Question

A solid, homogeneous sphere with a mass of m₀, a radius of r₀ and a density of ρ₀ is placed in a container of water. Initially the sphere floats and the water level is marked on the side of the container. What happens to the water level, when the original sphere is replaced with a new sphere which has different physical parameters? (Answer choices per question: rises, falls, stays the same, rises or stays the same, falls or stays the same)
1. The new sphere has a mass of m < m₀ and a radius of r = r₀.

2. The new sphere has a density of ρ > ρ₀ and a radius of r = r₀.

3. The new sphere has a mass of m = m₀ and a density of ρ > ρ₀.

**Please explain how you got the answers also :) Thank you!

Answer 1

To get answers we will use only this facts:

When object is floating:

Mass of water displaced = Mass of object

When object is submerged:

Volume of water displaced = Volume of the object

Answer 1:

As mass is less m < m₀ therefore less water will be displaced------implies that----->water level will fall

Answer 2:

Radius is same( r = r₀) implies volume is also same as volume is given by :

Where r is the radius

but the density is large( ρ > ρ₀) therefore mass will increase as density is given by:

As density increases and volume remains the same therefore from above equation the mass will increase

This means that more water needs to be displaced to keep the sphere afloat , if it doesn't float and sinks completely even then more water will be displaced resulting in rise in water level

Answer 3:

Here mass is same(m = m₀ ) but the density increases (ρ > ρ₀ )

Now there could be two cases which can occur:

1. Sphere floats

2. Sphere submerges

1. Sphere floats:As the mass is same therefore the water displaced by it will be same implies the water level will remain the same.

2. Sphere submerges: As the density is less then before and mass is same--> implies that the volume of the sphere will decrease because:

From above relation it is clear that if mass remains the same and density is increased , therefore the volume will decrease to obey the equality.

Now when the sphere is submerged under water then the amount of water displaced by the sphere will be equal to the volume of the sphere, as the volume of the sphere is less then before ---> implies that the water level will fall.

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