Kinetic and Static Friction

Learning Goals

After this lesson, you should be able to:

• distinguish static and kinetic friction from the relative motion at an interface;
• determine friction direction without relying only on an object's velocity;
• apply \(f_s\le\mu_sN\) and \(f_k=\mu_kN\) correctly;
• calculate the normal force before calculating friction;
• analyze horizontal and inclined-surface problems;
• interpret the transition from sticking to sliding on a force graph; and
• design a procedure to measure friction coefficients with uncertainty.

1. Friction Describes an Interface

Friction is exerted by one surface on another and lies parallel to their common interface. Its direction opposes their relative sliding or the sliding that would occur without friction.

This is more precise than saying “friction always opposes motion.” A car can accelerate forward because the road exerts forward static friction on a drive tire that would otherwise slip backward at the contact point.

2. Static Friction Adjusts

When two surfaces do not slide relative to one another, static friction takes whatever value is required by the force equations up to a maximum:

If a \(20\,\mathrm N\) horizontal push leaves a box at rest and the maximum available static friction is \(50\,\mathrm N\), the actual friction is \(20\,\mathrm N\), not \(50\,\mathrm N\).

3. Kinetic Friction During Sliding

In the simple dry-sliding model,

The dimensionless coefficient \(\mu_k\) depends on the pair of materials and their surface condition. For many dry material pairs, \(\mu_k<\mu_s\), so maintaining sliding requires less force than initiating it.

5. Determine Direction Before Magnitude

1. Identify the two surfaces in contact.
2. Imagine the relative sliding at the interface if friction were removed.
3. Draw friction on each surface opposite that relative sliding tendency.
4. Use static or kinetic magnitude rules only after deciding whether the surfaces stick or slide.

The friction forces on the two surfaces form a Newton's-third-law pair: equal magnitude, opposite direction, different receivers.

6. Normal Force Comes from the Force Equation

The normal force \(N\) is the perpendicular contact force exerted by a surface. It is not automatically \(mg\). Calculate it from the perpendicular component of Newton's second law:

On a horizontal surface with no other vertical forces and \(a_y=0\), \(N=mg\). An angled pull, angled push, vertical acceleration, or incline changes this relationship.

7. Worked Example: Does the Box Move?

Question: A \(10.0\,\mathrm{kg}\) box rests on a horizontal floor with \(\mu_s=0.50\) and \(\mu_k=0.30\). A \(40\,\mathrm N\) horizontal force is applied. Does it move?

Only \(40\,\mathrm N\) of static friction is required, which is below the maximum.

Answer: The box remains at rest, and the actual static friction is \(40\,\mathrm N\) opposite the applied force.

8. Worked Example: After Sliding Begins

Question: Apply \(60\,\mathrm N\) horizontally to the same box. Find its acceleration after it begins sliding.

Because \(60\,\mathrm N>49\,\mathrm N\), static friction cannot prevent motion. Then

Answer: The box accelerates at approximately \(3.1\,\mathrm{m\,s^{-2}}\) in the applied-force direction.

9. Angled Pull Changes Friction

A force \(F\) pulling at angle \(\theta\) above horizontal reduces the normal force when vertical acceleration is zero:

Thus kinetic friction becomes \(f_k=\mu_k(mg-F\sin\theta)\). Pushing downward at an angle instead increases \(N\) and therefore increases the modeled friction magnitude.

10. Friction on an Incline

Choose axes parallel and perpendicular to a ramp of angle \(\theta\). With no acceleration perpendicular to the surface,

The downslope component of weight is \(mg\sin\theta\). Static friction can hold the object only if

At the critical angle for impending downward sliding, \(\mu_s=\tan\theta_c\).

11. Worked Example: Sliding Down a Ramp

Question: A block slides down a \(30^\circ\) incline with \(\mu_k=0.20\). Find its acceleration along the ramp.

Answer: The block accelerates down the ramp at approximately \(3.2\,\mathrm{m\,s^{-2}}\). Its mass cancels within this simple model.

12. Contact Area and Model Limits

For the AP simple-friction model, \(f_k=\mu_kN\) does not depend on the apparent contact area. Turning the same rigid block onto another face leaves \(N\) and the predicted friction unchanged.

This is an empirical approximation. Soft tires, lubricated surfaces, adhesives, high speeds, deformation, and fluid resistance can require more detailed models. Coefficients should be measured for the actual material pair and conditions.

13. Friction Can Enable Motion

Friction does not always slow an object's center of mass. Static friction can:

• push a walking person forward;
• accelerate a rolling drive wheel without slipping;
• hold a book against a tilted surface;
• allow an object to accelerate with a moving conveyor belt.

The correct direction comes from the impending or actual relative motion at the interface, not from a memorized “opposite velocity” rule.

14. Investigation: Measure \(\mu_s\) and \(\mu_k\)

Place a block on an adjustable incline and increase the angle slowly.

1. Record the angle \(\theta_c\) at which the block first begins to slide.
2. Estimate \(\mu_s=\tan\theta_c\) and repeat enough trials to report variation.
3. Adjust the ramp until the block slides at approximately constant velocity.
4. At that angle, use \(\mu_k=\tan\theta_k\).
5. Repeat with added mass and with the block on a different face.
6. Compare coefficients using angular uncertainty and discuss surface wear or contamination.

Use a catch barrier so the block cannot leave the apparatus.

15. Common Traps and Key Takeaways

• Do not set every static friction force to \(\mu_sN\): that is only its maximum.
• Check whether surfaces slide relative to each other before selecting static or kinetic friction.
• Calculate \(N\) from perpendicular dynamics; it is not universally equal to weight.
• Friction opposes relative sliding, not necessarily the object's velocity or acceleration.
• Coefficients are dimensionless and depend on the material pair and conditions.
• Apparent contact area is absent from the simple dry-friction equations.
• Keep vectors and magnitudes distinct: the formulas for \(f_s\) and \(f_k\) give magnitudes after direction is established.