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Theme A — Space, Time and Motion

A.4.3 Angular Displacement, Velocity and Acceleration — IB Physics HL Revision Notes & Topical Questions

5 min read 8 parts Lesson 45 of 63 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Lesson summary

A.4.3 Angular Displacement, Velocity and Acceleration — IB PHYSICS HL notes: Define angular displacement, velocity and acceleration and relate them to linear quantities.Angular displacement θ = the change in angle through which a rigid body rotates about a fixed point.It is measu…

  • It is measured in radians
  • Linear displacement along the arc
  • Angular velocity ω
  • Linear speed v
  • For a full cycle θ
  • Angular acceleration α

Part 1 of 6Angular displacement

Angular displacement
Angular displacement θ
Angular displacement θ the change in angle through which a rigid body rotates about a fixed point.
It is measured in radians
  • It is measured in radians an angle is arc length divided by radius.
Linear displacement along the arc
  • Linear displacement along the arc s = rθ.
  • s is the arc length, r the distance from the axis and θ the angle in radians.

Part 2 of 6Angular velocity

1Angular velocity ω
Angular velocity ω the rate of change of angular displacement with time.
  • ω = Δθ/Δt, in rad s-1.
2Linear speed v
ωr — where r is the distance from the axis.
3For a full cycle θ
2π — so ω = v/r = 2πf = 2π/T.
4Linear speed v
2πfr = 2πr/T.

Part 3 of 6Angular acceleration

Angular acceleration
Angular acceleration α
Angular acceleration α the rate of change of angular velocity with time.
  • α = Δω/Δt, in rad s-2, with Δω = ωf − ωi.
Link to linear acceleration
  • α = Δv/(rΔt) = a/r, so a = αr.
Angular acceleration is not centripetal acceleration
  • Angular acceleration is not centripetal acceleration they are different quantities.

Part 4 of 6Linear and angular variables compared

VariableLinearAngular
Displacements = rθθ = s/r
Velocityv = ωrω = v/r
Accelerationa = αrα = a/r
1
Each angular quantity equals its linear counterpart divided by r.
2All points on a rotating rigid body share ω and α
  • All points on a rotating rigid body share ω and α their linear speeds depend on r.

Part 5 of 6Graphs of rotational motion

Angular displacement
Angular displacement the area under the angular velocity–time graph.
Angular velocity
Angular velocity the gradient of the angular displacement–time graph.
  • It is also the area under the angular acceleration–time graph.
Angular acceleration
Angular acceleration the gradient of the angular velocity–time graph.
Interpret these graphs as for linear motion.

Part 6 of 6More from the specification

1Rotational dynamics and torque
  • Rotational dynamics and torque applying a tangential force at distance r produces a resultant torque that induces angular acceleration on an object.
2Moment of inertia and mass distribution
  • Moment of inertia and mass distribution moment of inertia measures rotational resistance,
    • increasing significantly when object mass is distributed further from rotation axes.
3Couple torque without linear translation
  • Couple torque without linear translation a couple consisting of two equal opposite parallel forces generates pure rotational torque without net linear acceleration.
4Torque, Couple, and Rotational Motion
  • A torque is the rotational analogue of force, defined as τ = F·r sin θ; a couple consists of two equal and opposite parallel forces separated by a perpendicular distance, exerting a pure turning torque on an object.
    • (or moment of a force)
5Moment of Inertia and Rotational Dynamics
  • The moment of inertia I quantifies an object's resistance to rotational acceleration;
    • the rotational version of Newton's second law states that resultant torque equals moment of inertia multiplied by angular acceleration (τ_net = I·α).
6Conservation of Angular Momentum
  • In the absence of an external resultant torque, total angular momentum L = I·ω of a rotating system remains constant;
    • decreasing the moment of inertia increases angular velocity ω.
7Equilibrium
  • State with zero net force.
8Perpendicular
  • At right angles.
9Impulse
  • 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)
10Constant
  • Not changing.
11Rotational equilibrium
  • Στ = 0; no angular acceleration.
12Angular impulse
  • Product of net torque and the time it acts, τΔt.
13Kinetic energy
  • K = ½mv2, energy of motion.
14Perpendicular distance
  • Distance from the axis to the line of motion, r⊥ = r sinθ.
15Torque
  • Rotational effect of a force, τ = rF sinθ.
16Moment
  • Turning effect of a force.
17Inertia
  • Resistance of an object to changes in its motion.
18Momentum
  • P = mv, a vector.
19Resultant
  • Vector sum of two or more vectors.
20Force
  • A push or pull measured in newtons.
21Couple
  • A pair of equal opposite forces with different lines of action.
22Clockwise
  • Turning in the direction of a clock's hands.

Key terms

TermMeaning
Angular displacementChange in angle in radians through which a body rotates.
Angular velocityRate of change of angular displacement.
Angular accelerationRate of change of angular velocity.
RadianAngle equal to arc length divided by radius.
Arc lengthDistance along a circular path, s = rθ.
FrequencyNumber of rotations per second.

Avoid these mistakes

1
Using degrees instead of radians in s = rθ or v = ωr.
2
Confusing angular acceleration with centripetal acceleration.
3
Forgetting that linear speed depends on r while ω does not.
4
Using total length instead of arc length for s.
5
Mixing up rad s-1 and revolutions per second.

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/s2 downwards (air resistance negligible).
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