Circular Motion Formula

The Formula

a = \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

a_c is centripetal acceleration in m/s², v is tangential speed in m/s, r is the radius in metres, \omega is angular velocity in rad/s, and T 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 directed toward the centre of the circle.

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

Formal View

For uniform circular motion with speed v and radius r, the centripetal acceleration is a_c = \frac{v^2}{r} = \omega^2 r, directed radially inward. The period is T = \frac{2\pi r}{v} = \frac{2\pi}{\omega}.

Worked Examples

Example 1

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

Solution

  1. 1
    Centripetal acceleration is directed toward the center of the circle: a_c = \frac{v^2}{r}.
  2. 2
    a_c = \frac{10^2}{50} = \frac{100}{50} = 2 \text{ m/s}^2
  3. 3
    This acceleration changes the direction of velocity, not its magnitude.

Answer

a_c = 2 \text{ m/s}^2 \text{ toward center}
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 \times 10^6 \text{ m} with a period of 5400 \text{ s}. What is its orbital speed and 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.
  • Drawing the acceleration vector tangent to the circle instead of pointing toward the centre — centripetal acceleration is always radially inward.
  • Confusing period (T, time for one revolution) with frequency (f, revolutions per second) — they are reciprocals: T = 1/f.

Why This Formula Matters

Circular motion governs planetary orbits, satellite trajectories, amusement park rides, and the operation of centrifuges. Understanding it is essential for designing safe curves on roads and engineering rotating machinery.

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 directed toward the centre of the circle.

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?

a_c is centripetal acceleration in m/s², v is tangential speed in m/s, r is the radius in metres, \omega is angular velocity in rad/s, and T is the period in seconds.

Why is the Circular Motion formula important in Physics?

Circular motion governs planetary orbits, satellite trajectories, amusement park rides, and the operation of centrifuges. Understanding it is essential for designing safe curves on roads and engineering rotating machinery.

What do students get wrong about Circular Motion?

The acceleration points toward the center, not along the motion.

What should I learn before the Circular Motion formula?

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

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