Static Friction

Coulomb Friction Model

$f_s \leq \mu_s N$

where:

• fs = static friction force (opposes the tendency of motion)
• μs = coefficient of static friction
• N = normal force

At the verge of motion (impending motion): fs = μsN

Kinetic (Dynamic) Friction

$f_k = \mu_k N$

where μk < μs (kinetic friction is less than maximum static friction).

Typical Friction Coefficients

Inclined Plane Problems

For a block of weight W on an incline at angle θ:

Forces parallel to incline: W sin θ (downhill) Forces perpendicular to incline: N = W cos θ

Impending motion (sliding down): $\mu_s = \tan \theta$

Friction angle: φ = arctan(μs)

• Block slides when θ > φ
• Block stays when θ < φ
• At θ = φ, motion is impending

Pulling a Block on an Incline

For a force P applied at angle α above the incline to pull a block up:

$P = \frac{W(\sin \theta + \mu_s \cos \theta)}{\cos \alpha + \mu_s \sin \alpha}$

The optimal pull angle (minimum force) occurs at α = φ = arctan(μs).

Wedge Problems

Wedges are inclined planes used to lift heavy loads. Analysis involves:

1. Draw FBD of each body (wedge and the object being lifted)
2. All friction forces oppose the direction of impending motion
3. Apply equilibrium equations to each body
4. Normal forces must be positive (compression)

Belt Friction (Flat Belt)

For a flat belt wrapped around a drum:

$\frac{T_2}{T_1} = e^{\mu \beta}$

where:

• T₂ = tension on the tight side (higher tension)
• T₁ = tension on the slack side (lower tension)
• μ = coefficient of friction between belt and drum
• β = angle of wrap in radians (NOT degrees!)

Example: A belt wraps 180° (π radians) around a drum with μ = 0.3. If T₁ = 100 N: T₂ = 100 × e^(0.3×π) = 100 × e^0.9425 = 100 × 2.566 = 256.6 N