Friction, Inclined Planes, Circular Motion and Gravitation

• Friction acts along the surface, opposing sliding (or the tendency to slide).
• Static friction adjusts to match the applied force, up to a maximum $\mu_s N$. Below that limit the body stays at rest.
• Kinetic (dynamic) friction acts while sliding: $f_k = \mu_k N$, usually a little smaller than the static maximum.
• $N$ is the normal reaction, the surface's push at right angles to it. On a level surface with no other vertical forces, $N = mg$.
• To a good approximation friction does not depend on the area of contact.

On a slope at angle $\theta$ to the horizontal:

• Weight component down the slope: $mg\sin\theta$.
• Weight component into the slope: $mg\cos\theta$, balanced by the normal reaction, so $N = mg\cos\theta$ (less than $mg$).
• Frictionless slope: $a = g\sin\theta$, whatever the mass.
• With kinetic friction, for a body sliding down: $a = g(\sin\theta - \mu_k\cos\theta)$.

• Centripetal force: the resultant force towards the centre needed for circular motion, $F = mv^2/r = m\omega^2 r$.
• It is supplied by a real force: string tension, friction, gravity, a normal reaction.
• It is at right angles to the velocity, so it changes the direction but not the speed, and does no work.
• If it vanishes, the body moves off in a straight line along the tangent. There is no outward "centrifugal" force on the body in a free-body diagram.

• Law of gravitation: $F = Gm_1m_2/r^2$, where $r$ is the distance between the centres and $G \approx 6.67 \times 10^{-11}\ \text{N m}^2\text{ kg}^{-2}$.
• The force on each mass is the same size (a third-law pair).
• It is an inverse-square law: $F \propto 1/r^2$.
• At distance $r$ from the centre of a planet of mass $M$: $g = GM/r^2$. At a height $h$ above a planet of radius $R$, $r = R + h$, so $g_h = g\,R^2/(R + h)^2$.