Buoyant Force Question: Cylinder and Cone in Water - Homework Solution

[Titan97]

Homework Statement

(I made this question myself)
What happens to the cylinder (in orange color) kept inside a beaker filled with water like this:

The cylinder just passes through a hole on the bottom of the beaker. Assume that water can't leak through the hole. Will the cylinder move up? Or will it move down?

Homework Equations

None

The Attempt at a Solution

I initially thought that the cylinder will move up because of buoyant force. But then, all the forces on the cylinder are downwards. The pressure acting on the cylinder is downward and so is its weight. So will the cylinder move down?

Also, if the cylinder is replaced by a cone like this

Will the cone remain at rest?

 

[andrewkirk]

The buoyant force on an object is the integral of pressure over the surface of the object. If the object is surrounded by liquid on all sides then it's fairly straightforward to show that that force will be upwards. But that is not the case here. Do you think you'd be able to do the integration for the first case? You don't even need to perform the integration to know the general direction of the buoyant force.

 

[Dick]

As you say, if all forces on the plug are downward then it will move down. Are you saying there is a vacuum below the plug? In the second picture won't the container exert a force on the plug? It can't fit going downwards.

 

[andrewkirk]

Are you saying there is a vacuum below the plug? In the second picture won't the container exert a force on the plug? It can't fit going downwards.

Not necessarily a vacuum. It could be air or other gas. All that's needed is for it to have a lower density than the liquid.
Whether the container exerts a force on the cylinder/plug depends on the construction. For instance the plug could be a cylinder that is free to slide up and down in a circular hole in the container base. There would probably need to be a membrane over the cylinder to stop water leaking out, but which would still allow the liquid to apply pressure to the cylinder.
The cylinder would keep going down until it fell out or was stopped. If it fell out then the liquid would run out of the container unless the membrane was strong enough to hold it. The cylinder could be stopped from falling out completely by a bolt or rim that prevents it passing entirely through the hole.

 

[Titan97]

@Dick , I am not saying the cone will move downwards. But can't it stay at rest?

 

[Dick]

@Dick , I am not saying the cone will move downwards. But can't it stay at rest?

It might and it might not. Depends on the density of the plug, the fluid and pressure of whatever you specify is outside of the container. Like andrewkirk said, you have to compute all of the forces and sum them to see if the total gives a net up or down force. It will depend on the numbers, it could be either. You can't tell just from looking at the picture.