AP Physics C Mechanics- 2.7 Kinetic and Static Friction- Study Notes- New Syllabus
AP Physics C Mechanics- 2.7 Kinetic and Static Friction – Study Notes
AP Physics C Mechanics- 2.7 Kinetic and Static Friction – Study Notes – per latest Syllabus.
Key Concepts:
- Kinetic Friction
- Magnitude of the Kinetic Friction Force
- Static Friction
- Magnitude of the Static friction
Kinetic Friction
Kinetic friction is the resistive force that acts between two surfaces that are sliding past each other. It opposes the relative motion of the surfaces and converts some of the system’s kinetic energy into thermal energy.
Direction of the Kinetic Friction Force
The kinetic friction force always acts in a direction opposite to the relative motion between the two surfaces in contact.
Key Idea:
- If a block slides to the right, friction acts to the left.
- If a car’s tires skid forward on ice, friction acts backward along the direction of motion.
This opposing direction ensures that kinetic friction reduces the relative motion between the surfaces.
Dependence on Contact Area
The magnitude of kinetic friction does not depend on the area of contact between two surfaces. Instead, it depends on the nature of the surfaces (through \( \mathrm{\mu_k} \)) and the normal force.
Explanation:
- Doubling the contact area reduces the pressure per unit area but increases total area — both effects cancel out.
- Thus, \( \mathrm{F_k} \) remains the same as long as \( \mathrm{F_N} \) and \( \mathrm{\mu_k} \) remain unchanged.
Example
When a hockey puck slides across the smooth ice of a rink, the puck continues moving forward because of its inertia, but it gradually slows down and eventually stops. This happens because the surface of the ice and the bottom of the puck exert a kinetic friction force on each other.
▶️ Answer / Explanation
Step 1: As the puck moves forward, the molecules on the ice and on the puck’s surface rub against each other, generating microscopic resistance.
Step 2: The frictional force acts opposite to the direction of motion of the puck — backward along the ice surface — slowing it down over time.
Step 3: The magnitude of this kinetic friction does not depend on the area of contact between the puck and the ice but only on the smoothness of the ice (coefficient of friction) and the normal force (the puck’s weight pressing on the ice).
Result: The puck eventually stops because the kinetic friction continuously removes its kinetic energy, converting it into heat. This illustrates that kinetic friction always acts opposite motion and is independent of the contact area.
Magnitude of the Kinetic Friction Force
The kinetic friction force opposes the motion of an object sliding across a surface. Its magnitude is directly proportional to the normal force and depends on the coefficient of kinetic friction between the two materials in contact.
Relevant Equation:
\( \mathrm{\vec{F}_{f,k} = \mu_k \vec{F}_N} \)
or in magnitude form:
\( \mathrm{F_{f,k} = \mu_k F_N} \)
- \( \mathrm{F_{f,k}} \): kinetic friction force (N)
- \( \mathrm{\mu_k} \): coefficient of kinetic friction (unitless)
- \( \mathrm{F_N} \): normal force exerted by the surface (N)
Kinetic friction depends on how strongly the two surfaces press against each other (through \( \mathrm{F_N} \)) and how “rough” or “sticky” the surfaces are (through \( \mathrm{\mu_k} \)).
Coefficient of Kinetic Friction (\( \mathrm{\mu_k} \))
The coefficient of kinetic friction is a dimensionless constant that depends on the material composition and texture of the two surfaces in contact.
- Smooth surfaces have small \( \mathrm{\mu_k} \) (e.g., ice on steel: 0.03).
- Rough surfaces have larger \( \mathrm{\mu_k} \) (e.g., rubber on concrete: 0.8).
Note: The coefficient of kinetic friction is usually less than the coefficient of static friction (\( \mathrm{\mu_k < \mu_s} \)) because less force is required to keep an object moving than to start its motion.
The Normal Force (\( \mathrm{F_N} \))
The normal force is the perpendicular contact force exerted by a surface on an object in contact with it. It acts away from the surface and balances the component of other forces perpendicular to that surface.
- For a horizontal surface: \( \mathrm{F_N = mg} \)
- For an inclined plane: \( \mathrm{F_N = mg\cos\theta} \)
Physical Meaning: The stronger the normal force pressing the surfaces together, the greater the frictional resistance between them.
Example
A \( \mathrm{15\,kg} \) box slides across a floor with a coefficient of kinetic friction \( \mathrm{\mu_k = 0.25.} \) Find the frictional force acting on the box.
▶️ Answer / Explanation
Step 1: Find the normal force:
\( \mathrm{F_N = mg = (15)(9.8) = 147\,N.} \)
Step 2: Use \( \mathrm{F_{f,k} = \mu_k F_N.} \)
\( \mathrm{F_{f,k} = (0.25)(147) = 36.75\,N.} \)
Step 3: Direction:
Opposite the direction of motion.
Result: The kinetic friction force is \( \mathrm{36.8\,N} \) directed opposite to the motion of the box.
Example
A \( \mathrm{5.0\,kg} \) block slides down a \( \mathrm{30^\circ} \) incline. The coefficient of kinetic friction is \( \mathrm{\mu_k = 0.20.} \) Find the frictional force acting on the block.
▶️ Answer / Explanation
Step 1: Find the normal force on the incline:
\( \mathrm{F_N = mg\cos\theta = (5)(9.8)\cos(30^\circ) = 42.4\,N.} \)
Step 2: Calculate kinetic friction:
\( \mathrm{F_{f,k} = \mu_k F_N = (0.20)(42.4) = 8.5\,N.} \)
Step 3: Direction:
Up the incline (opposite motion).
Result: The kinetic friction force on the block is \( \mathrm{8.5\,N} \), acting up the incline.
