Question

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.

The new sphere has radius R > R₀ and density ρ < ρ₀

The new sphere has radius R = R₀ and mass M < M₀

The new sphere has mass M < M₀ and density ρ = ρ₀

The new sphere has density ρ = ρ₀ and mass M > M₀

The new sphere has mass M = M₀ and density ρ > ρ₀

The new sphere has mass M = M₀ and density ρ < ρ₀

Answer 1

here,

FOR MASS
m = (4/3)*d*Pi*r^3 -----------(1)

where
m = mass of sphere
d = is density of sphere
r = radius of sphere

m is diretly proportinal to both density and radius

FOR DENSITY :

d = m /(4/3 * pi * r^3) ------------(2)

density is directly proportional to mass but inversely proportional to radius of sphere

FOR RADIUS :

r = cuberoot((3*m)/(4*d*Pi)) -----------(3)

r is directly proportional to mass but inversely propotinal to density of sphere

By following above theory answer to following query will be :

1.) The new sphere has radius R > R0 and density ρ < ρ0
radius inc, density dec so mass might go up, down, or stay same,
R or F or U

2.) The new sphere has radius R = R0 and mass M < M0
same radius, less mass therefore less density so It will
Fall(F)

3.) The new sphere has mass M < M0 and density ρ = ρ0
density same but mass dec so radius inc so it will Fall(F)

4.) The new sphere has density ρ = ρ0 and mass M > M0
same density it will Fall or remian unchanged ( R )

5.) The new sphere has mass M = M0 and density ρ > ρ0
same mass: U

6.) The new sphere has mass M = M0 and density ρ < ρ0
same mass: U