# Friction and Tension

> AP Physics 1 · Unit 2: Dynamics
> Source: https://www.owlsprep.com/study/ap-physics-1-u2-friction-and-tension/

Covers static and kinetic friction, tension in ideal ropes and fixed pulleys, and analysis of connected object systems for AP Physics 1 dynamics. Includes step-by-step worked examples and exam-specific tips.

**Prerequisites:** [Newton's three laws of motion](https://www.owlsprep.com/study/ap-physics-1-u2-newtons-laws/); Drawing and interpreting free-body diagrams; Vector resolution of force components

## Learning objectives

- Distinguish between static and kinetic friction
- Calculate friction forces for stationary and sliding objects
- Analyze tension in ideal ropes and fixed pulleys
- Solve connected system problems involving friction and tension

## Core Definitions of Friction and Tension

Friction is a contact force that opposes relative motion between two solid surfaces in contact, while tension is a pulling force transmitted through a flexible stretched medium (e.g., a rope, string, or cable). This subtopic makes up roughly a third of AP Physics 1 Unit 2: Dynamics, which accounts for 12–18% of your total AP exam score, appearing regularly in both multiple-choice and free-response sections.

**Friction** — Contact force that opposes relative (or impending relative) motion between two solid surfaces in contact

*Notation:* $f$, $f_s$ (static), $f_k$ (kinetic)

*Example:* A crate sliding across a floor slows down due to kinetic friction opposing its motion

**Tension** — Pulling force transmitted along a flexible stretched medium, acting equally on both ends of the medium

*Notation:* $T$

*Example:* A rope holding a stationary hanging mass pulls upward on the mass with tension equal to the weight

For AP Physics 1, we almost always assume ideal ropes (massless, inextensible) and ideal pulleys (massless, frictionless) unless explicitly stated otherwise. This simplifies analysis because tension is uniform along an ideal rope.

## Static and Kinetic Friction

Friction is split into two categories based on whether surfaces are moving relative to each other. Static friction acts when there is no relative motion, and adjusts its magnitude to exactly oppose the parallel component of the applied force, up to a maximum threshold. Kinetic friction acts when surfaces slide relative to each other, and has a constant magnitude for a given surface pair and normal force.

The formula for maximum static friction is:

$$f_{s,max} = \mu_s N$$

where $\mu_s$ is the dimensionless coefficient of static friction (dependent on the two surface materials), and $N$ is the magnitude of the normal force perpendicular to the contact surface. Kinetic friction follows the formula:

$$f_k = \mu_k N$$

For any pair of surfaces, $\mu_k < \mu_s$, which means it takes more force to start moving an object than to keep it moving at constant speed. A common misconception is that normal force always equals an object’s weight; this is only true for horizontal surfaces with no additional vertical forces. $N$ must always be calculated from Newton’s second law in the direction perpendicular to the contact surface.

**Worked example:** A 12 kg wooden crate rests on a horizontal concrete floor, with $\mu_s = 0.6$ and $\mu_k = 0.4$. What is the magnitude of friction when a horizontal 50 N force pushes on the stationary crate?

1. Calculate the normal force: no vertical acceleration, so

   $$N = mg = 12 \times 9.8 = 117.6 \text{ N}$$
2. Calculate maximum static friction

   $$f_{s,max} = \mu_s N = 0.6 \times 117.6 = 70.56 \text{ N}$$
3. Compare the applied force to the maximum threshold: $50 \text{ N} < 70.56 \text{ N}$, so the crate remains stationary
4. For stationary objects not at the sliding threshold, static friction matches the applied parallel force

   $$f_s = 50 \text{ N}$$

> **tip**
>
> Always compare the applied force to $f_{s,max}$ before assuming friction is kinetic. AP exam questions regularly trick students into automatically using $f_k$ when the object is not moving.

