Finding the Gravitational Acceleration of a sphere with two cavities.

[striker300]

Homework Statement

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.

Homework Equations

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

The Attempt at a Solution

 

[Filip Larsen]

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.

 

[striker300]

But I don't think I'm looking for the gravitational force, I'm looking for the acceleration.

 

[Filip Larsen]

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.

 

[striker300]

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).

 

[Filip Larsen]

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.

 

[striker300]

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.