow spherical shell of uniform density. I assume you know this to be zero every where inside.
     
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  7. ImaHamster2 Registered Senior Member

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    Dinosaur posted: “I assume you know this to be zero every where inside.”

    Yep. Given the excellent quality of James R. prior posts, that is a pretty safe assumption. Hehe.
     
  8. (Q) Encephaloid Martini Valued Senior Member

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    I know that the correct formula is GMm/d^2.

    James R is correct.

    Why not G*(Mass1 + Mass2)/d^2?

    Principia - Newton's Law of Universal Gravitation.

    Every particle in the universe exerts a force on every other particle along the line joining their centers. The magnitude of the force is directly proportional to the product of the masses of the two particles, and inversely proportional to the square of the distances between them.
     
  9. Dinosaur Rational Skeptic Valued Senior Member

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    Okay folks: I give up. There is no answer to my question.
     
  10. James R Just this guy, you know? Staff Member

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    Dinosaur,

    I thought I answered you already. Here it is again:

    The gravitational potential energy of a system of two masses separated by distance r is:

    U = -GMm / r<sup>2</sup>

    The force is the negative gradient of the potential energy:

    F = -dU/dr

    or:

    F = Gmm / r<sup>2</sup>.

    If your book says something different, I'm afraid it's wrong.
     
  11. (Q) Encephaloid Martini Valued Senior Member

    Messages:
    20,855
    dinosaur

    Okay folks: I give up. There is no answer to my question.

    Is this the question?

    Why not G*(Mass1 + Mass2)/d^2?

    The magnitude of the force is directly proportional to the product of the masses of the two particles...

    The product ! To get the product you must multiply and not add the magnitude of force of the masses.

    What other answers do you need?

    What other questions do you have?
     
  12. Elmo Registered Senior Member

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    Q

    well, I'm sure that vocabulary lesson has really cleared things up! Nice one Q!

    Please Register or Log in to view the hidden image!

     
  13. ChristCrusher Registered Senior Member

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    well, its not reallly THAT hard to do, but I am not going to the trouble of doing it here........


    solve Kepler's equations of motion for an equal period system in polar coordinates.

    angular momentum conservation leads to a conclusion that of course, in line torque is 0

    anyways, consider a center of mass in the kepler system and solve for centripetal accelerations, setting forces equal







    if that is to hard for you, you could always look at the UNITS of G and simply conclude that you need a mass product.

    of course, that is not scientific, nor mathematical really.....

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    , but then again, this question was rather simple.
     
  14. allant Version 1.0 Registered Senior Member

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    88
    Hi,

    The guys are all correct but not explaining it in the way I think you require. May be this will help.


    Assume the two masses A and B are made up of bits. Now each bit of A is going to be attracted to each bit of B.

    So lets assume A is made up of 10 bits of 1 gram each and b has 20 bits of 1 gram each and assume the force between any two bits is G/d^2.

    Now we can add the force up.

    G/d^2 between the first bit off A and the first bit of B

    There are 20 bits in B so there are 20G/d^2 between the first bit of A and all the 20 bits of B.

    Now there are 10 bits of A all the same with 20G/d^2 force with all the 20 bits of B.

    The total of the force is F = G*(10*20)/d^2 or F= (G*M*m)/d^2.

    Basically what I have done above is done the calculas numerically in bits of 1 gram with a fixed distance. This works out Ok if A and B are far apart compared to the size of A and B.

    The key point is that the force is attraction between each any every "part" of A and B.

    It is not a force/field somehow beaming out of B and A. Think about if B was the earth. By your thinking the earth would be beaming out a huge force and a 1kg or a 2kg weight would make no difference to the total. A 1kg weight would weight almost the same as a 2 kg weight by your rule. i.e ( 1+Earthweight) is almost equal to ( 2 + Earth weight)

    Does this make it clear ?
     

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