Static and Kinetic Friction: Principles, Applications, and Problem Solving

Part 1 | Space, Time, and Motion

Equilibrium and Force Diagrams

Understanding the conditions for equilibrium is fundamental in classical mechanics. When an object is at rest or moving with constant velocity, the net force acting on it is zero.

• Equilibrium Conditions:

  • $\sum F_x = 0$ (Sum of forces in the x-direction is zero)

  • $\sum F_y = 0$ (Sum of forces in the y-direction is zero)

• Free-Body Diagram:

  • Shows all forces acting on an object (e.g., normal force $F_N$, applied forces $F_{A1}$, $F_{A2}$, friction $F_f$, weight $F_W$).

  • At rest: $\vec{v} = 0$, $\vec{a} = 0$

Part 15 | Static and Kinetic Friction

Introduction to Friction

Friction is a resistive force that opposes the relative motion or attempted motion between two surfaces in contact. It is classified as either static (preventing motion) or kinetic (opposing ongoing motion).

• Static Friction ($F_{fs}$): The force that must be overcome to start moving an object at rest.

• Kinetic Friction ($F_{fk}$): The force that opposes the motion of an object already in motion.

Factors Affecting and Not Affecting Friction

The magnitude of friction depends on certain physical properties and conditions.

• Affecting Friction:

  • Material/Surfaces: The types of materials in contact determine the coefficient of friction.

  • Normal Force ($F_N$): The perpendicular force pressing the surfaces together.

• Not Affecting Friction:

  • Surface Area: The area of contact does not significantly affect friction for most macroscopic cases.

  • Speed: For dry friction, speed is not a major factor (except at very high speeds or for lubricated surfaces).

Friction Force Equations

The frictional force is proportional to the normal force and depends on the coefficient of friction.

• General Formula: $F_f = k F_N$ where $k$ is the coefficient of friction.

• Static Friction: $F_{fs} \leq \mu_s F_N$ $\mu_s$ is the coefficient of static friction.

• Kinetic Friction: $F_{fk} = \mu_k F_N$ $\mu_k$ is the coefficient of kinetic friction.

Example: Refrigerator Problem

When pushing a refrigerator with increasing force, the friction force matches the applied force up to a maximum value (static friction). Once exceeded, the refrigerator moves and kinetic friction applies.

• At rest: Friction force equals the applied force, up to the maximum static friction.

• When motion starts: Friction force drops to the kinetic value, which is typically less than the maximum static friction.

• Example:

  • Push with 50 N, 100 N, 150 N: Friction force = applied force (object does not move).

  • Push with 200 N: Object moves, friction force = kinetic friction (less than 200 N).

Graphical Representation of Friction

The relationship between applied force and friction force can be visualized in a graph:

• Static Friction Region: Friction force increases linearly with applied force, up to a maximum.

• Kinetic Friction Region: Once motion starts, friction force becomes nearly constant.

Microscopic Origin of Friction

Friction arises from microscopic interactions between surfaces, including roughness and adhesive forces.

• Contact Points: Microscopic peaks and valleys interlock, resisting motion.

• Direction: Friction always acts opposite to the direction of motion or attempted motion.

Coefficients of Friction: Table and Comparison

The coefficient of friction varies depending on the materials in contact. The following table summarizes typical values:

Surface Pair	Coefficient of Static Friction ($\mu_s$)	Coefficient of Kinetic Friction ($\mu_k$)
Wood on wood	0.4	0.2
Ice on ice	0.1	0.03
Metal on metal (lubricated)	0.15	0.06
Steel on steel (unlubricated)	0.74	0.57
Rubber on dry concrete	1.0	0.8
Rubber on wet concrete	0.3	0.15
Teflon on Teflon in air	0.04	0.04
Lubricated ball bearings	0.16	0.16
Synovial joints (human limbs)	0.01	0.01

Additional info: Table values are approximate and may vary with conditions.

Problem Solving with Friction

Solving friction problems involves identifying all forces, drawing free-body diagrams, and applying Newton's laws.

• Example 1: Dresser Problem

  • Given: $m = 45$ kg, $\mu_s = 0.77$, $\mu_k = 0.59$, $F_A = 325$ N

  • Calculate normal force: $F_N = mg = 45 \times 9.8 = 441$ N

  • Maximum static friction: $F_{fs, max} = \mu_s F_N = 0.77 \times 441 = 339.57$ N

  • Since $F_A < F_{fs, max}$, the dresser does not move.

• Example 2: Car Braking Problem

  • Given: $\mu_k = 0.72$, $g = 9.8$ m/s$^2$

  • Maximum deceleration: $a = -\mu_k g = -0.72 \times 9.8 = -7.056$ m/s$^2$

  • Use kinematic equations to find stopping distance or time.

• Example 3: Crate in Truck

  • Truck accelerates from rest to 21.6 m/s in 13.5 s

  • Acceleration: $a = \frac{21.6}{13.5} = 1.6$ m/s$^2$

  • Minimum $\mu_s$ to prevent sliding: $\mu_s = \frac{a}{g} = \frac{1.6}{9.8} = 0.163$

Summary Table: Static vs. Kinetic Friction

Type	Symbol	Equation	When Applies
Static Friction	$F_{fs}$	$F_{fs} \leq \mu_s F_N$	Object at rest
Kinetic Friction	$F_{fk}$	$F_{fk} = \mu_k F_N$	Object in motion

Key Points

• Friction opposes motion and is proportional to the normal force.

• Static friction prevents motion up to a maximum value; kinetic friction acts during motion and is usually less than maximum static friction.

• Coefficients of friction depend on material pairs and are determined experimentally.

• Free-body diagrams are essential for analyzing forces and solving friction problems.

Applications

• Designing tires and road surfaces for optimal grip.

• Calculating safe acceleration and braking distances for vehicles.

• Engineering moving parts to minimize unwanted friction.

Additional info: These principles are foundational for further study in mechanics, engineering, and applied physics.