Kinetic and Static Friction - AP Physics C: Mechanics Study Guide

Getting Started

We consider a system of two objects in contact, where microscopic interactions at their interface create a resistive force. This force, known as friction, opposes the relative motion or tendency of motion between the surfaces. The core question is how to mathematically model this complex interaction to predict the dynamics of the system, both when it is held in static equilibrium and when it is sliding.

What You Should Be Able to Do

Upon completing this section, you should be able to:

• Construct and solve the vector equations of motion ($\sum \vec{F}=m\vec{a}$) for systems involving both static and kinetic friction.

• Determine the threshold conditions for an object to transition from a static to a kinetic state.

• Analyze systems where the normal force, and consequently the friction force, is a function of position or another variable, requiring the use of integration to find quantities like work.

• Set up and interpret differential equations of motion for objects subject to velocity-dependent forces, including friction, to predict their trajectory over time.

Key Concepts & Mechanisms

This section analyzes friction through the lens of Dynamics as Cause, where forces dictate the resulting motion of a system according to fundamental laws.

System & Preconditions

The system consists of a rigid body of mass $m$ in contact with a surface. We make several key idealizations for the standard model of dry friction:

1. Rigid Body: The object does not deform under applied forces.

2. Empirical Model: The model is not derived from first principles but is an empirical approximation of complex surface interactions.

3. Independence of Area and Velocity: The friction force is assumed to be independent of the macroscopic contact area and, for kinetic friction, independent of the relative speed between the surfaces.

Key Steps / Relations

The analysis of any system with friction follows a clear, logical progression rooted in Newton's Second Law.

1. System Representation: Begin by drawing a Free-Body Diagram (FBD) for the object of interest. Identify all forces acting on it: gravitational force (${\vec{F}}_g$), applied forces (${\vec{F}}_{app}$), the normal force from the surface (${\vec{F}}_N$), and the friction force (${\vec{F}}_f$). Choose a coordinate system, typically with one axis perpendicular and one parallel to the surface of contact.

2. Governing Law (Perpendicular): Apply Newton's Second Law for the components perpendicular to the surface. Since the object is not accelerating into or out of the surface, the net force in this direction is zero.

   $\sum F_{\perp }=0$

   This equation is always used to solve for the magnitude of the normal force, $F_N$. Note that $F_N$ is not universally equal to $mg$; its value depends on other forces with components perpendicular to the surface (e.g., on an incline or with an angled applied force).

3. Governing Law (Parallel) & Friction Model: Apply Newton's Second Law for the components parallel to the surface. The friction force ${\vec{F}}_f$ is included here. The specific model for ${\vec{F}}_f$ depends on the state of motion.

   $\sum F_{\parallel }=m{a}_{\parallel }$

   • Case 1: Static Friction ($\vec{v}=0$)

     If the object is at rest, the static friction force, ${\vec{F}}_{f,s}$, is a responsive force. It adjusts its magnitude and direction to oppose the net applied force parallel to the surface, maintaining equilibrium ($a_{\parallel }=0$). Its magnitude is bounded by a maximum value determined by the coefficient of static friction, ${\mu }_s$, and the normal force.

     $|{\vec{F}}_{f,s}|\le {\mu }_s{F}_N$

     To solve a problem, you first assume the system is in equilibrium and solve for the required $F_{f,s}$. You then check if this required force is less than or equal to the maximum possible value, ${\mu }_s{F}_N$. If it is, the object remains static. If it is not, the object will begin to slide.

   • Case 2: Kinetic Friction ($\vec{v}\ne 0$)

     If the object is sliding, the friction force becomes kinetic friction, ${\vec{F}}_{f,k}$. Its magnitude is constant and given by the coefficient of kinetic friction, ${\mu }_k$. Its direction always opposes the object's velocity vector, $\vec{v}$.

     $|{\vec{F}}_{f,k}|={\mu }_k{F}_N$

     This value is substituted directly into the parallel component of Newton's Second Law to solve for the object's acceleration.

Outputs & Effects

The output of this analysis is a prediction of the object's motion.

• Static Equilibrium: The object remains at rest because the net applied parallel force is insufficient to overcome the maximum static friction.

• Impending Motion: This is the critical state where $|{\vec{F}}_{f,s}|={\mu }_s{F}_N$. Any infinitesimal increase in the driving force will initiate sliding.

• Kinetic Motion: The object slides with an acceleration determined by the net force, which includes the constant kinetic friction force. This can lead to constant acceleration (if other forces are constant) or non-uniform acceleration if the normal force or other applied forces vary.

Regulation & Limits

The standard friction model is an approximation. In reality, friction can depend on velocity, and the transition from static to kinetic is not instantaneous. A key feature of this model is that, generally, ${\mu }_s>{\mu }_k$. This means a larger force is required to start an object moving than to keep it moving. The static condition, $|{\vec{F}}_{f,s}|\le {\mu }_s{F}_N$, acts as a stability condition. The system remains in equilibrium as long as the state resides within the bounds of this inequality.

Key Models & Diagrams

The decision process for applying the correct friction model can be visualized as a flowchart.

Friction Analysis Flowchart

graph TD

    A[Start: Analyze an object on a surface] --> B{Is the object sliding?};

    B -- No --> C[Assume static equilibrium: a = 0];

    C --> D[Use ΣF_parallel = 0 to find the required static friction, F_fs_req];

    D --> E{Is F_fs_req ≤ μs * F_N?};

    E -- Yes --> F[Conclusion: Object remains static. The actual friction force is F_fs_req.];

    E -- No --> G[Conclusion: Object slips. The static assumption was wrong.];

    G --> H;

    B -- Yes --> H[Object is in motion. Use the kinetic friction model.];

    H --> I[Set friction force magnitude: |F_fk| = μk * F_N];

    I --> J[Direction of F_fk opposes velocity vector v];

    J --> K[Use ΣF_parallel = m*a_parallel to find the object's acceleration.];

Key Components & Evidence

• Normal Force (${\vec{F}}_N$): The perpendicular contact force exerted by a surface on an object. Its magnitude is determined by the condition $\sum F_{\perp }=0$. Units: Newtons (N).