## Tension in Ideal Ropes and Pulleys

Tension is a pulling force that acts along the length of a rope, pulling equally on both objects connected to the rope. For AP Physics 1, all ropes and pulleys are assumed ideal unless stated otherwise, with the following properties:

- Ideal rope: massless and inextensible. Inextensible means all connected objects have the same magnitude of acceleration, even if acceleration directions differ. Massless means net force on the rope is zero, so tension is uniform along the rope.
- Ideal fixed pulley: massless and frictionless. It only changes the direction of tension, not its magnitude, so tension is equal on both sides of the pulley.

**Worked example:** A 5 kg mass hangs vertically from an ideal rope that runs over a fixed ideal pulley, connected to an 8 kg block resting on a frictionless horizontal table. What is the magnitude of tension in the rope?

1. Assign acceleration: the hanging mass accelerates downward, the block accelerates to the right, with equal magnitude $a$
2. Write Newton's second law for the 8 kg block (horizontal direction)

   $$\sum F = T = 8a$$
3. Write Newton's second law for the 5 kg hanging mass (downward as positive)

   $$\sum F = mg - T = 5a = 49 - T$$
4. Substitute $T=8a$ into the second equation and solve for $a$

   $$49 - 8a = 5a \rightarrow 13a = 49 \rightarrow a \approx 3.77 \text{ m/s}^2$$
5. Solve for tension

   $$T = 8 \times 3.77 \approx 30.2 \text{ N}$$

> **tip**
>
> If a pulley is accelerating (e.g., a movable pulley in a system), you must include forces on the pulley itself in your analysis; only fixed ideal pulleys have equal tension on both sides.

## Combined Tension-Friction Connected Systems

Most AP Physics 1 problems involving both friction and tension are connected object systems, where one or more objects rest on a frictional surface, connected by a rope and pulley to a hanging object. Follow this systematic approach to solve these problems:

1. Draw a separate free-body diagram for every object in the system
2. Resolve forces into components aligned with the direction of possible motion
3. Write Newton's second law for each object, using equal tension and equal acceleration magnitude for ideal systems
4. Check if the system is stationary or accelerating by comparing the applied pulling force to maximum static friction, then solve the system of equations

**Worked example:** Block A (mass 4 kg) rests on a horizontal table, connected by an ideal rope over a fixed ideal pulley to hanging Block B (mass 3 kg). $\mu_s = 0.35$ and $\mu_k = 0.25$ between Block A and the table. Is the system stationary, or does it accelerate? If it accelerates, what is the tension?

1. Calculate maximum static friction on Block A

   $$f_{s,max} = \mu_s m_A g = 0.35 \times 4 \times 9.8 = 13.72 \text{ N}$$
2. Compare to the pulling force from Block B: the required tension for equilibrium would equal $m_B g = 29.4 \text{ N}$. Since $29.4 \text{ N} > 13.72 \text{ N}$, static friction cannot hold the system, so it accelerates
3. Write Newton's second law for Block A (right positive)

   $$T - f_k = m_A a, \quad f_k = \mu_k N = 9.8 \text{ N} \rightarrow T - 9.8 = 4a$$
4. Write Newton's second law for Block B (down positive)

   $$29.4 - T = 3a$$
5. Add equations to eliminate tension, then solve for $a$ and $T$

   $$19.6 = 7a \rightarrow a = 2.8 \text{ m/s}^2, \quad T = 21 \text{ N}$$

> **tip**
>
> Always confirm the direction of friction: friction opposes impending or actual motion, so if the system is pulling a block up an incline, friction acts down the incline, and vice versa.

**Check your understanding**

Test your understanding of friction with an angled applied force:

1. A 10 kg box rests on a horizontal surface with $\mu_s = 0.5$ and $\mu_k = 0.3$. A person pulls the box with a 30 N force at an angle of 30° above the horizontal. What is the magnitude of friction acting on the box?

   - 0 N
   - ~26 N
   - ~36 N
   - ~41 N

   *Why:* First calculate the reduced normal force from the upward pull component, then check if the applied horizontal force is less than maximum static friction. Since it is, static friction equals the applied horizontal component, giving ~26 N.

