AP Physics 1: Kinetic and Static Friction

Understanding friction is not just an academic exercise; it’s the key to explaining why objects stay put, start moving, and eventually slow down. From the tires on your car to a book sliding off a tilted desk, frictional forces dictate motion in our everyday world. Mastering the distinction between static and kinetic friction, and knowing how to calculate these forces, is a fundamental skill for solving a vast array of AP Physics 1 problems and forms the basis for more advanced engineering concepts.

The Nature of Frictional Force

Friction is a force that opposes relative motion or the attempt at motion between two surfaces in contact. It arises from electromagnetic interactions between the atoms of the two surfaces at their microscopic points of contact. A crucial first step is to understand that the frictional force depends on two factors: the nature of the surfaces and how hard they are pressed together. We quantify the "nature of the surfaces" with a unitless constant called the coefficient of friction ($\mu$). The "how hard they are pressed together" is the normal force ($F_N$), which is the perpendicular contact force exerted by a surface on an object. The maximum possible magnitude of the frictional force is proportional to this normal force, giving us the foundational relationship $f_{max}=\mu F_N$. Importantly, friction does not depend on the surface area of contact for most physics problems at this level.

Static Friction: The Force That Holds Things in Place

Static friction is the force that prevents two surfaces from starting to slide relative to each other. Its most critical characteristic is that it is a responsive or variable force. It adjusts, up to a maximum value, to exactly counteract any applied force trying to cause motion. Think of it like a stubborn, sticky boot that grips the ground until you push hard enough to make it slip.

The magnitude of static friction can be anywhere from zero up to a maximum value: $0\le f_s\le f_{s,max}$. The maximum static frictional force is given by $f_{s,max}={\mu }_s{F}_N$, where ${\mu }_s$ is the coefficient of static friction. For example, if you gently push a heavy box with a force of 10 N and it doesn’t move, the static friction force is exactly 10 N opposing you. Push with 20 N, and static friction becomes 20 N. It matches your push until you exceed its maximum possible value, ${\mu }_s{F}_N$. At that instant, the object begins to move.

Kinetic Friction: The Force of Sliding Motion

Once an object is in motion, the friction it experiences changes. Kinetic friction (or sliding friction) is the force that opposes the relative motion of two surfaces sliding past each other. Unlike static friction, kinetic friction has a nearly constant magnitude. It is given by the equation $f_k={\mu }_k{F}_N$, where ${\mu }_k$ is the coefficient of kinetic friction.

A universal rule of thumb is that for the same pair of surfaces, ${\mu }_k$ is less than ${\mu }_s$ (${\mu }_k<{\mu }_s$). This means that the kinetic frictional force is less than the maximum static frictional force. This is why it’s often harder to start pushing an object than to keep it moving. The force drops at the moment the object starts to slide, which can lead to jerky motion if the applied force isn't adjusted.

The Transition and Problem-Solving Strategy

The moment when an applied force equals the maximum static friction is the transition point from rest to motion. Your problem-solving strategy should always begin by checking for impending motion. First, calculate the maximum possible static friction: $f_{s,max}={\mu }_s{F}_N$. Then, compare this to any applied force trying to move the object parallel to the surface. If the applied force is less than $f_{s,max}$, the object remains at rest, and the actual static friction force equals the applied force. If the applied force equals or exceeds $f_{s,max}$, the object accelerates, and you immediately switch to using the kinetic friction model: $f_k={\mu }_k{F}_N$ for all subsequent motion.

Consider a 5 kg box on a horizontal surface where ${\mu }_s=0.5$ and ${\mu }_k=0.3$.

1. Normal Force: $F_N=mg=(5\text{ kg})(9.8{\text{ m/s}}^2)=49\text{ N}$.
2. Max Static Friction: $f_{s,max}={\mu }_s{F}_N=(0.5)(49\text{ N})=24.5\text{ N}$.
3. Scenario A: You push with 20 N. Since $20\text{ N}<24.5\text{ N}$, the box doesn't move. Friction is $f_s=20\text{ N}$.
4. Scenario B: You push with 30 N. Since $30\text{ N}>24.5\text{ N}$, the box accelerates. The frictional force is now kinetic: $f_k={\mu }_k{F}_N=(0.3)(49\text{ N})=14.7\text{ N}$.

Applying Friction to Inclined Planes

Inclined plane problems integrate friction with component vector analysis. The key change is calculating the normal force. On a frictionless incline, $F_N=mg\cos \theta$. When friction is present, this normal force is still $F_N=mg\cos \theta$. This value is then used in the friction equations.

The force trying to pull the object down the incline (the parallel component) is $F_{\parallel }=mg\sin \theta$. To determine if an object will slide or remain at rest, compare this $F_{\parallel }$ to the maximum static friction $f_{s,max}={\mu }_s(mg\cos \theta )$.

• If $mg\sin \theta <{\mu }_{s}mg\cos \theta$, the object is held by static friction. Simplifying, this condition is $\tan \theta <{\mu }_s$.
• If the angle $\theta$ is increased until $\tan \theta ={\mu }_s$, the object is on the verge of slipping.
• If $\tan \theta >{\mu }_s$, the object accelerates down the incline, and kinetic friction $f_k={\mu }_{k}mg\cos \theta$ acts up the incline.

Example: A block rests on a plank tilted at $20^{\circ }$. Given ${\mu }_s=0.4$, will it slip? Check: $\tan (20^{\circ })\approx 0.364$. Since $0.364<0.4$, the condition $\tan \theta <{\mu }_s$ is true. Therefore, static friction is sufficient to hold the block, and it does not slip.

Common Pitfalls

1. Using the Wrong Coefficient: The most frequent error is using ${\mu }_k$ to analyze a stationary object or ${\mu }_s$ for a sliding object. Always ask: "Is it moving?" If at rest, use the static friction model (variable up to a max). If moving, use kinetic friction (constant value).
2. Misidentifying the Normal Force: On a horizontal surface, $F_N$ often equals weight ($mg$), but this is not universal. On an incline, $F_N=mg\cos \theta$. If an extra force pushes down on or lifts the object, the normal force changes. Always solve for $F_N$ by applying Newton's Second Law in the perpendicular direction, where acceleration is zero.
3. Assuming $f=\mu N$ Always Works: Remember, $f_s={\mu }_s{F}_N$ gives only the maximum possible static friction. The actual static force is often less than this maximum. The equation $f_k={\mu }_k{F}_N$, however, does give the actual kinetic friction.
4. Ignoring the Transition: When a problem involves an object starting from rest, you must first check if the applied force overcomes static friction. Do not assume motion begins immediately. Calculate $f_{s,max}$ first as a critical step.

Summary

• Friction opposes motion and is calculated with $f=\mu F_N$, where $F_N$ is the normal force.
• Static friction ($f_s$) is variable ($0\le f_s\le {\mu }_s{F}_N$) and prevents motion from starting. It matches the applied force until it reaches its maximum.
• Kinetic friction ($f_k$) is constant ($f_k={\mu }_k{F}_N$) and acts on objects already in motion. For most materials, ${\mu }_k<{\mu }_s$.
• Problem-solving requires a two-step check: First, compare applied forces to $f_{s,max}$ to determine if motion occurs. If yes, switch to using $f_k$ for the net force analysis.
• On an incline, the normal force is $mg\cos \theta$. The object will remain at rest if $mg\sin \theta <{\mu }_{s}mg\cos \theta$, which simplifies to $\tan \theta <{\mu }_s$.
• Always carefully determine the correct normal force for the situation; it is not always equal to the object's weight.