Topic 2.7 Kinetic and Static Friction: model kinetic friction as proportional to the normal force, treat static friction as adjustable up to a maximum, and apply both to decide whether and how an object slides.

What this topic is asking

The College Board (Topic 2.7) wants you to model kinetic friction as proportional to the normal force, to treat static friction as a self-adjusting force up to a maximum, and to use both to decide whether an object slides and how it accelerates. Friction is the force that most often completes a free-body diagram, and a classic exam move is to compare an applied force with the static maximum to decide which friction model applies.

Kinetic friction

Kinetic friction has a fixed magnitude once an object slides: it depends only on the normal force and the surfaces, not on how fast the object moves or on the contact area. The normal force is whatever the perpendicular equilibrium requires, $N = mg$ on level ground, but $N = mg\cos\theta$ on an incline, or modified by a vertical component of an applied force. Always solve for $N$ first, because $f_k$ depends on it.

Static friction

Static friction acts when surfaces are in contact but not sliding. Unlike kinetic friction, it is not a fixed value: it adjusts itself to exactly oppose whatever force is trying to start the motion, up to a maximum. The maximum is

and while the object stays put, $f_s$ equals the applied force, which may be anywhere from zero up to that ceiling. The instant the applied force exceeds $\mu_s N$, the object breaks free and begins to slide, at which point kinetic friction (usually smaller, since $\mu_s > \mu_k$) takes over. This is why it takes a bigger push to start an object moving than to keep it moving.

Will it move?

The standard procedure for any friction problem: draw the free-body diagram, find the normal force from the perpendicular balance, and compute $f_{s,max} = \mu_s N$. Compare the net force tending to cause sliding (an applied push, or the gravity component down an incline) with $f_{s,max}$. If it is smaller, the object stays in equilibrium and static friction equals the driving force. If it is larger, the object slides, and you switch to $f_k = \mu_k N$ to find the net force and the acceleration. On an incline, the block begins to slide when $mg\sin\theta > \mu_s mg\cos\theta$, that is when $\tan\theta > \mu_s$.

Try this

Q1. A $10$ kg crate needs a horizontal force of $40$ N to start sliding on a level floor. Calculate $\mu_s$ ($g = 9.8$ m/s squared). [2 points]

• Cue. At the verge, $40 = \mu_s mg = \mu_s(10)(9.8)$, so $\mu_s = 40/98 = 0.41$.

Q2. State the minimum incline angle at which a block with $\mu_s = 0.30$ begins to slide. [2 points]

• Cue. It slides when $\tan\theta > \mu_s$, so $\theta = \tan^{-1}(0.30) = 17^\circ$.