Unit 6: Force and Motion - II

6.1 - Friction

When a force tends to slide a body along a surface, a frictional force from the surface acts on the body. The frictional force is parallel to the surface and directed so as to oppose the sliding. It is due to bonding between the body and the surface. If the body does not slide, the frictional force is a static frictional force $\vec f_s$ . If there is sliding, the frictional force is a kinetic frictional force $\vec f_k$ .

If a body does not move, the static frictional force $\vec f_s$ and the component of $\vec F$ parallel to the surface are equal in magnitude, and $\vec f_s$ is directed opposite that component. If the component increases, $\vec f_s$ also increases.

The magnitude of $\vec f_s$ has a maximum value, $\vec f_{s,max}$ given by:
$f_{s,max} = \mu_sF_N$
where $\mu_s$ is the coefficient of static friction and $F_N$ is the magnitude of the normal force. If the component of $\vec F$ parallel to the surface exceeds $\vec f_{s,max}$ , the body slides on the surface.

If the body begins to slide on the surface, the magnitude of the frictional force rapidly decreases to a constant value $f_k$ given by:
$f_k = \mu_kF_N$
where $\mu_k$ is the coefficient of kinetic friction.

6.2 - The Drag Force and Terminal Speed

Drag Force

When there is relative motion between air (or some other fluid) and a body, the body experiences a drag force $\vec D$ that opposes the relative motion and points in the direction in which the fluid flows relative to the body. The magnitude of $\vec D$ is related to the relative speed $v$ by an experimentally determined drag coefficient $C$ according to:

D = \frac{1}{2} C \rho A v^2

where $\rho$ is the fluid density (mass per unit volume) and $A$ is the effective cross-sectional area of the body (the area of a cross section taken perpendicular to the relative velocity $\vec v$ ).

Terminal Speed

When a blunt object has fallen far enough through the air, the magnitudes of the drag force $\vec D$ and the gravitational force $\vec F_g$ on the body become equal. The body then falls at a constant terminal speed $v_t$ given by:

v_t = \sqrt{\frac{2F_g}{C \rho A}}

6.3 - Uniform Circular Motion

If a particle moves in a circle or a circular arc of radius $R$ at constant speed $v$ , the particle is said to be in uniform circular motion. It then has a centripetal acceleration with magnitude given by:

a = \frac{v^2}{R}

This acceleration is due to a net centripetal force on the particle, with magnitude given by:

F = \frac{mv^2}{R}

where $m$ is the particle's mass. The vector quantities $\vec a$ and $\vec F$ are directed toward the center of the curvature of the particle's path.