The possible effects on the water level

[litz057]

Homework Statement

A small solid sphere of mass M0, of radius R0, and of uniform density ρ0 is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls, U-Unchanged), when that sphere is replaced by a new solid sphere of uniform density.
1. The new sphere has radius R < R0 and density ρ > ρ0.
2. The new sphere has density ρ = ρ0 and mass M < M0.
3. The new sphere has mass M = M0 and density ρ < ρ0.
4. The new sphere has radius R = R0 and density ρ < ρ0.
5. The new sphere has density ρ = ρ0 and mass M > M0.
6. The new sphere has radius R < R0 and mass M = M0.

Homework Equations

Archimedes Principle

The Attempt at a Solution

1. R or F or U (Density increased, but size decreased; thus, the unknown net result on mass)
2. F (mass decreased)
3. U (the mass did not change)
4. F (the radius is the same, but the density decreases; thus, mass decreases)
5. R (the density remains the same, but the mass increases= mass decreases)
6. U (the mass did not change)

I am not sure where I went wrong. Please help me understand my mistakes.

 

[AlephNumbers]

For 3. if the mass stays the same but the density decreases, what does that imply about the volume of the sphere?

 

[AlephNumbers]

Ooops. Ignore my last post. Didn't read the question carefully enough. I think where you went wrong was 6.

 

[AlephNumbers]

The important thing to realize is that the amount of fluid the sphere will displace is limited by its volume; Over a certain threshold the density, and by extension the mass, of the object will no longer have an effect on the amount of fluid the object will displace. This threshold is the density of the fluid the object is being submerged in. So the radius could decrease by a magnitude that would require the density to increase past the density of water in order to maintain the same amount of mass.

 

[litz057]

The important thing to realize is that the amount of fluid the sphere will displace is limited by its volume; Over a certain threshold the density, and by extension the mass, of the object will no longer have an effect on the amount of fluid the object will displace. This threshold is the density of the fluid the object is being submerged in. So the radius could decrease by a magnitude that would require the density to increase past the density of water in order to maintain the same amount of mass.

Does that means that it is hard to determine what will exactly happen to the level of the water? The water level might fall due to the smaller volume, which is related to the decrease in the radius. So it is hard to determine if the water level will fall or be unchanged.

 

[AlephNumbers]

Yes, it could either remain unchanged or decrease. But it cannot increase, since the mass stays the same.

 

[litz057]

Yes, it could either remain unchanged or decrease. But it cannot increase, since the mass stays the same.

When I was attempting this, I only factored in the mass staying the same and not the change in volume. Thank you so much for your help and making me realize my mistakes! I really appreciate it!

 

[AlephNumbers]

You are very much welcomed.