Question

A hollow sphere of inner radius 8.69 cm and outer radius 9.99 cm floats half-submerged in a liquid of density 927.00 kg/m3. (a) What is the mass of the sphere? (b) Calculate the density of the material of which the sphere is made.

Answer 1

a)
`m_(sph) = 3.768\,kg`, b)
`\rho_(sph) = 4897.795\,(kg)/(m^(3))`

Step-by-step explanation:

According to the Archimedes' Principle, the buoyancy force is equal the weight of the displaced fluid. Then, this equation of equilibrium is constructed by the Newton's Laws:

`\Sigma F = \rho_(w)\cdot V_(disp)\cdot g -\rho_(sph)\cdot V_(sph)\cdot g = 0`

After some algebraic handling:

`\rho_(w)\cdot V_(disp) = \rho_(sph)\cdot V_(sph)`

a) The mass of the sphere is:

`m_(sph) = \rho_(w)\cdot V_(disp)`

`m_(sph) = (927\,(kg)/(m^(3)) )\cdot \left[(4)/(3)\pi\cdot (0.099\,m)^(3) \right]`

`m_(sph) = 3.768\,kg`

b) The density of the sphere is:

`\rho_(sph) = (m_(sph))/(V_(sph))`

`\rho_(sph) = (3.768\,kg)/((4)/(3)\pi\cdot [(0.099\,m)^(3)-(0.087\,m)^(3)])`

`\rho_(sph) = 4897.795\,(kg)/(m^(3))`