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Theme A — Space, Time and Motion

A.4.9 Rotational Kinetic Energy — IB Physics HL Revision Notes & Topical Questions

5 min read 8 parts Lesson 51 of 63 1 quiz

📚 IB Physics HL Hub / Topical Revision
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pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
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Lesson summary

A.4.9 Rotational Kinetic Energy — IB PHYSICS HL notes: Calculate rotational kinetic energy and apply it to rolling bodies and bodies on slopes.A rotating body has kinetic energy associated with its angular velocity.Ek = ½Iω2, the rotational analogue of ½mv2.Ek = L2/2I, the analog…

  • Rolling motion
  • Without slipping, each point has a different linear velocity
  • Solid cylinder (I
  • Solid sphere (I
  • For a solid sphere mgh
  • A rolling body reaches a lower speed than a sliding one

Part 1 of 6Rotational kinetic energy

A rotating body has kinetic energy associated with its angular velocity.
Ek
½Iω2, the rotational analogue of ½mv2.
Ek
L2/2I, the analogue of p2/2m.
I is in kg m2, ω in rad s-1 and L in kg m2 s-1.

Part 2 of 6Rolling without slipping

1Rolling motion
Rolling motion rotation combined with translation.
2
With enough friction the object rolls without slipping.
  • The contact point is at rest.
  • The centre of mass has speed v = ωr.
  • The top point moves at 2v = 2ωr.
3
When a disc slips, every point has the same linear velocity and ω is zero.
4Without slipping, each point has a different linear velocity
  • Without slipping, each point has a different linear velocity proportional to r from the centre.

Part 3 of 6Total kinetic energy of a rolling body

Ek total
½mv2 + ½Iω2.
Use v
ωr to write it in terms of one speed.
Solid cylinder (I
½mr2) — Ek total = ¾mv2.
Solid sphere (I
(2/5)mr2) — Ek total = (7/10)mv2.

Part 4 of 6Rolling down a slope

1
At the top the body has Ep = mgh and is at rest.
2
At the bottom the potential energy has become translational and rotational kinetic energy.
3
Equate mgh = ½mv2 + ½Iω2.
4
Substitute I and use v = ωr to find v or ω.
1For a solid sphere mgh
(7/10)mv2.
2A rolling body reaches a lower speed than a sliding one
  • A rolling body reaches a lower speed than a sliding one because some energy goes into rotation.

Part 5 of 6Changes of rotational energy

Changes of rotational energy
Combining flywheels changes I and ω together.
  • For a flywheel of ½MR2 with a second of ½M and ½R placed on top, the moment of inertia rises to (9/16)MR2.
  • At 2/3 the angular velocity, Ek becomes ½ of the original.
Compute Ek before and after using ½Iω2
  • Compute Ek before and after using ½Iω2 and track both I and ω.
Rotational energy can be converted to other forms, e.g. thermal energy through friction.

Part 6 of 6More from the specification

More from the specification
Rotational kinetic work in flywheel systems
  • Rotational kinetic work in flywheel systems calculating half moment of inertia times angular velocity squared gives energy stored in a flywheel system.
Rotational kinetic energy
  • E equals one half I omega squared, where I is moment of inertia and omega angular speed.
Torque and work
  • Torque and work work done by a torque equals torque times angular displacement.
Angular acceleration
  • Angular acceleration resultant torque divided by moment of inertia gives it.
Rolling
  • Rolling a rolling object has both translational and rotational kinetic energy, so it accelerates down a slope more slowly than a sliding one.
Conservation of energy
  • Conservation of energy on a slope, loss of gravitational energy equals the gain in kinetic energy of both types.
Equilibrium
  • State with zero net force.
Perpendicular
  • At right angles.
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)
Radius
  • Distance from the centre to the circular path.
Constant
  • Not changing.
Rotational equilibrium
  • Στ = 0; no angular acceleration.
Angular impulse
  • Product of net torque and the time it acts, τΔt.
Perpendicular distance
  • Distance from the axis to the line of motion, r⊥ = r sinθ.
Torque
  • Rotational effect of a force, τ = rF sinθ.
Acceleration
  • Rate of change of velocity.
Resultant
  • Vector sum of two or more vectors.
Force
  • A push or pull measured in newtons.
Distance
  • Total length of path travelled, a scalar.
Displacement
  • Change in position.
Couple
  • A pair of equal opposite forces with different lines of action.
Clockwise
  • Turning in the direction of a clock's hands.

Key terms

TermMeaning
Rotational kinetic energyEnergy of rotation, ½Iω^2.
Rolling without slippingRolling with the contact point at rest, v = ωr.
Moment of inertiaResistance to change in rotational motion.
Angular velocityRate of change of angular displacement.
Translational kinetic energyEnergy of motion of the centre of mass, ½mv2.
FlywheelRotating disc used to store rotational energy.

Avoid these mistakes

1
Using only ½mv2 for a rolling body — rotational energy is also present.
2
Forgetting that the top of a rolling wheel moves at 2v.
3
Using ω in rev/s or degrees/s in ½Iω^2.
4
Assuming the same Ek for all rolling shapes — it depends on I.
5
Treating a sliding and a rolling body as having equal speeds at the bottom of a slope.

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