# Circular Motion

> AP Physics C: Mechanics · AP Physics C: Mechanics
> Source: https://www.owlsprep.com/study/ap-physics-c-mech-u2-circular-motion/

This module covers uniform and non-uniform circular motion, centripetal acceleration derivation, force analysis for standard circular paths, and exam-focused problem solving strategies.

**Prerequisites:** [Newton's Second Law of Motion](https://www.owlsprep.com/study/ap-physics-c-mech-u2-newtons-second-law/); [2D Vector Operations for Kinematics](https://www.owlsprep.com/study/ap-physics-c-mech-u1-vector-operations/)

## Learning objectives

- Distinguish between tangential and centripetal acceleration for uniform and non-uniform circular motion
- Derive the centripetal acceleration formula using vector subtraction of velocity vectors
- Apply Newton's second law to solve force problems for horizontal, vertical, and banked circular paths
- Analyze minimum speed constraints for vertical circular loops and frictionless banked curve conditions

## Uniform Circular Motion Core Principles

**Uniform Circular Motion** — Motion along a fixed-radius circular path where tangential speed is constant, so only the direction of velocity changes over time

*Notation:* UCM

*Example:* A car driving around a perfectly circular race track at a steady 15 m/s

$$a_c = \frac{v_t^2}{r} = \omega^2 r$$

**Worked example:** A 0.4 kg remote controlled car travels at a constant 2.2 m/s on a circular track of radius 1.2 m. Calculate the magnitude of its centripetal acceleration.

1. Identify given values: tangential speed $v_t = 2.2$ m/s, radius $r = 1.2$ m
2. Substitute directly into the centripetal acceleration formula

   $$a_c = \frac{(2.2)^2}{1.2}$$
3. Calculate final value, rounding to 2 significant figures

   $$a_c = 4.0 \text{ m/s}^2$$

**Check your understanding**

Test your understanding of UCM basics:

1. What is the direction of centripetal acceleration for an object in UCM?

   - Tangent to the path
   - Pointing towards the center of the circle
   - Pointing away from the center of the circle
   - Opposite to the direction of motion

   *Why:* Centripetal acceleration is always radial and points inward to change the direction of velocity.

## Derivation of Centripetal Acceleration

> **Exam tip:** College Board explicitly awards points for full vector derivation of centripetal acceleration on FRQs, so memorize every step.

## Force Analysis for Circular Paths

**Worked example:** A 0.3 kg mass attached to a 0.8 m string is swung in a perfectly horizontal circle at 3 m/s. Calculate the tension in the string.

1. Note that tension is the only force providing the full centripetal net force
2. Set tension equal to mass multiplied by centripetal acceleration

   $$F_T = m \frac{v^2}{r}$$
3. Substitute all given values

   $$F_T = 0.3 \times \frac{(3)^2}{0.8} = 3.4 \text{ N}$$

## Frictionless Banked Curve Scenarios

**Worked example:** A highway curve of radius 70 m is designed for traffic moving at 25 m/s with no friction. Calculate the required bank angle.

1. Use the derived formula for frictionless banked curves

   $$\tan\theta = \frac{v^2}{rg}$$
2. Substitute values $v=25$ m/s, $r=70$ m, $g=9.8$ m/s²

   $$\tan\theta = \frac{625}{70 \times 9.8} \approx 0.91$$
3. Take inverse tangent to find the angle

   $$\theta \approx 42^\circ$$

## Common pitfalls

- **Wrong:** Adding 'centripetal force' as a separate independent force in free body diagrams
  - Why it fails: Centripetal force is the net radial force, not a new force distinct from tension, gravity, or friction
  - Correct: Sum all existing radial forces and set the total equal to $mv^2/r$
- **Wrong:** Ignoring tangential acceleration for non-uniform circular motion
  - Why it fails: Students only use centripetal acceleration when speed is changing, leading to incorrect total acceleration values
  - Correct: Calculate total acceleration as the vector sum of tangential and centripetal components
- **Wrong:** Assume $a_c = g$ at the top of every vertical circular loop
  - Why it fails: This only holds true at the minimum possible speed where normal force equals zero
  - Correct: Include normal force in your net force calculation unless minimum speed is explicitly requested
- **Wrong:** Using degrees per second for angular velocity in $a_c = \omega^2 r$
  - Why it fails: The formula only returns correct values when angular velocity is measured in radians per second
  - Correct: Convert all angular quantities to rad/s before plugging into kinematic equations
- **Wrong:** Pointing friction away from the center on banked curve problems
  - Why it fails: Friction points inward to prevent the car from sliding up the bank at high speeds
  - Correct: Resolve friction parallel to the bank, with its radial component pointing towards the path center

## Cheatsheet

| Scenario | Net Radial Force | Key Formula | Common Constraint |
| --- | --- | --- | --- |
| Uniform Horizontal Circle | Tension / Static Friction | $F_{net} = mv^2/r$ | Speed is constant |
| Top of Vertical Loop | Gravity + Normal Force | $mg + N = mv^2/r$ | $N \geq 0$ for no fall |
| Bottom of Vertical Loop | Normal Force - Gravity | $N - mg = mv^2/r$ | $N > mg$ always |
| Frictionless Banked Curve | Horizontal Normal Component | $\tan\theta = v^2/(rg)$ | No sideways friction |

## What's next

Mastering circular motion is a critical bridge between Newtonian linear kinematics and the upcoming unit on rotational motion, where you will extend these concepts to angular momentum, torque, and rigid body rotation. You will frequently combine circular motion principles with the work-energy theorem to solve multi-step FRQ problems that appear on nearly every AP Physics C exam. This foundation will also help you analyze gravitational orbital motion, a high-yield topic that makes regular appearances on both multiple choice and free response sections. Practice applying these skills to the linked sub-topics to build fluency before your full unit assessment.

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