Difficulty: Intermediate | Prerequisites: Newton's laws of motion, basic trigonometry, vectors
Circular motion is movement along a curved path; rotational motion is spinning around a fixed axis. Both are governed by quantities that mirror linear mechanics: angular velocity replaces speed, centripetal force replaces net force, and moment of inertia replaces mass. Master the formulas linking these quantities and you can solve nearly every problem in this unit.
Circular motion
Motion of an object along a circular path. Think of it as any trajectory that traces out a circle or arc, such as a planet orbiting the sun or a car rounding a bend.
Rotational motion
Motion of an object spinning around a fixed axis. In simple terms, this means the object stays in place but turns, like a wheel on an axle or the Earth rotating once per day.
Angular velocity (ω)
A vector quantity describing how fast an object rotates around a fixed axis, defined as the change in angular displacement over time. Measured in radians per second (rad/s). Think of it as the rotational equivalent of linear velocity.
Centripetal force
The net inward force that keeps an object moving in a circle, directed towards the centre of the circular path. Given by F = mv²/r. In simple terms, this is whatever force (tension, gravity, friction) is pulling the object inward so it curves instead of flying off in a straight line.
Moment of inertia (I)
A scalar quantity describing an object's resistance to changes in rotational motion, analogous to mass in linear motion. It depends on both the object's mass and how that mass is distributed relative to the axis of rotation. Think of it as "rotational mass": the further the mass sits from the axis, the harder the object is to spin or stop.
Uniform circular motion occurs when an object moves at constant speed along a circular path. The speed is constant, but the velocity is continuously changing direction, which means the object is accelerating.
Centripetal acceleration points towards the centre of the circle and has magnitude a = v²/r. This acceleration does not change the object's speed, only its direction.
Centripetal force is the net force responsible for centripetal acceleration. It is not a new type of force; it is whatever real force (gravity, tension, friction, normal force) happens to point inward.
For a car on a flat curve, friction provides the centripetal force.
For a planet in orbit, gravity provides it.
For a ball on a string, tension provides it.
Period (T) is the time for one complete revolution. It relates to speed by v = 2πr/T.
Frequency (f) is the number of revolutions per second, f = 1/T. Angular velocity connects to both: ω = 2πf = 2π/T.
In non-uniform circular motion, the object's speed changes as well as its direction. There is a tangential component of acceleration in addition to the centripetal component.
Angular displacement (θ) measures how far an object has rotated, in radians. One full revolution = 2π radians = 360°.
Angular velocity (ω) = dθ/dt, the rate of change of angular displacement. Direction follows the right-hand rule: curl your fingers in the direction of rotation and your thumb points along the angular velocity vector.
Angular acceleration (α) = dω/dt. A positive α means the object is speeding up its rotation (if ω is positive); a negative α means it is slowing down.
Moment of inertia (I) depends on mass distribution relative to the axis.
Point mass: I = mr²
Solid disc or cylinder (about central axis): I = ½mr²
Solid sphere (about diameter): I = ⅖mr²
Thin ring or hoop (about central axis): I = mr²
The parallel axis theorem lets you shift axes: I = I_cm + md², where d is the distance between the centre-of-mass axis and the new axis.
Torque (τ) is the rotational equivalent of force: τ = r × F, or τ = rF sin θ. Net torque drives angular acceleration via Newton's second law for rotation: τ_net = Iα.
Rotational kinematic equations mirror their linear counterparts, swapping x for θ, v for ω, a for α:
ω = ω₀ + αt
θ = ω₀t + ½αt²
ω² = ω₀² + 2αθ
Circular motion
Centripetal acceleration: a_c = v²/r = ω²r
Centripetal force: F_c = mv²/r = mω²r
Speed from period: v = 2πr/T
Angular velocity from frequency: ω = 2πf
Rotational motion
Torque: τ = rF sin θ
Newton's second law (rotation): τ_net = Iα
Rotational kinetic energy: KE_rot = ½Iω²
Angular momentum: L = Iω
Relationship between linear and angular quantities: v = ωr, a_tangential = αr
Moment of inertia reference
Shape | Axis | I |
|---|---|---|
Point mass | Distance r from axis | mr² |
Thin ring / hoop | Central axis | mr² |
Solid disc / cylinder | Central axis | ½mr² |
Solid sphere | Through diameter | ⅖mr² |
Thin rod (length L) | Through centre | ¹⁄₁₂mL² |
Thin rod (length L) | Through one end | ⅓mL² |
Car on a curved road: Friction between the tyres and the road provides the centripetal force. On a banked curve, a component of the normal force contributes as well, reducing the reliance on friction and allowing higher speeds.
Figure skater spinning: When a skater pulls their arms in, their moment of inertia decreases. Because angular momentum (L = Iω) is conserved (no external torque), their angular velocity increases and they spin faster. Extending the arms reverses the effect.
