Circular Motion

Understanding uniform circular motion and centripetal acceleration

4 min read

Prerequisites

Before studying this topic, make sure you understand:

What is Circular Motion?

Circular motion occurs when an object moves along a circular path. Even if the speed is constant, the velocity is constantly changing direction.

Thor's Hammer Spin!
Remember how Thor spins Mjolnir in circles before throwing it in the Thor movies? That’s circular motion! Even though the hammer moves at constant speed, its velocity (direction) keeps changing — pointing tangent to the circle. The centripetal force? That’s the tension in his arm pulling the hammer toward the center!
Key Insight
In circular motion, even at constant speed, the object is accelerating because velocity (a vector) is changing direction!

Interactive Demo

See how velocity (green, tangent) and centripetal acceleration (orange, towards center) behave:

Angular Quantities

Angular Displacement ($\theta$)

The angle swept by the radius vector.

  • Unit: radian (rad)
  • 1 revolution = 2π rad = 360°

Angular Velocity ($\omega$)

Rate of change of angular displacement.

$$\omega = \frac{d\theta}{dt}$$
  • Unit: rad/s
  • Direction: Along axis of rotation (right-hand rule)

Angular Acceleration ($\alpha$)

Rate of change of angular velocity.

$$\alpha = \frac{d\omega}{dt}$$
  • Unit: rad/s²

Relation Between Linear and Angular Quantities

For a particle at distance $r$ from the center:

LinearAngularRelation
Arc length $s$$\theta$$s = r\theta$
Linear velocity $v$$\omega$$v = r\omega$
Tangential acceleration $a_t$$\alpha$$a_t = r\alpha$
Easy Conversion
To convert angular to linear: multiply by radius $r$

Uniform Circular Motion (UCM)

In uniform circular motion:

  • Speed is constant
  • Angular velocity is constant
  • Acceleration exists but is directed toward center

Key Formulas

$$\boxed{v = r\omega}$$ $$\boxed{T = \frac{2\pi}{\omega} = \frac{2\pi r}{v}}$$ $$\boxed{f = \frac{1}{T} = \frac{\omega}{2\pi}}$$

where:

  • $T$ = Time period (time for one revolution)
  • $f$ = Frequency (revolutions per second)

Centripetal Acceleration

The acceleration directed toward the center of the circle:

$$\boxed{a_c = \frac{v^2}{r} = \omega^2 r = v\omega}$$

Direction

Always toward the center (hence “centripetal” = center-seeking)

Why It Exists

Even though speed is constant, velocity direction changes. This change in velocity direction requires acceleration toward the center.

Velocity and Acceleration Relationship

In UCM:

  • Velocity is tangent to the circle
  • Acceleration is perpendicular to velocity (toward center)
  • This is why speed doesn’t change!

Centripetal vs Centrifugal

CentripetalCentrifugal
Real forcePseudo force
Exists in inertial frameAppears in rotating frame
Toward centerAway from center
Causes circular motionAppears to oppose centripetal
Interstellar Spinning Space Station
In Interstellar (2014), the Endurance spacecraft spins to create artificial gravity. From inside, astronauts feel pushed outward (centrifugal force) — but that’s a pseudo force! From outside, it’s the spacecraft walls pushing them inward (centripetal force) that keeps them in circular motion. Christopher Nolan got the physics right!
Common Misconception
Centrifugal force is NOT a real force! It’s a pseudo force that appears when you’re in the rotating reference frame (like inside a merry-go-round).

Non-Uniform Circular Motion

When speed changes while moving in a circle:

Two Components of Acceleration

  1. Centripetal acceleration ($a_c$): Toward center, changes direction

    $$a_c = \frac{v^2}{r}$$

  2. Tangential acceleration ($a_t$): Along tangent, changes speed

    $$a_t = \frac{dv}{dt} = r\alpha$$

Total Acceleration

$$\vec{a} = \vec{a}_c + \vec{a}_t$$ $$|a| = \sqrt{a_c^2 + a_t^2}$$

Direction makes angle $\phi$ with radius:

$$\tan\phi = \frac{a_t}{a_c}$$

Angular Kinematic Equations

For constant angular acceleration ($\alpha$ = constant):

Linear FormAngular Form
$v = u + at$$\omega = \omega_0 + \alpha t$
$s = ut + \frac{1}{2}at^2$$\theta = \omega_0 t + \frac{1}{2}\alpha t^2$
$v^2 = u^2 + 2as$$\omega^2 = \omega_0^2 + 2\alpha\theta$

Summary Table

QuantitySymbolFormulaUnit
Angular displacement$\theta$—rad
Angular velocity$\omega$$\frac{d\theta}{dt}$rad/s
Angular acceleration$\alpha$$\frac{d\omega}{dt}$rad/s²
Linear velocity$v$$r\omega$m/s
Centripetal acceleration$a_c$$\frac{v^2}{r} = \omega^2 r$m/s²
Time period$T$$\frac{2\pi}{\omega}$s
Frequency$f$$\frac{1}{T}$Hz

Practice Problems

Problem 1

A particle moves in a circle of radius 2 m with constant speed 4 m/s. Find centripetal acceleration and time period.

Solution: $a_c = \frac{v^2}{r} = \frac{16}{2} = $ 8 m/s²

$T = \frac{2\pi r}{v} = \frac{2\pi \times 2}{4} = $ π s ≈ 3.14 s

Problem 2

A wheel starts from rest and accelerates at 2 rad/s². Find angular velocity and angle turned after 5 seconds.

Solution: $\omega = \omega_0 + \alpha t = 0 + 2 \times 5 = $ 10 rad/s

$\theta = \omega_0 t + \frac{1}{2}\alpha t^2 = 0 + \frac{1}{2} \times 2 \times 25 = $ 25 rad

Problem 3

A car moves on a circular track of radius 100 m. At a certain instant, its speed is 20 m/s and is increasing at 3 m/s². Find total acceleration.

Solution: $a_c = \frac{v^2}{r} = \frac{400}{100} = 4$ m/s²

$a_t = 3$ m/s²

$a = \sqrt{a_c^2 + a_t^2} = \sqrt{16 + 9} = $ 5 m/s²

Further Study

Now that you’ve mastered Kinematics, move on to: