The complete lecture — every circle comes alive in the live panel on the right as you read. Watch the velocity arrow hug the tangent, see the string get cut, tilt a motorway curve, and read the tension change around a vertical loop.
1 — Angular displacement & the radian
On a circle of radius r, motion is best described by the angle the radius sweeps. The angular displacement θ is measured in radians: one radian is the angle whose arc length equals the radius.
Angle & arcθ (rad) = s / r → s = r θ
360° = 2π rad · 180° = π rad · 1 rad ≈ 57.3°
example — fan blade
A 0.60 m fan blade turns 1.5 rev. θ and arc?
θ = 1.5 × 2π = 9.42 rad → s = rθ = 5.65 m
Exam point: the radian is a ratio of two lengths — it is dimensionless.
2 — Angular velocity ω, period & frequency
- Angular velocity ω — radians swept per second: ω = Δθ/Δt (rad/s).
- Period T — time for one revolution; frequency f = 1/T (rev/s, Hz).
The ω familyω = 2π/T = 2πf · rpm × 2π/60 = rad/s
clock second hand: ω = 2π/60 = 0.105 rad/s
Exam point: every point of a rotating wheel — hub to rim — has the same ω, but not the same speed.
3 — Linking v and ω: v = rω
Differentiate s = rθ: the linear speed along the arc is v = rω (radians only!). The velocity vector is always tangent to the circle, and in uniform circular motion its magnitude is constant while its direction changes every instant — so the body accelerates without speeding up.
The bridge equationv = r ω · tangential acceleration aₜ = r α
same ω, double the r → double the v (rim of a tonga wheel vs half-way)
example — tonga wheel
r = 0.70 m, ω = 10 rad/s. Rim speed?
v = rω = 7.0 m/s — and 3.5 m/s half-way in: same ω, half the r.
4 — No inward force, no circle
Circular motion is a forced state: something must pull the body towards the centre every instant. Remove that pull — cut the string — and Newton's first law takes over: the body continues in a straight line along the tangent, with whatever velocity it had at that moment.
Exam point: "centrifugal force" is fictitious — felt only inside the rotating frame. In the ground frame nothing pushes outward: mud from a tonga wheel, sparks from a grinding wheel and the cut ball all leave tangentially, never radially.
5 — Centripetal acceleration: a = v²/r = rω²
Turning the velocity vector needs an acceleration that points to the centre. In time Δt the radius sweeps Δθ and the velocity vector turns by the same Δθ — the triangle of radii and the triangle of velocities are similar:
Derivation sketch & resultΔv/v = Δs/r → Δv = (v/r)·vΔt → a = Δv/Δt = v²/r
a = v²/r = rω² = vω = 4π²r/T² — towards the centre
example — motorway curve
v = 20 m/s, r = 100 m.
a = v²/r = 400/100 = 4.0 m/s² ≈ 0.4 g — the push you feel in your seat.
6 — Centripetal force F = mv²/r — who provides it
Centripetal forceFₛ = m v²/r = m r ω² — always towards the centre
| Motion | Provider of Fₛ |
|---|
| stone on a string | tension |
| car on a flat curve | friction (tyres–road) |
| Moon / satellite | gravity |
| car on a banked road | component of the normal reaction |
example — tension
0.5 kg stone, r = 0.8 m, 2 rev/s.
ω = 4π = 12.57 rad/s → F = mrω² = ≈ 63 N
7 — Banking of roads: tan θ = v²/rg
On a flat curve only friction turns the car — risky in rain. Tilt the road by θ and the normal reaction N leans towards the centre and does the turning itself:
Resolve NN cos θ = mg (vertical) · N sin θ = mv²/r (horizontal)
divide: tan θ = v²/(r g) → design speed v = √(rg tanθ)
θ is independent of mass — the same motorway bank serves a Mehran and a loaded truck. The same physics tilts a turning aeroplane and a leaning cyclist.
example — motorway bank
r = 320 m, v = 25 m/s.
tanθ = 625/(320×9.8) = 0.199 → θ ≈ 11.3°
8 — Motion in a vertical circle
Whirl a bucket of water vertically: gravity helps the centripetal pull at the top and opposes it at the bottom, so the tension changes around the loop:
Tension around the looptop: T = mv²/r − mg (slackest) · bottom: T = mv²/r + mg (tightest)
minimum speed at the top (T = 0): v = √(gr) · full loop: v ≥ √(5gr) at the bottom
example — bucket of water
r = 1.0 m. Minimum top speed so no water spills?
v = √(gr) = √9.8 = 3.13 m/s — an easy arm swing.
Real world: on a chand-raat Ferris wheel you feel heaviest at the bottom and lightest at the top — your seat's normal force obeys the same two formulas.
9 — Centrifuge & spin dryer · recap
A centrifuge spins a mixture at high ω: every particle needs F = mrω² towards the axis. Denser particles need more than the liquid can supply, so they drift to the wall — separation by density (cream separator, blood centrifuge).
In a washing-machine spin, the perforated drum pushes the clothes inward onto the circle, but at each hole nothing pushes the water — so it continues along the tangent, out through the hole. At 1200 rpm and r = 0.25 m, a = rω² ≈ 400 g — clothes come out merely damp.
- θ = s/r (radian, dimensionless); ω = 2π/T = 2πf; v = rω.
- a = v²/r = rω² towards the centre; a ∝ v², a ∝ 1/r.
- F = mv²/r is a job — tension, friction, gravity or N's component does it; cut it → tangent.
- Banking: tanθ = v²/rg, mass-independent.
- Vertical circle: T = mv²/r ∓ mg (top/bottom); minimum top speed √(gr).
- Centrifuge/spin dryer: the unsteered leaves on the tangent; a = rω² can reach hundreds of g.