Friction: static and kinetic

Example 1: kinetic friction

Problem A 20.0 kg box slides across a floor where $\mu_k = 0.30$. What is the friction force?

1. Flat floor, so the normal force equals the weight.
   $$F_N = mg = (20.0\ \text{kg})(9.8\ \text{m/s}^2) = 196\ \text{N}$$
2. Kinetic friction.
   $$f_k = \mu_k F_N = (0.30)(196\ \text{N}) = 58.8\ \text{N}$$
3. $\mu_k$ has 2 significant figures, so round to 2.

Answer 59 N, opposite the sliding

Example 2: does it move?

Problem A 30.0 kg crate sits on a floor with $\mu_s = 0.50$. You push it sideways with 120 N. Does it move? What is the friction force?

1. Find the most static friction can give.
   $$f_{s,\max} = \mu_s mg = (0.50)(30.0\ \text{kg})(9.8\ \text{m/s}^2) = 147\ \text{N}$$
2. Your 120 N push is less than 147 N, so static friction holds the crate still.
3. Static friction only matches the push. It does not use its full maximum.
   $$f_s = 120\ \text{N}$$

Answer It does not move, and friction is 120 N

Example 3: once it slides

Problem You push the same 30.0 kg crate with 200. N. The kinetic coefficient is $\mu_k = 0.40$. What is its acceleration?

1. 200. N is more than the 147 N static maximum, so the crate starts to slide.
2. Kinetic friction.
   $$f_k = (0.40)(30.0)(9.8) = 117.6\ \text{N}$$
3. Net force and acceleration.
   $$a = \frac{200.\ \text{N} - 117.6\ \text{N}}{30.0\ \text{kg}} = 2.75\ \text{m/s}^2$$

Answer 2.7 m/s²

Example 4: measuring $\mu_s$ with a ramp

Problem A block on a board starts to slide when the board is tilted to 31°. What is $\mu_s$?

1. Right at the slipping point, the pull down the slope equals the maximum static friction.
   $$mg\sin\theta = \mu_s\, mg\cos\theta$$
2. Mass and g cancel.
   $$\mu_s = \tan\theta = \tan 31^\circ = 0.601$$

Answer $\mu_s = 0.60$