## Common pitfalls

- **Wrong:** Using $f_s = \mu_s N$ for static friction when the object is not at the point of sliding
  - Why it fails: Students memorize the maximum static friction formula and apply it to all static friction cases, forgetting static friction adjusts to match the applied force
  - Correct: Only use $f_s = \mu_s N$ if the problem states the object is just about to slide; for all other stationary cases, use $f_s = F_{applied,parallel}$
- **Wrong:** Assuming normal force $N$ equals the object's weight $mg$ in all cases
  - Why it fails: Students generalize from simple horizontal surface problems to all cases, including angled forces and inclines
  - Correct: Always calculate $N$ from Newton's second law in the direction perpendicular to the surface, accounting for angled applied forces or inclines before calculating friction
- **Wrong:** Assigning different acceleration magnitudes to connected objects on an ideal inextensible rope
  - Why it fails: Students confuse different acceleration directions with different magnitudes of acceleration
  - Correct: For any two objects connected by an ideal rope, set the magnitude of acceleration equal when writing your system of equations
- **Wrong:** Changing the magnitude of tension when it goes around an ideal fixed pulley
  - Why it fails: Students assume pulleys change tension magnitude, when they only change direction for ideal fixed pulleys
  - Correct: For any ideal massless, frictionless fixed pulley, tension has the same magnitude on both sides of the pulley
- **Wrong:** Using kinetic friction when the applied force is less than maximum static friction
  - Why it fails: Students rush to use the kinetic friction formula without checking if motion actually occurs
  - Correct: Always compare the net applied force trying to move the object to $f_{s,max}$ first; only use $f_k$ if the applied force exceeds $f_{s,max}$

## Cheatsheet

| Category | Formula/Rule | Key Notes |
| --- | --- | --- |
| Maximum Static Friction | $f_{s,max} = \mu_s N$ | Only applies when object is just about to slide; $f_s \leq f_{s,max}$ for all stationary objects |
| Kinetic Friction | $f_k = \mu_k N$ | Applies when surfaces slide relative to each other; $\mu_k < \mu_s$ for all surface pairs |
| Static Friction (non-maximum) | $f_s = F_{applied,parallel}$ | Matches the parallel applied force for stationary objects not at the sliding threshold |
| Tension in ideal rope | $T_1 = T_2$ | Equal tension magnitude at both ends of a massless inextensible rope |
| Connected object acceleration | $\|a_1\| = \|a_2\|$ | Equal magnitude acceleration for all objects connected by an ideal inextensible rope |
| Tension over ideal fixed pulley | $T_{left} = T_{right}$ | Ideal fixed pulleys only change tension direction, not magnitude |
| Static friction direction | Opposes impending relative motion | Points opposite to the direction the object would slide if friction were removed |
| Kinetic friction direction | Opposes actual relative motion | Points opposite to the direction the object is sliding relative to the surface |

## What's next

Mastering friction and tension is the foundation for all subsequent dynamics problems in AP Physics 1, and these concepts are immediately applied to nearly all future units. In Unit 3: Circular Motion and Gravitation, friction provides the centripetal force for objects like cars turning on flat roads, and tension acts as the centripetal force for objects moving in vertical circles. Friction also appears later in energy problems, where it does non-conservative work that changes the total mechanical energy of a system. In rotational dynamics, analyzing rolling motion without slipping relies entirely on static friction to provide the torque needed for rotation. Solid skills here will make all more complex force problems much easier to solve.

- [Inclined Planes and Atwood Machines](https://www.owlsprep.com/study/ap-physics-1-u2-inclined-planes-and-atwood-machines/)
- [Circular Motion and Gravitation Overview](https://www.owlsprep.com/study/ap-physics-1-u3-overview/)
- [Acceleration in Uniform Circular Motion](https://www.owlsprep.com/study/ap-physics-1-u3-acceleration-in-uniform-circular-motion/)

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