Circular Motion

Circular motion basics

An object moves in a circle of radius r at speed v. Even at constant speed, its velocity vector is constantly changing direction — so it's accelerating.

The acceleration points toward the center of the circle:

a_centripetal = v² / r = ω² · r

where ω = v/r is the angular speed in radians/second.

By Newton's second law, this requires a centripetal force:

F_centripetal = m · a = m · v² / r = m · ω² · r

The force always points from the object toward the center. WITHOUT this force, the object flies off tangentially (Newton's first law).

Sources of centripetal force

Setting	Centripetal force comes from
Ball on a string (whirled in circle)	Tension in the string
Car turning a corner	Friction between tires and road
Banked curve (race track)	Horizontal component of normal force
Roller coaster loop	Normal force from track + gravity at top
Satellite orbiting Earth	Earth's gravity
Electron orbiting nucleus (classical)	Electrostatic attraction
Spinning carousel rider	Inward push from seat/wall
Earth orbiting Sun	Sun's gravity

Period, frequency, angular velocity

For uniform circular motion:

Quantity	Symbol	Formula	Unit
Period (time per revolution)	T	2π/ω = 2πr/v	seconds
Frequency (revolutions per sec)	f	1/T = ω/(2π)	Hz
Angular velocity	ω	2π/T = v/r	rad/s
Linear speed	v	ωr = 2πr/T	m/s
Centripetal acceleration	a	v²/r = ω²r	m/s²
Centripetal force	F	mv²/r = mω²r	N

Banked curves

On a flat road, friction is the only horizontal force — it must provide all the centripetal force. Maximum cornering speed is limited by μ:

μmg ≥ mv²/r  →  v_max = √(μgr)

On a banked curve angled at θ, the normal force tilts inward, providing horizontal component N·sin θ. For cars to corner at design speed without relying on friction:

tan θ = v² / (gr)

For a Daytona banking of 31° at v = 88 m/s (200 mph), required radius:

r = v²/(g·tan θ) = 88²/(9.81 × 0.601) ≈ 1313 m

Real Daytona is 366 m radius — banking handles MUCH higher speeds, but cars still rely on tire friction at the limit.

Satellites and orbits

A satellite in circular orbit at radius r:

F_gravity = F_centripetal
GMm/r² = mv²/r
v = √(GM/r)

where G = 6.674 × 10⁻¹¹ N·m²/kg², M = 5.972 × 10²⁴ kg (Earth's mass).

Object	Orbital radius	Orbital speed	Period
ISS (LEO)	6,778 km	7.66 km/s	92 min
GPS satellite	26,560 km	3.87 km/s	11 h 58 min
Geostationary	42,164 km	3.07 km/s	23 h 56 min
Moon (Earth orbit)	384,400 km	1.02 km/s	27.3 days
Earth (Sun orbit)	1.496 × 10⁸ km	29.78 km/s	365.25 days

JavaScript — circular motion calculations

// Centripetal acceleration
function centripetalAccel(v, r) {
  return (v * v) / r;
}

// Period and frequency
function period(v, r) { return 2 * Math.PI * r / v; }
function angularVelocity(v, r) { return v / r; }

// Required centripetal force
function centripetalForce(mass, v, r) {
  return mass * v * v / r;
}

// Banked curve angle for speed v and radius r (no friction)
function bankAngle(v, r, g = 9.81) {
  return Math.atan((v * v) / (g * r));
}

// Orbital velocity for circular orbit
const G = 6.674e-11;  // N·m²/kg²
const M_EARTH = 5.972e24; // kg
const R_EARTH = 6.371e6;  // m
function orbitalVelocity(altitude_km) {
  const r = R_EARTH + altitude_km * 1000;
  return Math.sqrt(G * M_EARTH / r);
}

console.log(`ISS speed: ${orbitalVelocity(408).toFixed(0)} m/s`);  // ~7,668
console.log(`GPS speed: ${orbitalVelocity(20180).toFixed(0)} m/s`); // ~3,874
console.log(`Geo speed: ${orbitalVelocity(35786).toFixed(0)} m/s`); // ~3,074

// Daytona banking analysis
const dayBank = bankAngle(88, 366); // 200 mph at 366 m radius
console.log(`Daytona ideal bank for 200 mph: ${(dayBank * 180/Math.PI).toFixed(1)}°`);
// Real banking is 31°; if going faster, friction helps

// Maximum unbanked corner speed
function maxFlatSpeed(r, mu = 0.7, g = 9.81) {
  return Math.sqrt(mu * g * r);
}
console.log(`Max speed on flat 50m corner: ${maxFlatSpeed(50).toFixed(1)} m/s`); // ~18.5 m/s = 41 mph

Where circular motion shows up

• Orbital mechanics. Satellites, planets, moons, particle accelerators (synchrotrons).
• Vehicle dynamics. Cars cornering, trains on tracks, motorcycles leaning, banked curves.
• Amusement park rides. Roller coasters, ferris wheels, spinning rides — all centripetal force in action.
• Centrifuges. Lab equipment that separates components by density via large centripetal acceleration.
• Atomic and subatomic. Bohr model has electrons in circular orbits (classical picture); particle accelerators use magnetic fields for circular paths.
• Astronomy. Galactic rotation, star clusters, accretion disks — everything that orbits something else.
• Sports. Tennis ball spin, baseball pitch curveballs, gyroscopic stability of bicycles.

Common mistakes

• Thinking constant speed = no acceleration. Velocity is a vector. Constant speed but changing direction = nonzero acceleration. The object IS accelerating — toward the center.
• Confusing centripetal with centrifugal. Centripetal is REAL (gravity, tension, friction) and points inward. Centrifugal is fictitious, appears in the rotating frame of the object, points outward.
• Treating "centrifugal force" as a real outward force. Only in the rotating frame is it useful (as a bookkeeping device). In the inertial frame, it doesn't exist.
• Ignoring friction or banking when calculating max corner speed. On flat road, max v depends on μ. On banked curve at design speed, friction isn't even needed; above design speed, friction helps; below, gravity helps pull you down (or you slide inward).
• Forgetting tangential acceleration. If speed isn't constant, there's tangential a in addition to centripetal a.
• Mixing up linear and angular quantities. ω is angular (rad/s), v is linear (m/s). v = ωr connects them.