Circular Motion & Rotational Dynamics

Classical Mechanics & Thermodynamics

Course 1 · Chapter 5 · Circular Motion & Rotational Dynamics

Every idea in Chapters 1–4 — motion, force, energy, momentum — was built around objects travelling in straight lines. Nothing about the real world stays that simple: wheels turn, planets orbit, figure skaters spin. This chapter gives circular and rotational motion their own real, parallel toolkit — and reintroduces a familiar name from Chapter 4 in a new role.

Centripetal Force: Why Circular Motion Needs a Force at All

By Chapter 2's own First Law, an object with no net force keeps moving in a straight line. An object moving in a circle is constantly changing direction, which means it is constantly accelerating — even at a perfectly constant speed — and Chapter 2's Second Law says acceleration requires a net force. That force, always pointing inward toward the centre of the circle, is called centripetal force:

F_c = mv²/r

Here m is mass, v is the object's speed along the circular path, and r is the radius of the circle. A larger speed or a tighter radius both demand a larger inward force — which is exactly why cornering fast on a small-radius bend feels far more dramatic than the same speed on a gentle, wide curve.

💡 Huygens, Again Christiaan Huygens — the same Dutch physicist who corrected Descartes' flawed momentum in Chapter 4 — derived the real mathematical description of centripetal force in 1659, nearly three decades before Newton's 1687 Principia. Newton himself coined the actual term "centripetal force," but built directly on Huygens' own earlier mathematical foundation — a real pattern across this course, where much of what gets popularly credited to Newton alone was genuinely developed collaboratively, or independently by others first.

Worked Example: A Car Rounding a Curve

A 1,000 kg car rounds a curve of radius 50 m at a constant 15 m/s. What centripetal force is required?

F_c = mv²/r
F_c = 1000 × 15² / 50
F_c = 1000 × 225 / 50
F_c = 4500 N

In practice, this inward force is supplied by friction between the tyres and the road (or, on a banked track, partly by the banking itself) — if the required force exceeds what friction can actually provide, the car skids outward rather than following the curve.

Torque: Rotational Force

A force applied off-centre to a rigid body doesn't just push it — it can also make it rotate. That rotational effectiveness is torque:

τ = rF sin(θ)

Here r is the distance from the pivot (rotation axis) to the point where the force is applied, F is the force, and θ is the angle between the force and the lever arm. Torque is measured in newton-metres (N·m).

Worked Example: Turning a Bolt with a Wrench

A mechanic applies a 40 N force perpendicular to a wrench 0.3 m long. What torque results?

τ = rF sin(90°)
τ = 0.3 × 40 × 1
τ = 12 N·m

This is exactly why a longer wrench (or a longer spanner handle) makes a stubborn bolt easier to turn: the same applied force produces more torque at a greater distance from the pivot.

Moment of Inertia: Rotational Mass

Just as ordinary mass measures resistance to linear acceleration, moment of inertia (I) measures resistance to rotational (angular) acceleration. For a single point mass at distance r from the rotation axis:

I = mr²

Real rigid bodies distribute their mass differently, giving each common shape its own real formula:

Shape (about central axis)	Moment of Inertia
Point mass at radius r	I = mr²
Thin hoop / ring	I = mr²
Solid disk / cylinder	I = ½mr²
Solid sphere	I = ⅖mr²

Notice a hoop and a solid disk of the same mass and radius have different moments of inertia — the hoop's mass sits entirely at the maximum radius, while the disk's mass is spread inward too, giving it less rotational inertia for the same total mass. This is a real, measurable effect: a solid disk and a hoop released together down an incline will not reach the bottom at the same time, even with identical mass and radius.

Angular Momentum & Its Conservation

Just as linear momentum is p = mv, rotational systems have their own conserved quantity, angular momentum:

L = Iω

Here ω (omega) is angular velocity. In a closed system with no external torque, total angular momentum stays exactly constant — the direct rotational counterpart of Chapter 4's conservation of linear momentum.

Worked Example: A Spinning Figure Skater

A figure skater spinning with arms extended has a moment of inertia of 4 kg·m² and an angular velocity of 3 rad/s. She pulls her arms in, reducing her moment of inertia to 1.6 kg·m². What is her new angular velocity?

Since no external torque acts on her (ignoring the small friction of the skate blade), angular momentum is conserved:

I₁ω₁ = I₂ω₂
(4)(3) = (1.6)(ω₂)
12 = 1.6ω₂
ω₂ = 7.5 rad/s

Pulling her arms in reduces her moment of inertia by 60%, and her spin rate increases correspondingly — from 3 rad/s to 7.5 rad/s, a real, visible effect any figure skating fan has watched countless times.

💡 The Same Principle, at a Genuinely Astronomical Scale The identical physics governs the extremely fast spin of collapsed stellar remnants — white dwarfs, neutron stars, and black holes. When a massive star's core collapses at the end of its life (the real process behind the supernovae Astronomy Fundamentals Chapter 5 covered), its moment of inertia shrinks dramatically while its angular momentum stays conserved, spinning the resulting compact object up to extraordinary rotation rates — some real neutron stars (pulsars) are measured spinning hundreds of times per second. The same conservation law also very slowly transfers angular momentum between the Earth and Moon, gradually slowing Earth's own rotation while pushing the Moon into a slightly larger orbit over geological time.

⚠ Linear and Rotational Quantities Are Genuine Parallels, Not Interchangeable Linear and rotational mechanics share the same underlying mathematical structure, but the quantities are not literally the same thing and cannot be mixed directly — a force (N) and a torque (N·m) have different units for good reason, and mass (kg) and moment of inertia (kg·m²) are not interchangeable even for the same object, since moment of inertia depends on how far the mass sits from the specific axis being considered.

Linear vs. Rotational Quantities: A Direct Parallel

Linear Quantity	Rotational Equivalent	Relationship
Mass (m)	Moment of inertia (I)	I = mr² for a point mass
Force (F)	Torque (τ)	τ = rF sin(θ)
Momentum (p = mv)	Angular momentum (L = Iω)	Both conserved in a closed system
Newton's Second Law (F = ma)	Rotational form (τ = Iα)	α is angular acceleration

Hands-On Exercises

Exercise 1

A 0.3 kg ball is swung in a horizontal circle on a 1.2 m string at a constant speed of 6 m/s. Calculate the centripetal force the string must provide.

→ Solution

Exercise 2

A cyclist applies a force of 250 N to a bicycle pedal, perpendicular to the 0.17 m pedal crank arm. Calculate the torque produced.

→ Solution

Exercise 3

A solid disk of mass 2 kg and radius 0.25 m spins with an angular velocity of 10 rad/s. Calculate its moment of inertia and its angular momentum. (Use I = 1/2 m r^2 for a solid disk.)

→ Solution

Quick Reference

Centripetal force: F_c = mv²/r, always points toward the centre
Torque: τ = rF sin(θ), measured in N·m
Moment of inertia: I = mr² for a point mass; depends on mass distribution and axis
Angular momentum: L = Iω, conserved in a closed system with no external torque
Real history: Huygens derived centripetal force mathematically in 1659; Newton later named it and built on Huygens' work directly

Next chapter: Gravity Revisited — a direct real callback to Astronomy Fundamentals Chapter 3, where Kepler's laws and Newton's law of universal gravitation return from mechanics' own side, using the tools this course has now built.