# How do you derive #eta=(2r^2g(rho_s-rho_l))/(9v)# for a sphere obeying stokes law in a liquid, falling at terming velocity so that #F+U=W#?

##### 1 Answer

Use the fact that three forces are in equilibrium - weight, buoyant force and the Stoke's drag force.

#### Explanation:

A perfectly smooth sphere of radius

The constant weight,

Initially the drag force is small. So the ball starts accelerating down. But the drag force increases in strength with speed. At some point the sum of the buoyant force and viscous drag equilibrate with the weight and the ball achieves a constant terminal velocity

**Dynamic Equilibrium Condition**:

**Weight**: Writing the mass of the ball in terms of its volume and the density of the material (

**Buoyant Force** : By the **Archimedes Principle** the buoyant force acting on an object is equal to the weight of the liquid it displaces,

**Stoke's Drag Force**: The upward viscous drag force on a smooth sphere of radius

Expanding (1),