Nature of Static Friction
Static friction is the resistive force that acts between two surfaces that are not moving relative to each other. It prevents motion from starting when an external force is applied but is not yet large enough to overcome this frictional resistance.
\( \mathrm{F_{s} \leq \mu_s F_N} \)
- \( \mathrm{F_s} \): static frictional force (N)
- \( \mathrm{\mu_s} \): coefficient of static friction (unitless)
- \( \mathrm{F_N} \): normal force (N)
Key Idea:
- Static friction adjusts its magnitude to exactly balance the applied force up to a maximum value.
- When the applied force exceeds this maximum, motion begins, and kinetic friction takes over.
Maximum Static Friction:
\( \mathrm{F_{s,max} = \mu_s F_N} \)
This is the threshold value that must be overcome to initiate motion between the surfaces.
Comparison Between Static and Kinetic Friction
For any given pair of materials, the coefficient of static friction (\( \mathrm{\mu_s} \)) is typically greater than the coefficient of kinetic friction (\( \mathrm{\mu_k} \)).
\( \mathrm{\mu_s > \mu_k} \)
Physical Explanation:
- When surfaces are stationary, microscopic irregularities (asperities) have time to interlock tightly, requiring greater force to initiate motion.
- Once motion begins, these interlocks are continuously broken and re-formed, producing less resistance on average — hence, smaller \( \mathrm{\mu_k} \).
It is generally harder to start an object moving than to keep it moving — this is why \( \mathrm{\mu_s} \) is larger than \( \mathrm{\mu_k} \).
Static vs. Kinetic Friction
| Property | Static Friction | Kinetic Friction |
|---|---|---|
| When It Acts | Between surfaces not moving relative to each other | Between surfaces sliding relative to each other |
| Equation | \( \mathrm{F_s \leq \mu_s F_N} \) | \( \mathrm{F_k = \mu_k F_N} \) |
| Coefficient | \( \mathrm{\mu_s} \), typically greater | \( \mathrm{\mu_k} \), typically smaller |
| Role | Prevents motion from starting | Opposes motion once sliding begins |
| Relative Magnitude | Usually larger (\( \mathrm{\mu_s > \mu_k} \)) | Usually smaller |
Example
Imagine a heavy crate resting on a warehouse floor. When a worker begins to push the crate, it doesn’t move immediately. The force of static friction between the crate and the floor balances the worker’s applied force, preventing motion. As the worker pushes harder, the static friction force increases proportionally — up to a certain limit.
▶️ Answer / Explanation
Step 1: Initially, the worker’s push is less than the maximum static friction force, so the crate remains stationary. The frictional force exactly matches the applied force — \( \mathrm{F_s = F_{applied}} \).
Step 2: As the worker increases the push, static friction grows until it reaches \( \mathrm{F_{s,max} = \mu_s F_N.} \)
Step 3: Once the worker’s push slightly exceeds \( \mathrm{F_{s,max}}, \) the crate begins to move. At that instant, static friction transitions into kinetic friction, which is smaller because \( \mathrm{\mu_k < \mu_s.} \)
Result: This example shows that static friction adjusts to resist motion up to its limit and that initiating motion requires a larger force than maintaining motion.
Magnitude of the Static friction
Static friction automatically adjusts its magnitude and direction to prevent an object from slipping or sliding on a surface. It acts in the direction opposite to any applied force that tends to initiate motion.
Relevant Equation:
\( \mathrm{|\vec{F}_{f,s}| \leq \mu_s \vec{F}_N} \)
Explanation:
- If the applied force is small, static friction exactly matches it to keep the object at rest.
- As the applied force increases, static friction increases proportionally — up to its maximum value.
- Once this limit is exceeded, the object begins to move, and kinetic friction takes over.
Meaning of Slipping and Sliding
Slipping and sliding refer to situations where the surfaces of two objects are moving relative to one another. Static friction acts to prevent this motion from beginning. Once slipping starts, the frictional force transitions into kinetic friction. Static friction operates only up to the point just before motion begins; beyond that, kinetic friction dominates.
Maximum Static Friction
There exists a maximum threshold of static friction that can resist applied forces before motion begins. This maximum value depends on the nature of the surfaces and the normal force between them.
Derived Equation:
\( \mathrm{F_{f,s,max} = \mu_s F_N} \)
Interpretation:
- \( \mathrm{F_{f,s,max}} \): largest possible static frictional force before motion starts.
- \( \mathrm{\mu_s} \): coefficient of static friction (unitless).
- \( \mathrm{F_N} \): normal force pressing the two surfaces together.
When the applied force exceeds \( \mathrm{F_{f,s,max}} \), the object begins to slide, and the resisting force becomes kinetic friction.
Example
A \( \mathrm{20\,kg} \) box rests on a horizontal floor. The coefficient of static friction between the box and the floor is \( \mathrm{\mu_s = 0.4.} \) A person applies a horizontal force that gradually increases in magnitude. Determine: (a) the maximum static friction force before the box begins to slide, and (b) whether the box will move if a \( \mathrm{60\,N} \) force is applied.
▶️ Answer / Explanation
Step 1: Compute the normal force:
\( \mathrm{F_N = mg = (20)(9.8) = 196\,N.} \)
Step 2: Find the maximum static friction:
\( \mathrm{F_{f,s,max} = \mu_s F_N = (0.4)(196) = 78.4\,N.} \)
Step 3: Compare the applied force to \( \mathrm{F_{f,s,max}} \):
Applied force = \( \mathrm{60\,N < 78.4\,N.} \)
Step 4: Conclusion:
- The static frictional force will match the applied force of \( \mathrm{60\,N} \) to prevent motion.
- The box will not move because the applied force is less than the maximum static friction.
Result: The box remains at rest. Motion would begin only if the applied force exceeds \( \mathrm{78.4\,N.} \)