Friction

Overview

Friction is a force that opposes the relative motion or tendency of motion between two surfaces in contact.

Types of Friction

Static Friction ($f_s$)

Friction between surfaces that are not moving relative to each other.

$f_s \leq \mu_s N$

• Maximum static friction: $f_{s,\max} = \mu_s N$
• Acts to prevent motion from starting
• Can take any value from 0 to $\mu_s N$
• Direction: opposite to the applied force

Kinetic Friction ($f_k$)

Friction between surfaces that are sliding relative to each other.

$f_k = \mu_k N$

• Constant magnitude (independent of speed)
• Direction: opposite to the direction of motion
• Generally $\mu_k < \mu_s$ (easier to keep moving than to start)

Coefficients of Friction

Surface Pair	$\mu_s$	$\mu_k$
Steel on steel	0.74	0.57
Rubber on concrete (dry)	1.0	0.8
Wood on wood	0.5	0.3
Ice on ice	0.1	0.03
Teflon on steel	0.04	0.04

Key Properties

• Friction is independent of contact area
• Friction is proportional to normal force
• Kinetic friction is approximately independent of velocity
• $\mu_s > \mu_k$ for most materials

Friction on a Horizontal Surface

For an object on a horizontal surface with applied force $F$:

At rest (static friction):

$f_s = F \quad \text{(if } F < \mu_s N\text{)}$

$f_s = \mu_s N \quad \text{(maximum, about to move)}$

Moving (kinetic friction):

$f_k = \mu_k N$

Normal force on horizontal surface:

$N = mg \quad \text{(no other vertical forces)}$

Friction on an Inclined Plane

For an angle $\theta$ from horizontal:

$N = mg\cos\theta$

Component of weight parallel to incline:

$W_{\parallel} = mg\sin\theta$

Static friction (not sliding):

$f_s = mg\sin\theta \quad \text{(if } \theta < \theta_{\max}\text{)}$

Maximum angle before sliding:

$\tan(\theta_{\max}) = \mu_s$

Kinetic friction (sliding):

$f_k = \mu_k mg\cos\theta$

Acceleration down the incline:

$a = g(\sin\theta - \mu_k\cos\theta)$

Rolling Friction

Much smaller than sliding friction:

$f_r = \mu_r N$

Where $\mu_r$ is typically much smaller than $\mu_k$

Examples

Example 1: Box on Floor

A 50 kg box on a floor ($\mu_s = 0.5$, $\mu_k = 0.4$). What force is needed to start moving it? What acceleration once moving with 300 N force?

$f_{s,\max} = \mu_s mg = 0.5 \times 50 \times 9.8 = 245 \text{ N (force to start)}$

Once moving:

$f_k = \mu_k mg = 0.4 \times 50 \times 9.8 = 196 \text{ N}$

$\sum F = F - f_k = 300 - 196 = 104 \text{ N}$

$a = \frac{104}{50} = 2.08 \text{ m/s}^2$

Example 2: Inclined Plane

A block on a 25° incline ($\mu_k = 0.2$). Find acceleration.

$a = g(\sin 25° - \mu_k\cos 25°)$

$a = 9.8(0.423 - 0.2 \times 0.906)$

$a = 9.8(0.423 - 0.181) = 2.37 \text{ m/s}^2$

Example 3: Critical Angle

Find the maximum angle at which a block will remain stationary ($\mu_s = 0.6$).

$\tan(\theta_{\max}) = \mu_s = 0.6$

$\theta_{\max} = \tan^{-1}(0.6) = 31°$

Motion with Friction

Stopping Distance

For initial velocity $v_0$ on a horizontal surface:

$f_k = \mu_k mg = ma$

$a = -\mu_k g$

$\text{Stopping distance: } d = \frac{v_0^2}{2\mu_k g}$

Connected Objects

For two objects connected, friction on each object must be considered separately based on its normal force.

Important Notes

• Friction always opposes relative motion
• Static friction is a response force (adjusts as needed)
• Normal force is not always equal to weight
• Friction converts kinetic energy to thermal energy