A small solid sphere of mass M0, of radius R0, and of uniform density 0 is place

Question

A small solid sphere of mass M0, of radius R0, and of uniform density 0 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.
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 R F U R or U F or U R or F or U  The new sphere has mass M = M0 and radius R < R0 
 R F U R or U F or U R or F or U  The new sphere has density = 0 and radius R > R0 
 R F U R or U F or U R or F or U  The new sphere has radius R > R0 and mass M = M0 
 R F U R or U F or U R or F or U  The new sphere has radius R < R0 and density = 0 
 R F U R or U F or U R or F or U  The new sphere has radius R > R0 and density < 0 
 R F U R or U F or U R or F or U  The new sphere has radius R = R0 and density < 0 

Explanation / Answer

Buoyant force = volume of water displaced x density of water x g

and as sphere floats that means it is in equilibrium

so sphere comes to equilibrium when Fnet = Fb - mg = 0

volume of water displaced x density of water = total volume of sphere x density of sphere

and total volume of sphere x density = mass of sphere

1. M = M0

and R < R0

mass is same so buoyant force will also be same and volume displaced of water will also be same

so level will be unchanged.

2. rho = rho0

and R > R0

radius increases -> volume increase -> mass of sphere increases -> bnuoyant force increase -> water displaced
volume increases- > level rises.

3. Mass is same (as discussed in 1)

level unchanged

4. R < R0 and rho = rho0

radius decreases -> volume decrease -> mass of sphere decreases -> bnuoyant force decrease -> water displaced
volume decreases- > level Falls.

5. R > R0 and rho < rho

R increases so volume of sphere increases but density also changes (decreases)

so mass = volume x density

mass may increases, decreases or may be it remains same.

hence relation needed, how much R increases and by what factor rho is decreases.

6. now density decreases.

so mass of new sphere will be less.

mass of sphere decreases -> bnuoyant force decrease -> water displaced
volume decreases- > level Falls.

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