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Finding the Gravitational Acceleration of a sphere with two cavities.

  1. Apr 23, 2010 #1
    1. The problem statement, all variables and given/known data

    My problem asks to find the gravitational acceleration at a point (P) due to the sphere with two spherical cavities. I am given the radius of the sphere (R), the radius of each of the two cavities (r1 and r2, which are the same), the distance between the center of the cavities to the center of the sphere (x1 and x2, both the same as well), the distance the point is away from the center of the sphere (X), and the volumetric density of the sphere.


    2. Relevant equations

    Not sure if I just subtract the cavities out of the sphere's radius or what.

    3. The attempt at a solution

    SCAN0005.jpg
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Apr 24, 2010
  2. jcsd
  3. Apr 24, 2010 #2

    Filip Larsen

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    You are not far from a solution. If you for an instance imagine that the cavities are filled with matter of same density as the surrounding sphere, then think about how you can write the total gravitational force from the three parts as a sum of the force from each part (two of which you can write down explicitly) and what this total gravitational force can be equated to. Rearranging this equation should then give you the force you are looking for.
     
  4. Apr 24, 2010 #3
    But I don't think I'm looking for the gravitational force, I'm looking for the acceleration.
     
  5. Apr 24, 2010 #4

    Filip Larsen

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    I just used the term force to make sure there was a clear association with Newtons law of gravitation. Acceleration is, as you already seem to know well, just specific force for the test mass at P.

    However, your solution in the scan (which I first see now) is not quite right in the sign of some of the terms. Think about if the gravity of the cavitated sphere really should be larger than the gravity from the solid sphere. Also, the radius of the cavity spheres are not all correct either.
     
    Last edited: Apr 24, 2010
  6. Apr 25, 2010 #5
    Ok, so in the problem I redid the 3rd row a bit. For the mass of hollow sphere 1, I replaced x_1 with r_1 and for the mass of hollow sphere 2, I replaced x_2 with r_2. I also redid the denominators where (X-x_1)^(2) was changed to (X+x_1)^(2) and (X-x_2)^(2) was kept the same.

    For my answer I got 1.95*10^(-8) m/s^(2) comparing this to the gravitational acceleration if the sphere was completely solid, the result would be 7.2*10^(-6) m/s^(2).
     
  7. Apr 25, 2010 #6

    Filip Larsen

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    Gold Member

    I don't quite arrive at the same result.

    With the numbers you have written on your scan, I get the gravitational acceleration of the solid sphere to 1.91*10-8 m/s2 (around 2 orders of magnitude lower than your number) and subtracting the cavities I get 1.86*10-8 m/s2. The "surplus" acceleration from the two cavities I get to 2.79*10-10 m/s2 and 2.01*10-10 m/s2. Notice, that since the radius of the cavities is around 1/4 of the radius of the big sphere, you should expect their (far-field) gravity to be around (1/4)3 = 1/64.
     
  8. Apr 25, 2010 #7
    Ok I see how you got the results, I added the gravitational acceleration of the two cavities to the gravitational acceleration of the solid sphere rather than subtracting the sum of the cavities from the solid sphere.
     
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