Theme A — Space, Time and Motion
A.2.16 Angular Velocity — IB Physics HL Revision Notes & Topical Questions
5 min read 10 parts Lesson 27 of 63 1 quiz
Lesson summary
A.2.16 Angular Velocity — IB PHYSICS SL notes: Describe circular motion using the radian, angular displacement, angular velocity, period and frequency, and relate angular and linear speed.Straight line — an object at constant velocity has zero acceleration and no resultant force.…
- Uniform circular motion
- If the string is cut
- Arc length
- Full circle
- Degrees to radians
- Radians to degrees
Part 1 of 8Straight line and circular motion
Straight line
- Straight line an object at constant velocity has zero acceleration and no resultant force.
Circle
- Circle a string attached to a puck pulls it at 90° to the velocity, so the direction changes but the speed does not.
Uniform circular motion
- Uniform circular motion the speed is constant and the velocity changes direction, so the object accelerates.
If the string is cut
- If the string is cut the puck moves off along the tangent in a straight line.
Part 2 of 8Radians
Radian
Radian the angle subtended at the centre of a circle by an arc equal in length to the radius.
Arc length
- Arc length s = rθ.
Full circle
- 2π rad = 360°, so 180° = π rad and 90° = π/2 rad.
Degrees to radians
- Degrees to radians multiply by π/180.
Radians to degrees
- Radians to degrees multiply by 180/π.
Dimensionless
- Dimensionless radians have no unit in calculations.
Part 3 of 8Angular displacement and angular velocity
Angular displacement Δθ
Angular displacement Δθ the change in angle, in radians, of a body rotating around a circle.
- Δθ = s/r, where s is the arc length.
Angular speed ω
Angular speed ω the rate of change of angular displacement, ω = Δθ/Δt.
- It is a scalar with unit rad s−1.
Angular velocity
- Angular velocity a vector with unit rad s−1, whose magnitude is the angular speed.
Constant ω
- θ = ωt.
Whole body
- Whole body all points of a rotating rigid body have the same ω, though linear speeds differ.
Independent of radius
- ω does not depend on the length of the rotating line.
Part 4 of 8Period, frequency and linear speed
Period T
- Period T the time for one revolution, in s.
Frequency f
- Frequency f the number of revolutions per second, in Hz, with f = 1/T.
Angular speed
- ω = 2π/T = 2πf, since one revolution is 2π rad.What the symbols mean: ω = angular frequency (rad s−1) · f = frequency (Hz) · T = period (s)
Linear speed
- Linear speed v = ωr = 2πr/T, found from the circumference 2πr divided by T.What the symbols mean: v = speed (m s−1) · ω = angular speed (rad s−1) · r = radius (m) · T = period (s)
Direction of v
- Direction of v the velocity is tangent to the circle, perpendicular to the radius.
Points farther out
- Points farther out they move faster;
- the Earth's equator moves at about 465 m s−1 while all points have ω = 7.3 × 10−5 rad s−1.
Part 5 of 8Using the equations
Units
- Units angles must be in radians, so convert degrees first.
Revolutions per minute
- Revolutions per minute multiply by 2π/60 to get rad s−1.
Typical values
- Typical values clock second hand 0.105 rad s−1;
- hour hand 1.45 × 10−4 rad s−1;
- spin drum at 1200 rev per minute 126 rad s−1.
Wheel
- Wheel at 25 m s−1 with radius 0.30 m, ω = 83 rad s−1.
Chain drive
- Chain drive sprockets share the chain speed, so angular velocities are in inverse proportion to the radii.
Part 6 of 8Graph link
Horizontal position
- Horizontal position plotted against time for a body in a circle, it is a sine or cosine wave of period T.
Link
- Link this connects circular motion to simple harmonic motion.
Part 7 of 8Measuring ω
Stopwatch
- Stopwatch count revolutions over a long time and divide, to reduce the timing uncertainty.
Light gate
- Light gate use it with a marked rotating arm.
Part 8 of 8More from the specification
More from the specification
Centripetal Force in Circular Motion
- An object undergoing uniform circular motion requires a net centripetal force directed toward the centre of the circle: Fc = mv² / r = mrω²,
- where ω is angular velocity in rad s⁻¹.
Impulse and Change of Momentum
- Linear impulse J is the product of average resultant force and time interval (J = FΔt),
- which is mathematically equal to the change in momentum (Δp = mv - mu).
Conservation of Momentum in Collisions
- In any closed system with no external resultant forces, total linear momentum is strictly conserved across elastic and inelastic collisions.
- (Σ pinitial = Σ pfinal)
Fluid Resistance and Viscous Drag
- Objects moving through fluids experience viscous drag;
- according to Stokes' law for spherical particles of radius r, fluid resistance is Fd = 6πηrv,
- leading to terminal fall speed.
Fluid resistance
- Drag force from a fluid opposing motion.
Normal
- Line at 90° to a surface.
Tension
- Force transmitted along a string or rope.
Buoyancy
- Another name for upthrust.
Centripetal force
- Net force toward the centre, mv2/r.
Static friction
- Friction between surfaces at rest relative to each other; fs ≤ μsFN.
Kinetic energy
- K = ½mv2, energy of motion.
Centripetal acceleration
- Ac = v2/r toward the centre.
Normal reaction
- The contact force perpendicular to a surface.
Momentum
- P = mv, a vector.
Friction
- Force parallel to the surface opposing relative motion or its tendency.
Newton
- Unit of force, kg m s-2.
Weight
- Gravitational force on a mass.
Collision
- Short interaction with large internal forces.
Impulse
- J = FΔt = Δp.What the symbols mean: F = force (N) · Δt = time the force acts (s) · Δp = change in momentum (kg m s−1); FΔt is the impulse (N s)
Key terms
| Term | Meaning |
|---|---|
| Radian | Angle subtended at the centre by an arc equal in length to the radius |
| Angular displacement | Change in angle, in radians, of a body rotating around a circle |
| Angular velocity | Rate of change of angular displacement, ω = Δθ/Δt, in rad s−1 |
| Period | Time taken for one complete revolution |
| Frequency | Number of revolutions per second, f = 1/T |
| Linear (tangential) speed | Speed along the circular path, v = ωr |
Avoid these mistakes
1
Using degrees in v = ωr — angles must be in radians.
2
Thinking ω depends on the radius — v depends on r, ω does not.
3
Saying uniform circular motion has no acceleration — the velocity direction changes.
4
Using the period as the frequency — f = 1/T.
5
Forgetting to convert revolutions per minute — multiply by 2π/60.
Marked practiceQuizzes for this lesson
Last updated:
Physics 0625 & 9702 SUVAT Mastery
Graph Interpretation
Constant Acceleration Equations (SUVAT) & Slopes
v = u + at | s = ut + ½at2 | v2 = u2 + 2as | s = ½(u + v)t
- Displacement-Time Graph: Gradient = Velocity (curve = accelerating).
- Velocity-Time Graph: Gradient = Acceleration; Area under curve = Distance travelled / Displacement.
- Free Fall in vacuum:
a = g ≈ 9.81 m/s2downwards (air resistance negligible).