Stokes’ Law
Stokes’ Law illustrates the dynamics of a body falling through a viscous fluid, outlining how the viscous drag force (F) affects its motion. As the body falls, it drags the fluid, creating relative motion among fluid layers and resulting in a retarding force opposite to the motion. This force is proportional to the object's velocity and can be expressed mathematically as:
$$F = \frac{\pi}{6} \eta v$$
where \( \eta \) is the viscosity of the fluid and \( v \) is the velocity. The law leads to important implications regarding terminal velocity (the constant speed a falling body reaches when the forces acting on it are balanced).
For a sphere falling in a fluid, the terminal velocity \( v_t \) is derived from the balance of forces, given by:
$$v_t = \frac{2a^2 (\rho - \sigma) g}{9\eta}$$
Where:
- \( a \) is the radius of the sphere,
- \( \rho \) is the density of the sphere,
- \( \sigma \) is the density of the fluid,
- \( g \) is the acceleration due to gravity.
This relationship indicates that the terminal velocity depends on the square of the radius and inversely on the fluid's viscosity, highlighting the importance of fluid characteristics on falling objects. The example of a copper ball falling through oil is employed to illustrate these principles effectively.