What are the steps for calculating the center of mass of this object?

[Pochen Liu]

Homework Statement

Centre of mass of a sphere with a spherical hole

Relevant Equations

N/A

What are the steps to calculating the center of mass for this object? I don't want a numerical answer just the theory.

I understand I need to calculate COM for all axis, we know in the Y axis it's dead centre because of symmetry but I don't know how to do it for the others.

A hint would be most helpful :)

 

[collinsmark]

Problem Statement: Centre of mass of a sphere with a spherical hole
Relevant Equations: N/A

What are the steps to calculating the center of mass for this object? I don't want a numerical answer just the theory.

I understand I need to calculate COM for all axis, we know in the Y axis it's dead centre because of symmetry but I don't know how to do it for the others.

A hint would be most helpful :)

View attachment 243664

I don't think you actually need to find the COM to solve this problem.

What would the gravitational attraction be if the larger sphere (R = 4 cm) was completely solid, and had no hollow part?

What would the gravitational attraction be if instead of the hollow part there was a solid lead sphere (R = 2 cm) in the location where the hollow part would be, and the larger sphere (R = 4cm) didn't exist? (In other words, there's only a single R = 2 cm sphere, made of solid lead, in the location where the location where the hollowed out part would have been.) [Edit: well, I mean of course the small m = 0.431 kg sphere shown in yellow is still there. I just mean that the bulk of the larger, R = 2 cm sphere doesn't exist, and is instead replaced by a lead sphere where the hollow part would have been.]

What's the difference between these two forces?

 

[Pochen Liu]

I don't think you actually need to find the COM to solve this problem.

What would the gravitational attraction be if the larger sphere (R = 4 cm) was completely solid, and had no hollow part?

What would the gravitational attraction be if instead of the hollow part there was a solid lead sphere (R = 2 cm) in the location where the hollow part would be, and the larger sphere (R = 4cm) didn't exist? (In other words, there's only a single R = 2 cm sphere, made of solid lead, in the location where the location where the hollowed out part would have been.) [Edit: well, I mean of course the small m = 0.431 kg sphere shown in yellow is still there. I just mean that the bulk of the larger, R = 2 cm sphere doesn't exist, and is instead replaced by a lead sphere where the hollow part would have been.]

What's the difference between these two forces?

I see, so how would you put this formally?

I understand that it's the force of gravity from its mass, but why does this work?

 

[collinsmark]

It's basically just superposition.

Sphere with hollow section + mass where hollow section would be = solid sphere

Similarly,

Force from sphere with hollow section + force from mass where hollow section would be
= force from solid sphere.

Applying algebra,

Force from sphere with hollow section
= force from solid sphere - force from mass where hollow section would be

 

[Samanko]

 

[Samanko]

 

[Samanko]

If you have any question, please let me know.

 

[gneill]

Hello @Samanko,

Welcome to Pysics Forums!

Please note for the future that it is against Forum rules to post complete or near-complete solutions to homework problems before the Original Poster has arrived at a correct solution to the problem and any related questions they have by their own efforts. Helpers can only provide guidance via hints and suggestions or pointing out errors in the OP's attempts.

Check out the pinned thread, https://www.physicsforums.com/threads/guidelines-for-students-and-helpers.686781/

 

[Samanko]

Hello @Samanko,

Welcome to Pysics Forums!

Please note for the future that it is against Forum rules to post complete or near-complete solutions to homework problems before the Original Poster has arrived at a correct solution to the problem and any related questions they have by their own efforts. Helpers can only provide guidance via hints and suggestions or pointing out errors in the OP's attempts.

Check out the pinned thread, https://www.physicsforums.com/threads/guidelines-for-students-and-helpers.686781/

Yes
I'll be careful in the future.

 

[haruspex]

I don't think you actually need to find the COM to solve this problem.

.. and it wouldn’t help anyway.
@Pochen Liu , you cannot in general find the gravitational attraction to an object at a point merely by knowing its mass and the distance from the point to its mass centre. That only works for spheres that are either uniformly dense or consist of concentric uniform shells; and the point must be outside.