Density of a floating sphere

[throneoo]

Homework Statement

a sphere of uniform density and radius R is floating on water , partially immersed such that the distance between the top of the sphere and the water surface is R/2
find the density of the sphere

Homework Equations

Archimedes Principle

The Attempt at a Solution

One can deduce from the Archimedes Principle ,that the weight of the displaced water = the weight of the object

ρ_WaterV_{Displaced water}g=ρ_Object V_Objectg

which basically turns the problem into a mathematical problem involving finding the volume of the immersed part of the sphere.

Consider a circle of radius R centered at the origin ,

the required volume is ∫π(R²-x²)dx from -R to R/2 = 9πR³/8

Thus , ρ_Object=(9πR³/8)/(4πR³/3) * ρ_Water

=27/32 ρ_WaterI don't know if it's a legitimate method . It is suggested that I utilize the concept of hydrostatic pressure instead , but i have no idea how to do that.[/SUB]

 

[ehild]

It is legitimate and correct. The buoyant force is equal to the weight of the displaced fluid.