Circular Motion Formula

Circular motion is motion of an object along a circular path where the speed may be constant but the velocity is continuously changing direction.

The Formula

a=v2ra = \frac{v^2}{r} (centripetal acceleration)

When to use: A car on a circular track at constant speed is still accelerating—toward the center.

Quick Example

A merry-go-round, Earth orbiting the Sun, electrons in atoms.

Notation

aca_c is centripetal acceleration in m/s², vv is tangential speed in m/s, rr is the radius in metres, ω\omega is angular velocity in rad/s, and TT is the period in seconds.

What This Formula Means

Motion of an object along a circular path where the speed may be constant but the velocity is continuously changing direction, requiring a centripetal acceleration.

A car on a circular track at constant speed is still accelerating—toward the center.

Formal View

For uniform circular motion with speed vv and radius rr, the centripetal acceleration is ac=v2r=ω2ra_c = \frac{v^2}{r} = \omega^2 r, directed radially inward. The period is T=2πrv=2πωT = \frac{2\pi r}{v} = \frac{2\pi}{\omega}.

Worked Examples

Example 1

easy
A car drives around a circular track of radius 50 m50 \text{ m} at a constant speed of 10 m/s10 \text{ m/s}. What is the centripetal acceleration?

Answer

ac=2 m/s2 toward centera_c = 2 \text{ m/s}^2 \text{ toward center}

First step

1
Centripetal acceleration is directed toward the center of the circle: ac=v2ra_c = \frac{v^2}{r}.

Full solution

  1. 2
    ac=10250=10050=2 m/s2a_c = \frac{10^2}{50} = \frac{100}{50} = 2 \text{ m/s}^2
  2. 3
    This acceleration changes the direction of velocity, not its magnitude.
In circular motion, even at constant speed, the velocity direction changes continuously. This requires a centripetal (center-seeking) acceleration, which is always directed toward the center of the circle.

Example 2

medium
A satellite orbits Earth at a radius of 6.8×106 m6.8 \times 10^6 \text{ m} with a period of 5400 s5400 \text{ s}. What is its orbital speed and centripetal acceleration?

Example 3

medium
A car travels around a circular track of radius 50 m at 20 m/s. Find the centripetal acceleration.

Common Mistakes

  • Thinking that constant speed means zero acceleration — in circular motion, the direction is always changing, so there is always centripetal acceleration toward the centre. - Fix this by naming the system, checking "Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated?", and attaching units or direction to the final statement.
  • Drawing the acceleration vector tangent to the circle instead of pointing toward the centre — centripetal acceleration is always radially inward. - Fix this by naming the system, checking "Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated?", and attaching units or direction to the final statement.
  • Confusing period (TT, time for one revolution) with frequency (ff, revolutions per second) — they are reciprocals: T=1/fT = 1/f. - Fix this by naming the system, checking "Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated?", and attaching units or direction to the final statement.
  • Using circular motion from a keyword alone - Signal words like position, speed, velocity only point to a possible model; the system must match too.

Why This Formula Matters

Circular Motion helps students describe motion precisely instead of relying on everyday words like fast or slow. It prepares them to interpret graphs, choose equations, and connect motion to forces and energy.

Frequently Asked Questions

What is the Circular Motion formula?

Motion of an object along a circular path where the speed may be constant but the velocity is continuously changing direction, requiring a centripetal acceleration.

How do you use the Circular Motion formula?

A car on a circular track at constant speed is still accelerating—toward the center.

What do the symbols mean in the Circular Motion formula?

aca_c is centripetal acceleration in m/s², vv is tangential speed in m/s, rr is the radius in metres, ω\omega is angular velocity in rad/s, and TT is the period in seconds.

Why is the Circular Motion formula important in Physics?

Circular Motion helps students describe motion precisely instead of relying on everyday words like fast or slow. It prepares them to interpret graphs, choose equations, and connect motion to forces and energy.

What do students get wrong about Circular Motion?

Students often know a formula related to circular motion but skip the recognition step: Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated? That leads to a correct-looking substitution attached to the wrong physical model.

What should I learn before the Circular Motion formula?

Before studying the Circular Motion formula, you should understand: acceleration, velocity.

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