Newtonian Gravitation with an extended body

[freemind]

Hello,

I'm having trouble with the following question:

There is a spherical hollow inside a lead sphere of radius R ; 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 . With what gravitational force does the hollowed-out lead sphere attract a small sphere of mass m that lies a distance d from the center of the lead sphere, on the straight line connecting the centers of the spheres and of the hollow?

My guess is to integrate $$dF = G \frac {m} {(r_i)^2} \mbox {\em dM}$$ . Is this right?

 

[Doc Al]

I presume you are asking whether one could solve this by integrating the force from every element of mass in the hollowed out sphere. Sure. But lots of luck working out that integral.

An easier way is to view the hollowed-out sphere as composed of two spheres, one of which has negative mass.

 

[freemind]

Thanks a lot Doc Al. I'm now hitting my head against the wall for not using your method.

 

[mrdrew]

hi, sorry to revive an old thread, but i was wondering if someone could explain in more depth the answer to this problem. I've been struggling with it for a while, I guess I am not the strongest in physics. If anyone could help it would be much appreciated.

 

[Doc Al]

hi, sorry to revive an old thread, but i was wondering if someone could explain in more depth the answer to this problem.

Describe the problem you are having and where you get stuck and you'll get plenty of help. What don't you understand?

 

[mrdrew]

well the way i was going to go about it was to determine the density of the one large hollow sphere without the circle taking out of it, then find the mass of the hollow that is taken out of it. Then i was going to find the attraction between the hollowed out mass (minus the sphere taken out) and the tiny mass at the distance away. For some reason, when i go to calculate the density however, I get very large numbers. I am taking the mass (in grams) and dividing by the volume (found by 4/3*Pi*R^3). I feel like i am messing up something so simple. any help?

 

[Doc Al]

Since the only masses given are M and m, I'm not sure how you are calculating the density numerically.

It is certainly OK to calculate the density of the large sphere by dividing M by its volume, and then use that to find the mass of the hollowed out piece. But an easier way is to realize that the radius of the hollow sphere is half the radius of the large sphere. So you should be able to find the mass of the hollow sphere in terms of M by using a simple ratio.

 

[mrdrew]

i forgot to mention that my problem was the same, only with numerical values.
ill write it out for your sake here:
a spherical hollow inside a lead sphere of radius R = 4.00 cm; 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 = 2.95 kg. With what gravitational force does the hollowed-out lead sphere attract a small sphere of mass m = 0.431 kg that lies at a distance d = 9.00 cm from the center of the lead sphere, on the straight line connecting the centers of the spheres of the hollow

as for the ratio of the masses, i will attempt to do that now, though I am not sure as to how exactly, but I will give it a try. Thank you for your help, I'll let you know if i get it done! many thanks once again.

edit: Managed to wrangle the question in. Did a ratio of densities and then found teh answer to be somethin close to 8.1 x 10 ^ -10