Centrifuge: Spins samples at high angular velocity to separate substances by density. The required centripetal force increases with mass, so heavier particles migrate outward.
Spinning top / gyroscope: The top's moment of inertia and angular velocity together determine its stability. Precession (the slow wobble of the spin axis) occurs because of the torque due to gravity acting on the centre of mass.
Students often think centripetal force is a separate, special type of force. It is not. It is simply the label for whatever net inward force (gravity, tension, friction, normal force) happens to keep an object on a curved path.
Students often confuse centripetal and centrifugal force. Centripetal force is the real inward force; centrifugal force is a fictitious (pseudo) force that appears only in a rotating reference frame.
Students often believe that an object in circular motion is in equilibrium because its speed is constant. The object is accelerating (direction is changing), so the net force is non-zero.
Students often assume moment of inertia depends only on mass. It also depends on how that mass is distributed relative to the axis. A hollow cylinder and a solid cylinder of the same mass and radius have different moments of inertia.
⚠️ Problems asking "what provides the centripetal force?" are extremely common. Identify the real force (gravity, friction, tension, normal force) rather than answering "centripetal force."
⚠️ Know the moment-of-inertia table cold. Exams frequently ask you to compare rolling objects (solid vs hollow cylinder) on an incline, and the answer depends entirely on I.
⚠️ Conservation of angular momentum problems (figure skater, collapsing star, spinning disc catching a mass) appear regularly. Set L_initial = L_final and solve for the unknown.
⚠️ Be comfortable converting between linear and angular quantities: v = ωr, a_t = αr. Many problems mix the two and expect you to move fluently between them.
⚠️ For objects rolling without slipping, the total kinetic energy is KE = ½mv² + ½Iω², and v = ωr. Forgetting the rotational term is a common mark-losing error.
True or false: An object moving in a circle at constant speed has zero acceleration.
False. It has centripetal acceleration directed inward.
Fill in the blank: The centripetal force on an object of mass m moving at speed v in a circle of radius r is F = ____.
mv²/r
True or false: Moment of inertia depends only on an object's mass.
False. It also depends on how mass is distributed relative to the axis of rotation.
Fill in the blank: When a figure skater pulls in their arms, their angular velocity ____ because angular momentum is conserved.
Increases
True or false: Centripetal force is a distinct fundamental force, like gravity or electromagnetism.
False. It is the net inward force provided by whatever real force acts on the object.
Q: A 1,200 kg car rounds a flat curve of radius 50 m at 20 m/s. What is the centripetal force, and what provides it?
A: F = mv²/r = (1200)(20²)/50 = 9,600 N. On a flat curve, static friction between the tyres and the road provides this force.
Q: A solid disc and a thin hoop of equal mass and radius roll down the same incline from rest. Which reaches the bottom first, and why?
A: The solid disc reaches the bottom first. It has a smaller moment of inertia (½mr² vs mr²), so less of its gravitational potential energy goes into rotational kinetic energy and more into translational speed.
Q: A figure skater spins at 2 rev/s with arms extended (I = 4.0 kg·m²). She pulls her arms in, reducing her moment of inertia to 1.6 kg·m². What is her new angular velocity?
A: Conservation of angular momentum: I₁ω₁ = I₂ω₂. So ω₂ = (4.0 × 2)/1.6 = 5 rev/s.
Q: A ball on a string is swung in a vertical circle. At the very top of the circle, what forces contribute to the centripetal force?
A: At the top, both gravity and string tension point downward (toward the centre). The centripetal force equals mg + T. If the ball moves just fast enough to maintain the circle, T approaches zero and the centripetal force equals mg.
Q: Explain why a hollow cylinder rolls more slowly down an incline than a solid cylinder of the same mass and radius.
A: The hollow cylinder has I = mr² compared to ½mr² for the solid one. A larger fraction of its energy goes into rotation rather than translation, so it accelerates more slowly down the slope.
This material connects directly to Newton's laws of motion: centripetal force is simply Newton's second law applied to curved paths, and torque is the rotational form of F = ma. If Newton's laws feel shaky, revisit those notes first.
Angular momentum conservation ties into the broader principle of conservation laws (energy, linear momentum, angular momentum). Problems involving collisions on rotating platforms or orbiting bodies combine all three.
Rolling motion bridges this unit with translational kinematics and energy methods: you need both ½mv² and ½Iω² to write a correct energy equation for any object that rolls.
Circular motion, rotational motion, angular velocity, angular acceleration, centripetal force, centrifugal force, moment of inertia, torque, angular momentum, conservation of angular momentum, rotational kinetic energy, rolling without slipping, uniform circular motion, non-uniform circular motion, radians, period, frequency, banked curve, parallel axis theorem, right-hand rule, rotational kinematics, Newton's second law for rotation, spinning, revolving, orbital motion