- #1

brotherbobby

- 700

- 163

- Homework Statement
- A solid sphere of volume 20 cm and uniform density ##\rho_B## = 400 ##\text{kg m}^{-3}## floats in a vessel containing pure water. Calculate the depth to which the sphere is submerged.

- Relevant Equations
- 1. ##\text{Law of floatation}## : When a body floats in a liquid, the weight of the body is equal to that of the weight of the liquid displaced : ##w_B = \Delta w_L##.

2. The volume of a spherical body of radius ##r_B## is given by : ##V_B = \frac{4}{3}\pi r_B^3##.

3. The weight of a body is given by : ##w_B = m_B g = \rho_B V_B g## where ##\rho_B## is the density of the body and ##V_B,m_B## are the volume and mass of the body respectively.

The body has a radius of ##r_B = 0.2## m.

The volume of the body is given by ##V_B = \frac{4}{3}\times \pi \times 0.2^3 = 0.034\;\text{m}^3##.

The mass of the body : ##w_B = \rho_B V_B = 400\times 0.034 = 13.37\,\text{kg}##.

Hence the mass of the liquid displaced : ##\Delta m_L = 13.37\, \text{kg}## . (Law of floatation can also be "read off" as ##m_B = \Delta m_L##)

The volume of liquid displaced : ##\Delta V_L = \frac{\Delta m_L}{\rho_L} = \frac{13.37}{1000} = 1.34\times 10^{-3}\,\text{m}^3##.

This is the volume of the sphere below the blue line inside the water, as shown to the right.

**Question is, how to connect the volume of liquid displaced ##\mathbf{\Delta V_L}## to the volume of the body ##\mathbf{V_B}## ?**

I could proceed no further. Any help would be appreciated.

I could proceed no further. Any help would be appreciated.