Work Done in Circular Motion

Introduction

Circular motion is one of the most common types of motion in nature and technology. From planets orbiting the Sun, to cars taking a turn on a curved road, to a stone tied with a string moving in a circle, the concept of circular motion is everywhere.

But what about work done in circular motion? Does the centripetal force that keeps an object moving in a circle do work? Under what conditions is work done in such motion? How does energy transform in circular paths?

To answer these questions, we need to carefully apply the definition of work in physics to the case of circular motion.

Recap: Definition of Work

In physics, work is defined as: W=F⋅d⋅cos⁡θW = F \cdot d \cdot \cos \thetaW=F⋅d⋅cosθ

Where:

FFF = Force applied,
ddd = Displacement of the object,
θ\thetaθ = Angle between force and displacement.

👉 Work is done only when there is a displacement and a component of force along displacement.

Circular Motion Basics

Uniform Circular Motion (UCM):
- Speed = constant.
- Velocity changes because direction changes.
- Acceleration is centripetal, directed towards the center.
Non-Uniform Circular Motion:
- Speed is not constant.
- Both centripetal and tangential accelerations exist.

Forces in Circular Motion

Centripetal Force:
- Acts towards the center.
- Keeps the body in circular path.
- Examples: tension in string, friction on a car’s tires, gravitational force in planetary motion.
Tangential Force (if any):
- Acts along the tangent to the path.
- Changes the speed of the body.

Work Done by Centripetal Force

In uniform circular motion:

Displacement of the body at any instant is along the tangent.
Centripetal force is directed towards the center, i.e., perpendicular to displacement.

θ=90∘⇒W=Fdcos⁡90∘=0\theta = 90^\circ \quad \Rightarrow \quad W = Fd \cos 90^\circ = 0θ=90∘⇒W=Fdcos90∘=0

👉 Centripetal force does no work.

This is why in uniform circular motion, even though force and acceleration exist, kinetic energy remains constant.

Work Done by Tangential Force

If there is a tangential force (non-uniform circular motion), then:

Tangential force is along displacement.
Work is done → Kinetic energy changes.

W=Ft⋅dW = F_t \cdot dW=Ft⋅d

Where FtF_tFt = tangential component of force.

👉 Example: A car speeding up on a circular track → engine provides tangential force → work is done.

Work-Energy Theorem in Circular Motion

From work-energy theorem: Wnet=ΔKEW_{net} = \Delta KEWnet=ΔKE

In uniform circular motion: ΔKE=0\Delta KE = 0ΔKE=0, so Wnet=0W_{net} = 0Wnet=0.
In non-uniform circular motion: ΔKE≠0\Delta KE \neq 0ΔKE=0, so net work is done by tangential force.

Cases of Work in Circular Motion

1. Particle in Uniform Circular Motion (UCM)

Only centripetal force acts.
No work is done.
Speed and kinetic energy remain constant.

2. Particle in Non-Uniform Circular Motion

Centripetal + tangential forces act.
Tangential force does work → change in KE.
Example: Rotating fan blades speeding up.

3. Vertical Circular Motion (e.g., pendulum, roller coaster loop)

Gravity does work when object moves vertically.
KE and PE interchange.
At top: more PE, less KE.
At bottom: more KE, less PE.
Total mechanical energy conserved (ignoring friction).

4. Planetary Motion (Circular/elliptical)

Gravity acts as centripetal force.
Since force ⟂ displacement, no work done.
Speed remains constant in perfectly circular orbit.

Graphical Interpretation

Force vs. Displacement Curve
- For centripetal force in UCM: always ⟂ displacement → zero area under curve → no work.
Kinetic Energy vs. Time
- In UCM: constant horizontal line.
- In non-UCM: increasing or decreasing curve depending on tangential force.

Numerical Examples

Example 1: Stone in Uniform Circular Motion

A stone tied to a string rotates at constant speed. Show that work done by tension is zero.

Tension = Centripetal force.
Always perpendicular to displacement.
Work:

W=Fdcos⁡90∘=0W = Fd \cos 90^\circ = 0W=Fdcos90∘=0

👉 No work is done.

Example 2: Car on Circular Track with Acceleration

A car of mass 1000 kg increases speed from 10 m/s to 20 m/s while going around a circular track. Find work done by tangential force.

Initial KE:

KE1=12mv2=0.5×1000×100=50,000 JKE_1 = \tfrac{1}{2}mv^2 = 0.5 \times 1000 \times 100 = 50,000 \, JKE1=21mv2=0.5×1000×100=50,000J

Final KE:

KE2=0.5×1000×400=200,000 JKE_2 = 0.5 \times 1000 \times 400 = 200,000 \, JKE2=0.5×1000×400=200,000J

Work done:

W=ΔKE=200,000−50,000=150,000 JW = \Delta KE = 200,000 – 50,000 = 150,000 \, JW=ΔKE=200,000−50,000=150,000J

👉 Work done = 150 kJ.

Example 3: Vertical Circular Motion

A 2 kg body swings in vertical circle of radius 2 m. Find speed at bottom if released from rest at top.

At top: PE=mgh=2×9.8×4=78.4J,KE=0PE = mgh = 2 \times 9.8 \times 4 = 78.4 J, \quad KE = 0PE=mgh=2×9.8×4=78.4J,KE=0
At bottom: TotalEnergy=78.4J⇒KE=78.4JTotal Energy = 78.4 J \quad \Rightarrow KE = 78.4 JTotalEnergy=78.4J⇒KE=78.4J

12mv2=78.4⇒v=2×78.42=8.86 m/s\tfrac{1}{2}mv^2 = 78.4 \quad \Rightarrow \quad v = \sqrt{\tfrac{2 \times 78.4}{2}} = 8.86 \, m/s21mv2=78.4⇒v=22×78.4=8.86m/s

👉 Speed at bottom = 8.86 m/s.

Real-Life Applications

Satellites in Orbit – Gravity acts centripetally, but does no work → constant energy.
Roller Coasters – Work by gravity converts PE ↔ KE in vertical loops.
Vehicles on Curves – Friction provides centripetal force → no work in ideal case.
Centrifugal Machines – Separation by circular motion (cream from milk).
Sports – Hammer throw, discus throw involve circular motion before release.

Misconceptions

Centripetal force does work → ❌ Wrong. It is perpendicular to displacement.
No force in circular motion → ❌ Wrong. Force exists, but no work in UCM.
Energy is lost in circular motion → ❌ Not true unless friction/air resistance present.

Energy Dissipation in Real Circular Motion

In practical systems, forces like friction and air resistance are present. These do negative work, leading to:

Gradual loss of mechanical energy.
Conversion into heat or sound.
Example: A fan blade slows down after switching off.

Advanced Concepts

Work in Non-Inertial Frames
- In rotating frames, pseudo-forces appear.
- Work-energy theorem still applies with these forces.
Power in Circular Motion
- If tangential force FtF_tFt is present:
P=Ft⋅vP = F_t \cdot vP=Ft⋅v
- Instantaneous power depends on tangential component only.
Torque and Work
- In rotational motion:
W=τ⋅θW = \tau \cdot \thetaW=τ⋅θ
- Where τ\tauτ = torque, θ\thetaθ = angular displacement.

Summary Table

Case	Work Done
Uniform Circular Motion	0 (centripetal force ⟂ displacement)
Non-Uniform Circular Motion	Work done by tangential force → KE changes
Vertical Circular Motion	Gravity does work → KE ↔ PE exchange
Planetary Circular Motion	Work by gravity = 0
Real Systems	Friction/drag cause negative work → energy loss