Module 08: Friction – Physics 211 Study Notes

Friction Between Surfaces

Introduction to Friction

Friction is a force that arises when two surfaces move or attempt to move across each other. It acts parallel to the surfaces in contact and generally opposes the direction of applied force.

• Normal Force (N): The perpendicular force exerted by a surface on an object in contact with it.

• Applied Force (F): The force applied to move an object.

• Friction Force (f): The force that resists the relative motion of surfaces.

• Gravitational Force (mg): The weight of the object acting downward.

Example: When a car turns, friction from the ground on the tires provides the force necessary for the car to change direction (accelerate).

The Friction Coefficient

Definition and Properties

The coefficient of friction (μ) quantifies the ratio of the frictional force to the normal force between two surfaces:

$\mu = \frac{|f|}{N}$

• μ is empirical—it must be measured experimentally for each pair of materials.

• Higher μ means stronger frictional resistance.

• μ depends on surface materials and external factors (e.g., pressure, surface conditions).

Example: Lowering tire pressure increases the contact area, reducing ground pressure and potentially increasing grip, even if the materials remain the same.

Static and Kinetic Friction

Static Friction

Static friction prevents an object from starting to move. It adjusts up to a maximum value to oppose applied forces:

$|f_s| \leq \mu_s N$

• Static friction (f_s): The frictional force when the object is at rest.

• When the applied force is less than the maximum static friction, the object does not move.

• Once the applied force exceeds this maximum, the object begins to move.

Kinetic Friction

Once an object is moving, kinetic friction acts to oppose its motion. The kinetic friction force is generally less than the maximum static friction:

$f_k = \mu_k N$

• Kinetic friction (f_k): The frictional force when the object is sliding.

• μk (kinetic) is usually less than μs (static).

Example: A box at rest requires a larger force to start moving than to keep it moving at constant speed.

Comparison of Static and Kinetic Friction

Type	Symbol	Formula	When it Applies
Static	fs	$|f_s| \leq \mu_s N$	Object at rest
Kinetic	fk	$f_k = \mu_k N$	Object in motion

Applications: ABS Brakes

Why Do ABS Brakes Pulse?

Anti-lock Braking Systems (ABS) pulse the brakes to prevent the wheels from locking. This keeps the wheels rolling, maximizing the friction force between the tires and the road.

• When wheels are rolling, the point of contact with the ground is at rest relative to the ground.

• Static friction ($f_s$) is greater than kinetic friction ($f_k$), so rolling wheels provide more stopping force.

• ABS allows the driver to maintain steering control while braking.

Example: Brake pads work better when the wheels are rolling because the friction force from the ground is higher ($f_s > f_k$).

Modeling Friction: Validity and Limitations

How Good is the Friction Model?

• The friction model is not as fundamental as Newton's laws but works well in most practical situations.

• It provides a general idea of how friction behaves, though it may not be exact for all materials or conditions.

• At very high speeds or in quantum-scale systems, more complex models may be needed.

Worked Examples

Pushing a Box Against a Wall

To keep a box from sliding down a wall, the applied force must generate enough normal force so that static friction can balance the box's weight.

• Sum of forces in the vertical direction: $f_s = mg$

• Maximum static friction: $f_s = \mu_s N$

• Normal force equals applied force: $N = F$

• Therefore, $mg = \mu_s F$ so $F = \frac{mg}{\mu_s}$

Braking Distance

When a vehicle brakes and the wheels lock, kinetic friction determines the stopping distance. The stopping distance does not depend on the mass of the vehicle.

• Friction force: $f_k = \mu_k N = \mu_k mg$

• Newton's 2nd law: $ma = -\mu_k mg \implies a = -\mu_k g$

• Kinematic equation: $v^2 = 2a(x - x_0)$

• Stopping distance: $x - x_0 = \frac{v^2}{2\mu_k g}$

Example: Both a car and a truck traveling at the same speed will have the same stopping distance if their wheels are locked and the same coefficient of kinetic friction applies.

Box Sliding on an Inclined Plane

The box will start sliding when the component of gravity down the slope exceeds the maximum static friction.

• Down-slope force: $mg \sin \theta$

• Normal force: $N = mg \cos \theta$

• Maximum static friction: $f_s = \mu_s N = \mu_s mg \cos \theta$

• Sliding begins when: $mg \sin \theta = \mu_s mg \cos \theta$

• Therefore, $\tan \theta = \mu_s$

• Critical angle: $\theta_c = \tan^{-1} \mu_s$

Acceleration Down the Slope (After Sliding Begins)

• Net force: $mg \sin \theta - f_k$

• Kinetic friction: $f_k = \mu_k mg \cos \theta$

• Newton's 2nd law: $ma = mg \sin \theta - \mu_k mg \cos \theta$

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

Pulling a Box at an Angle

When a force is applied at an angle, both the normal force and friction are affected.

• Horizontal component: $F \cos \theta$

• Vertical component: $F \sin \theta$ (reduces normal force)

• Normal force: $N = mg - F \sin \theta$

• Kinetic friction: $f_k = \mu_k N = \mu_k (mg - F \sin \theta)$

• Net force: $F \cos \theta - f_k = ma$

Example: Pulling a sled with a rope at an angle reduces the normal force and thus the friction, making it easier to move.

Summary Table: Types of Friction

Type	Symbol	Formula	Typical Value
Static	μs	$|f_s| \leq \mu_s N$	0.3–1.0 (varies by material)
Kinetic	μk	$f_k = \mu_k N$	0.2–0.7 (varies by material)

Additional info: The values of μs and μk are empirical and depend on the specific materials and surface conditions involved.