Physics 211: Module 08 – Friction

Friction Between Surfaces

Introduction to Friction

When two surfaces are in contact and move (or attempt to move) relative to each other, a force known as friction acts at the interface. Friction opposes the relative motion and acts parallel to the surfaces in contact.

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

• Applied Force (F): The external force attempting to move the object.

• Friction Force (f): The force that opposes the applied force, acting parallel to the surface.

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

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

The Friction Coefficient

Definition and Properties

The coefficient of friction (μ) quantifies the amount of friction between two surfaces. It is defined as the ratio of the magnitude of the friction force to the normal force:

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

• μ is empirical and depends on the materials and surface conditions.

• Higher μ means stronger friction.

• μ can be affected by external factors such as surface roughness, cleanliness, and pressure.

Example: Reducing tire pressure increases the contact area, which can reduce ground pressure and affect the effective friction ("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 to match the applied force up to a maximum value:

$|f_s| \leq \mu_s N$

• As long as the applied force is less than the maximum static friction, the object remains at rest.

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

Kinetic Friction

Kinetic friction acts when an object is sliding over a surface. It is generally less than the maximum static friction and is given by:

$f_k = \mu_k N$

• μk is the coefficient of kinetic friction.

• Kinetic friction is usually constant for a given pair of surfaces and does not depend on velocity (for moderate speeds).

Comparison of Static and Kinetic Friction

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

Example: A box on a table will not move until the applied force exceeds the maximum static friction. Once moving, the friction force drops to the kinetic value.

Applications: ABS Brakes

Why Do ABS Brakes Pulse?

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

• Rolling wheels: The point of contact with the ground is at rest, allowing for higher static friction ($f_s > f_k$).

• Locked wheels: The tires slide, and only kinetic friction applies, which is lower.

• ABS helps maintain control and reduce stopping distance by keeping the tires in the static friction regime.

Modeling Friction: Validity and Limitations

How Good is the Friction Model?

• The friction model ($f = \mu N$) is empirical and not as fundamental as Newton's laws.

• It works well for many practical situations but may not be accurate at very high speeds, very small scales, or under unusual conditions (e.g., quantum effects).

• Always check if the model applies to your specific problem.

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 weight.

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

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

• Normal force equals the 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: $v^2 = 2a(x - x_0)$

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

Conclusion: Stopping distance is independent of mass and depends only on initial speed, friction coefficient, and gravity.

Box Sliding on an Inclined Plane

A box will start to slide down an incline when the component of gravity parallel to the slope exceeds the maximum static friction.

• Parallel component: $mg \sin \theta$

• 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, $\tan \theta = \mu_s$

• Box slides when $\theta > \tan^{-1}(\mu_s)$

Acceleration Down the Incline (After Sliding Starts)

• 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: $F \cos \theta - f_k = ma$

To move at constant velocity: $F = \frac{\mu_k mg}{\cos \theta + \mu_k \sin \theta}$

Summary Table: Static vs. Kinetic Friction

Property	Static Friction	Kinetic Friction
Symbol	$f_s$	$f_k$
Coefficient	$\mu_s$	$\mu_k$
Formula	$|f_s| \leq \mu_s N$	$f_k = \mu_k N$
When it acts	Object at rest	Object in motion
Typical magnitude	Greater	Smaller

Additional info: The notes have been expanded with standard academic context, including definitions, formulas, and worked examples, to ensure completeness and clarity for exam preparation.