Hollowed out sphere exerting gravitational force

AI Thread Summary
The discussion centers on the gravitational effects of a hollowed-out sphere, where it was established that the mass of the hollow section is M/8. The confusion arises from the incorrect assumption that the remaining mass (7M/8) can be directly substituted into the gravitational force equation. The correct approach involves using superposition, where the gravitational force from the hollow sphere is determined by subtracting the force of the removed mass from that of the full sphere. Additionally, the remaining mass lacks spherical symmetry, which means Newton's shell theorem does not apply, affecting the gravitational calculations. Ultimately, understanding the distinction between center of mass and center of gravity is crucial in this context.
jolly_math
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Homework Statement
A spherical hollow is made in a lead sphere of radius R, such that its surface touches the outside surface of the lead sphere and passes through its centre. The mass of the sphere before hollowing was M. With what force, according to the law of universal gravitation, will the hollowed lead sphere attract a small sphere of mass m, which lies at a distance d from the centre of the lead sphere on the straight line connecting the centres of the spheres and of the hollow?
(diagram below)
Relevant Equations
F = G*m1m2/r
1666650052639.png

I solved that the hollowed out mass is M/8, which is correct. I don't understand why it is incorrect to substitute the remaining mass (7M/8) back into the F = G*m1m2/r to produce the force. Why is the solution the force of the whole lead sphere minus the force of the “hole” lead sphere, which is
1666650187678.png
? What effect does this "hole" have?

Thank you.
 
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This idea is superposition. The force at mass ##m## due to a full sphere is greater than that due to the hollowed sphere because there is more mass. To get the force due to the hollowed sphere, you need to subtract the force due to the smaller sphere that you took out in order to create the cavity.
 
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kuruman said:
This idea is superposition. The force at mass ##m## due to a full sphere is greater than that due to the hollowed sphere because there is more mass. To get the force due to the hollowed sphere, you need to subtract the force due to the smaller sphere that you took out in order to create the cavity.
Is the force needed to create the cavity a gravitational force?
 
No, pure elbow grease ... :smile:

jolly_math said:
I don't understand why it is incorrect to substitute the remaining mass (7M/8) back into the F =
Because the remaining mass doesn't have its center of mass at the same position !

##\ ##
 
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BvU said:
Because the remaining mass doesn't have its center of mass at the same position !
Also, somewhat more subtly, because the remaining mass is no longer spherically symmetric. So Newton's shell theorem no longer applies, so the gravitational effect of the remaining mass is no longer guaranteed to be identical to that of an equal point mass positioned at a particular mass center.

Pithy version: Center of mass and center of gravity are not always synonymous.
 
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