# Statics and equilibrium of a rigid body [closed]

A vertical cylindrical container contains within it three identical spheres of equal weight P which are tangent to each other and also to the inner wall of the container. A fourth sphere, identical to the previous ones, is then superimposed on the three spheres as illustrated in dotted. Determine the respective intensities of the normal forces as a function of P which the vessel wall exerts on the three spheres.

Very interesting question, but I saw a resolution and could not understand why there are no contact forces between the base spheres. thanks in advance

• Without the top sphere, the bottom spheres are in casual contact with each other and the walls of the cylinder (no contact forces). However, each would exert a force of P on the bottom, if any, of the cylinder. With the top sphere in place, it would tend to push the bottom spheres apart applying forces to the side walls, but still not to each other. – Bob D Sep 24 '19 at 18:48
• thanks! so if there were the three spheres at the base, would there be no contact forces between them and not at the sides? why? – Helen Sep 24 '19 at 19:50
• I have decided to respond in the form of an answer rather than comment. Hope it helps. – Bob D Sep 24 '19 at 20:59

Very interesting question, but I saw a resolution and could not understand why there are no contact forces between the base spheres. thanks in advance

Here are my thoughts, though I admit they may be debatable.

I think we need to address the question in two parts. One with the upper sphere not in place and one with the upper sphere in place.

No sphere on top:

In my mind, this one is a bit tricky. We can say that the separation of the spheres from one another is "zero" if any attempt to bring them a tiny bit closer results in a "measurable" (non zero) force that tends to push them apart, and that increases rapidly the closer you attempt to bring them together. Conversely, any tiny increase in separation from this "zero" position would yield no measurable force between them. So approaching the zero separation from either direction we may consider the forces between the spheres approach zero in the limit.

We may arbitrarily define this zero separation as "touching" where the contact force is, in the limit, zero. One thought experiment would be to ask what would happen if the cylinder was removed? If there was no friction involved at any of the contacts, would the spheres separate, indicative of repulsive contact forces between them?

Sphere on top:

This is probably more straight forward. Starting with the assumption that the lower spheres are just "touching" (as defined above), clearly there will be a component of the weight of the upper sphere acting on each of the lower spheres that will tend to push the spheres apart and press them against the cylinder walls. Thus we may say that the forces between the bottom spheres should definitely be be zero due to influence of the top sphere. Once again this assumes frictionless contacts throughout.

Hope this helps.

• Thanks!! Excellent – Helen Sep 24 '19 at 21:13