Circular Motion Basics

1. Introduction

Motion is everywhere around us. From the rotation of the Earth around its axis, to the revolution of planets around the Sun, to a car turning on a curved road—all these are examples of circular motion.

Unlike straight-line motion, circular motion introduces new forces and concepts that are crucial to understand. The two most fundamental ideas are:

Centripetal acceleration – the inward acceleration that keeps a body moving in a circular path.
Centripetal force – the inward force responsible for producing centripetal acceleration.

These ideas are not only theoretical but also practical. Engineers design roads, airplanes, amusement park rides, and satellites based on these very principles.

In this post, we’ll cover the complete basics of circular motion, with derivations, formulas, solved examples, misconceptions, and real-life applications.

2. What is Circular Motion?

Definition:
When an object moves along a circular path, it is said to be in circular motion.

Examples:

A stone tied to a string and whirled in a circle.
A satellite orbiting the Earth.
A fan blade rotating.
A car taking a curved turn.

Circular motion can be:

Uniform Circular Motion (UCM): Speed is constant, but velocity keeps changing direction.
Non-Uniform Circular Motion: Both speed and direction change.

3. Why Velocity Changes in Circular Motion

Velocity is a vector quantity (has magnitude and direction). Even if speed remains constant in uniform circular motion, the direction of velocity changes continuously.

👉 That change in direction means the body is accelerating.

This acceleration is always directed toward the center of the circle. It is called Centripetal Acceleration.

4. Centripetal Acceleration

Derivation

Consider an object of mass mmm moving in a circle of radius rrr with constant speed vvv.

Velocity at two points on the circle has same magnitude but different directions.
Using vector subtraction and geometry, the magnitude of acceleration is:

ac=v2ra_c = \frac{v^2}{r}ac=rv2

Where:

aca_cac = centripetal acceleration
vvv = linear speed
rrr = radius of circle

👉 Centripetal acceleration is always directed toward the center of the circle.

5. Centripetal Force

According to Newton’s Second Law: F=maF = m aF=ma

Substitute centripetal acceleration: Fc=m⋅v2rF_c = m \cdot \frac{v^2}{r}Fc=m⋅rv2

👉 Centripetal force is the inward force that keeps the object in circular motion.

Sources of Centripetal Force in Real Life

Gravitational force → keeps planets in orbit.
Tension in string → keeps a stone whirling in circular path.
Friction between tires and road → keeps cars turning safely.
Normal force in a roller coaster loop → provides centripetal pull.
Magnetic force → makes charged particles move in circular paths.

6. Centrifugal Force – An Apparent Force

In circular motion, people often feel an outward force (e.g., when a car takes a sharp turn).

This is not a real force but an apparent (pseudo) force observed in a rotating frame of reference.

From ground frame: centripetal force pulls inward.
From rotating frame: a person feels pushed outward, called centrifugal force.

7. Angular Velocity and Relation with Linear Velocity

Angular velocity (ω\omegaω): Rate of change of angular displacement.

ω=θt\omega = \frac{\theta}{t}ω=tθ

Relation between linear velocity vvv and angular velocity:

v=rωv = r \omegav=rω

Substituting into centripetal acceleration:

ac=rω2a_c = r \omega^2ac=rω2

Centripetal force:

Fc=mrω2F_c = m r \omega^2Fc=mrω2

8. Work Done in Circular Motion

Centripetal force is always perpendicular to velocity.

👉 Therefore, work done by centripetal force is zero.

This is why an object can keep moving in a circle at constant speed without losing/gaining energy.

9. Banking of Roads

When vehicles take a turn on flat roads, friction provides centripetal force. But at high speeds, friction may not be enough.

👉 Roads are banked at an angle θ\thetaθ to reduce reliance on friction.

Condition: tan⁡θ=v2rg\tan \theta = \frac{v^2}{r g}tanθ=rgv2

Where:

vvv = speed of vehicle
rrr = radius of curve
ggg = acceleration due to gravity

10. Conical Pendulum

If a pendulum bob is whirled in a horizontal circle, the string makes an angle θ\thetaθ with vertical.

Tension provides:

Vertical component = weight (mgmgmg)
Horizontal component = centripetal force (mv2/rmv^2/rmv2/r)

Equations: Tcos⁡θ=mgT \cos \theta = mgTcosθ=mg Tsin⁡θ=mv2rT \sin \theta = \frac{mv^2}{r}Tsinθ=rmv2

11. Vertical Circular Motion

When a body moves in a vertical circle (like a swing or roller coaster loop), tension/normal force varies at different positions.

At the top: T+mg=mv2rT + mg = \frac{mv^2}{r}T+mg=rmv2

At the bottom: T−mg=mv2rT – mg = \frac{mv^2}{r}T−mg=rmv2

👉 Tension is maximum at the bottom and minimum at the top.

12. Real-Life Applications of Centripetal Force

Planetary motion: Sun’s gravity provides centripetal force.
Satellite orbits: Centripetal force balances gravitational pull.
Washing machines: Holes in drum let water escape due to lack of centripetal force.
Amusement park rides: Roller coasters, merry-go-rounds.
Banked roads & race tracks: Prevent skidding.
Cyclists leaning in turns: To balance centripetal force.

13. Solved Examples

Example 1:

A car of mass 1000 kg moves at 20 m/s around a curve of radius 50 m.
Find centripetal force. Fc=mv2r=1000×20250=8000 NF_c = \frac{m v^2}{r} = \frac{1000 \times 20^2}{50} = 8000 \, NFc=rmv2=501000×202=8000N

Example 2:

A stone tied to a string of length 1.5 m is whirled at 5 m/s. Find centripetal acceleration. ac=v2r=251.5=16.67 m/s2a_c = \frac{v^2}{r} = \frac{25}{1.5} = 16.67 \, m/s^2ac=rv2=1.525=16.67m/s2

Example 3:

At what speed should a road of radius 200 m be banked at 10∘10^\circ10∘? tan⁡θ=v2rg\tan \theta = \frac{v^2}{rg}tanθ=rgv2 tan⁡10∘=v2200×9.8\tan 10^\circ = \frac{v^2}{200 \times 9.8}tan10∘=200×9.8v2 v2=200×9.8×0.176=345v^2 = 200 \times 9.8 \times 0.176 = 345v2=200×9.8×0.176=345 v≈18.6 m/sv \approx 18.6 \, m/sv≈18.6m/s

14. Common Misconceptions

Centripetal vs centrifugal force: Centripetal is real, centrifugal is apparent.
Force direction: Always toward center, never outward.
Work done: Zero in uniform circular motion.

15. Key Formulas Recap

Centripetal acceleration:

ac=v2r=rω2a_c = \frac{v^2}{r} = r \omega^2ac=rv2=rω2

Centripetal force:

Fc=mv2r=mrω2F_c = \frac{m v^2}{r} = m r \omega^2Fc=rmv2=mrω2

Banking of roads:

tan⁡θ=v2rg\tan \theta = \frac{v^2}{r g}tanθ=rgv2

Work done by centripetal force = 0