• Static Friction Force (${\vec{F}}_{f,s}$): The contact force that prevents relative motion between surfaces. It is a variable force, opposing the tendency of motion, up to a maximum value. Units: Newtons (N).

• Kinetic Friction Force (${\vec{F}}_{f,k}$): The contact force that opposes relative motion between surfaces that are sliding past one another. It has a constant magnitude for a given normal force. Units: Newtons (N).

• Coefficient of Static Friction (${\mu }_s$): A dimensionless scalar property of two interacting surfaces that determines the maximum static friction force: $|{\vec{F}}_{f,s}{|}_{max}={\mu }_s{F}_N$.

• Coefficient of Kinetic Friction (${\mu }_k$): A dimensionless scalar property of two interacting surfaces that determines the kinetic friction force: $|{\vec{F}}_{f,k}|={\mu }_k{F}_N$. Typically, ${\mu }_k<{\mu }_s$.

• Newton's Second Law ($\sum \vec{F}=m\vec{a}$): The fundamental governing law of classical dynamics. It is applied in component form to solve for unknown forces (like $F_N$ or $F_{f,s}$) or acceleration.

• Work Done by Friction ($W_f$): Since friction is a non-conservative force, the work it does depends on the path taken. It is calculated by the line integral $W_f=\int {\vec{F}}_f\cdot d\vec{r}$. For kinetic friction along a path, this often simplifies to $W_f=-\int {\mu }_k{F}_N(s)ds$, where $s$ is the path length.

Skill Snapshots

Causation

• Driver → Change: An applied force parallel to a surface increases → The responsive static friction force increases to maintain equilibrium, up to its maximum limit.

• Driver → Change: The net parallel force exceeds the maximum static friction (${\mu }_s{F}_N$) → The object transitions from static equilibrium to kinetic motion (acceleration).

• Driver → Change: An object is sliding with velocity $\vec{v}$ → A kinetic friction force ${\vec{F}}_{f,k}$is generated in the $-\vec{v}$ direction, causing a change in the object's momentum.

Comparison

• Static vs. Kinetic Friction: Static friction is a variable force described by an inequality ($|{\vec{F}}_{f,s}|\le {\mu }_s{F}_N$), while kinetic friction is a constant magnitude force described by an equality ($|{\vec{F}}_{f,k}|={\mu }_k{F}_N$).

• Normal Force on Horizontal vs. Inclined Plane: For a block on a horizontal surface, $F_N=mg$. For the same block on an incline of angle $\theta$, the normal force is reduced to $F_N=mg\cos \theta$, which in turn reduces the friction force.

• Work by Friction vs. Work by Gravity: The work done by kinetic friction is always negative and path-dependent (non-conservative). The work done by gravity is path-independent and depends only on vertical displacement (conservative).

Change, Continuity, and Order

• Baseline: An object is at rest on a horizontal surface. The static friction force is zero.

• Change 1: A small horizontal force is applied. The static friction force grows to be equal and opposite to the applied force, and the object remains at rest.

• Change 2: The applied force is increased beyond the threshold ${\mu }_s{F}_N$. The object begins to accelerate, and the friction force abruptly drops in magnitude to the kinetic value, ${\mu }_k{F}_N$.

• Continuity: Throughout this entire process, assuming no vertical acceleration, the normal force exerted by the surface on the object remains constant.

Common Misconceptions & Clarifications

1. Misconception: The normal force is always equal to the object's weight, $mg$.

   Clarification: The normal force, $F_N$, is the value that makes the net force perpendicular to the surface zero. It equals $mg$ only on a horizontal surface with no other vertical forces. On an incline or with angled applied forces, $F_N$ will be different.

2. Misconception: The static friction force is always equal to ${\mu }_s{F}_N$.

   Clarification: This is the maximum possible static friction. The actual static friction force is a responsive force that takes on whatever value is necessary to prevent motion, from zero up to this maximum. You must solve for the required force first.

3. Misconception: Friction always opposes motion.

   Clarification: Friction opposes relative motion or the tendency of relative motion between surfaces. Static friction on the tires of an accelerating car actually points in the same direction as the car's motion, preventing the tires from slipping backward relative to the road.

4. Misconception: The coefficients of friction are universal constants.

   Clarification: The values of ${\mu }_s$ and ${\mu }_k$ are empirical properties that depend on the specific pair of materials in contact (e.g., wood on steel, rubber on asphalt) and the condition of the surfaces. They are not fundamental constants.

One-Paragraph Summary

Friction is a contact force that resists the relative motion or tendency of motion between surfaces, modeled in two distinct regimes: static and kinetic. Static friction is a variable, responsive force that prevents motion, governed by the inequality $|{\vec{F}}_{f,s}|\le {\mu }_s{F}_N$, where ${\mu }_s$ is the coefficient of static friction and $F_N$ is the normal force. Once the tendency of motion overcomes this maximum, the object begins to slide, and the force transitions to kinetic friction, which has a constant magnitude $|{\vec{F}}_{f,k}|={\mu }_k{F}_N$ and always opposes the velocity vector. The analysis of friction is a core component of dynamics, requiring careful application of Newton's Second Law in component form to determine the normal force and subsequently the object's state of equilibrium or acceleration. This empirical model, while an idealization, provides powerful predictive capabilities for a vast range of mechanical systems.