Gravitation, hollowed out sphere

[Puchinita5]

Homework Statement

Figure 13-39 shows a spherical hollow inside a lead sphere of radius R = 4.5 m; the surface of the hollow passes through the center of the sphere and “touches” the right side of the sphere. The mass of the sphere before hollowing was M = 380 kg. With what gravitational force does the hollowed-out lead sphere attract a small sphere of mass m = 22 kg that lies at a distance d = 17 m from the center of the lead sphere, on the straight line connecting the centers of the spheres and of the hollow?

Homework Equations

http://edugen.wiley.com/edugen/courses/crs1650/art/qb/qu/c13/fig13_39.gif

The Attempt at a Solution

I attempted to figure out the mass of the hollowed out section by taking the density of the whole sphere and setting that equal to what the density of the small sphere would be and solving for mass. I got the mass for the hollowed section to be 47.5, so I subtracted this from the original mass to get the final mass. I then used F=G(m1)(m2)/r^2 to solve for force, using the final mass of the hollowed sphere. This gave me the wrong answer. The only thing I can think is that the center of mass would no longer be at the spheres center, so the distance between the two objects should be different, but I don't know how I would calculate where the center of mass would now be. Any insight?

 

[Puchinita5]

ok nevermind, i just figured it out...i took a wild guess how to do it and it worked, though i am not really sure why. I used the force equation for the orignal sphere, and then for the hollowed sphere. I subtracted the force of the hollowed sphere from the force of the orignal, and got the right answer. Doesn't wholly make sense to me

 

[nlightner]

It seems like you are on the right track with your approach. However, there are a few things to consider when calculating the gravitational force between the hollowed-out sphere and the small sphere.

Firstly, when you hollow out a sphere, the mass distribution changes. The mass is no longer evenly distributed throughout the sphere, as it is now concentrated towards the edges. This means that the center of mass of the hollowed-out sphere will not be at the same location as the center of the original sphere. To calculate the new center of mass, you can use the formula for center of mass:

x_cm = (m1*x1 + m2*x2 + ... + mn*xn) / (m1 + m2 + ... + mn)

Where x_cm is the new center of mass, m1, m2, etc. are the masses of the different parts of the sphere, and x1, x2, etc. are the distances of those parts from the original center of the sphere.

Secondly, when calculating the gravitational force between the two spheres, you need to take into account the distance between their centers of mass, not their physical centers. This means that the distance between the two spheres will be different from the given distance of 17 m.

Once you have calculated the new center of mass and the correct distance between the two spheres, you can use the formula F = G(m1)(m2)/r^2 to calculate the gravitational force between them.

I hope this helps. Keep in mind that when dealing with complex systems like this, it's important to carefully consider all the factors involved and make sure your calculations are taking everything into account. Good luck!