Rotational and Circular Motion

Rotational and Circular Motion deals with bodies that rotate about an axis or move along curved paths. The PMDC MDCAT 2026 syllabus tests angular displacement, angular velocity, and the conversions between angular and linear quantities. Centripetal acceleration is a perennial favourite in MCQs.

PMC Table of Specifications. Three subtopics — Angular Displacement, Angular Velocity, and the Relation between Angular & Linear Quantities — with centripetal acceleration weaving through all of them.

Angular Displacement

Angular displacement θ is the angle (in radians) swept out by a rotating radius vector. SI unit: radian (rad). It is dimensionless but treated as a vector along the axis of rotation (right-hand rule).

For a particle traversing arc length s on a circle of radius r:

θ = s/r (radians)

Useful conversions:

1 revolution = 2π rad = 360°.
1 rad = 180°/π ≈ 57.3°.
π/2 rad = 90°; π rad = 180°.

Angular Velocity

Angular velocity ω is the rate of change of angular displacement:

ω = Δθ/Δt (instantaneous: ω = dθ/dt)

SI unit: rad/s. Direction is along the axis of rotation by the right-hand rule. Closely related quantities:

Time period T: Time for one revolution: T = 2π/ω.
Frequency f: Number of revolutions per second: f = 1/T = ω/(2π). SI unit: hertz (Hz).
Angular acceleration α: Rate of change of angular velocity: α = dω/dt. SI unit: rad/s².

Equations of rotational kinematics

Analogous to linear SUVAT, for constant α:

ω = ω₀ + αt
θ = ω₀t + ½αt²
ω² = ω₀² + 2αθ

Relation between Angular and Linear Quantities

For a particle at radius r from the rotation axis, the angular and linear quantities are connected by:

Linear vs Angular quantities — full mapping
Quantity	Linear	Angular	Relation
Displacement	s (m)	θ (rad)	s = rθ
Velocity	v (m/s)	ω (rad/s)	v = rω
Acceleration	a (m/s²)	α (rad/s²)	a_t = rα (tangential)
Acceleration (curved)	a_c (m/s²)	—	a_c = v²/r = rω² (centripetal)
Force / "force"	F = ma	τ = Iα	F_c = mv²/r = mrω²
Mass / inertia	m (kg)	I (kg·m²)	I = Σmr² for point masses
Momentum	p = mv	L = Iω	L = r × p
Kinetic energy	½mv²	½Iω²	—

Memory aid. "Multiply by r to go from angular to linear." s = rθ, v = rω, a_t = rα. Then for centripetal a_c = v²/r — the only one that divides by r.

Real sources of centripetal force

The centripetal force is not a new force — it is whichever force happens to point toward the centre. Recognise the source for each scenario:

Satellite / planet: gravity provides F_c → GMm/r² = mv²/r → orbital v = √(GM/r).
Car on a flat circular track: friction between tyres and road.
Object on a string (vertical / horizontal): tension.
Electron in Bohr atom: electrostatic Coulomb force from the nucleus.
Banked road (no friction): horizontal component of normal reaction; tanθ = v²/(rg).
Conical pendulum: horizontal component of tension; T cosθ = mg, T sinθ = mv²/r.

Centripetal vs centrifugal

Centripetal force is the real, inward force that keeps a body moving along a circular path. It is supplied by gravity, friction, tension, normal reaction, etc. Centrifugal force is a pseudo-force experienced only in a non-inertial (rotating) frame — it does not exist in an inertial frame.

Worked example — rotational kinematics

A wheel starts from rest and accelerates uniformly to 600 rpm in 10 s. Find (a) the angular acceleration, (b) the number of revolutions in this time.

Solution: Final ω = 600 rpm = 600/60 rev/s = 10 rev/s = 10 × 2π = 20π rad/s.

(a) α = (ω − ω₀)/t = (20π − 0)/10 = 2π rad/s².
(b) θ = ω₀t + ½αt² = 0 + ½ × 2π × 10² = 100π rad → revolutions = 100π/(2π) = 50 revolutions.

Common trap. In uniform circular motion the speed is constant but the velocity is not — its direction is constantly changing. Hence there is acceleration (centripetal, directed toward the centre) even when speed is unchanged.

Worked MCQs

Five MCQs that capture the high-yield testing patterns for this chapter.

Q1. An angle of 60° in radians is:

π/2
π/4
π/3
π/6

Convert: 60° × (π/180) = π/3 rad.

Q2. A wheel rotates at 300 rpm. Its angular velocity in rad/s is:

5
10π
20π
300

300 rpm = 5 rev/s. ω = 2πf = 2π × 5 = 10π rad/s.

Q3. The linear speed of a point on the rim of a disc of radius 0.2 m rotating at 50 rad/s is:

2.5 m/s
5 m/s
10 m/s
250 m/s

v = rω = 0.2 × 50 = 10 m/s.

Q4. A car of mass 1000 kg goes round a circular track of radius 50 m at 10 m/s. The centripetal force needed is:

200 N
1000 N
2000 N
5000 N

F_c = mv²/r = 1000 × 100 / 50 = 2000 N.

Q5. In uniform circular motion:

Speed and velocity are both constant
Speed and velocity are both variable
Speed is constant but velocity changes
Velocity is constant but speed changes

Magnitude (speed) is unchanged, but direction of motion is constantly changing — therefore velocity changes and there is centripetal acceleration.

Quick Recap

1 rev = 2π rad = 360°.
ω = dθ/dt; T = 2π/ω; f = 1/T.
v = rω, a_t = rα, a_c = v²/r = rω².
Centripetal force F_c = mv²/r directed toward centre.
Centrifugal is a pseudo-force, only in rotating frames.
Speed constant, velocity changes → centripetal acceleration exists.

Test yourself. Take a timed Circular Motion quiz or browse all Physics MCQs to lock these concepts in.