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Physics · University

Rotational dynamics and rolling constraints

Angular velocity is the time derivative of angular displacement. For a fixed axis and constant moment of inertia, net torque equals I times angular acceleration.

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Prerequisites

Newtonian mechanics, derivatives and energy.

Study notes

Rotation

Angular velocity is the time derivative of angular displacement. For a fixed axis and constant moment of inertia, net torque equals I times angular acceleration.

Energy

Rigid-body rotation about a fixed axis has kinetic energy Iω²/2. A rolling body generally has both translational and rotational kinetic energy.

Constraints

For rolling without slipping, the centre speed equals Rω. That kinematic condition does not imply friction always dissipates energy.

Model limits

The moment of inertia depends on mass distribution and the chosen axis. A point-mass formula cannot replace the inertia of an extended object without justification.

Worked examples

A fixed-axis rotor has I=3 kg m² and net torque 12 N m. Find its angular acceleration.

Use angular acceleration = torque/I = 12/3.

4 rad/s²

A rotor with I=2 kg m² spins at 5 rad/s. Find its rotational kinetic energy.

K=Iω²/2=(2)(25)/2.

25 J

Check your understanding

  1. A wheel of radius 0.4 m rolls without slipping at centre speed 2 m/s. Find ω.

  2. Which change definitely changes the moment of inertia about an axis?

    1. Changing angular speed only
    2. Renaming the axis
    3. Changing the clock origin
    4. Moving some mass farther from that axis

  3. Explain why a rolling solid cylinder has more kinetic energy than a point mass of equal mass moving at the same centre speed.

  4. Using I=mr²/2 for a solid cylinder, m=4 kg and centre speed 3 m/s, find its total rolling energy.

Answers and marking guidance

Answers and marking notes
  1. 5 rad/s

    The rolling constraint gives ω=v/R=2/0.4.

  2. Moving some mass farther from that axis

    Inertia is the mass-weighted squared distance from the specified axis.

  3. The cylinder has translational energy mv²/2 plus rotation about its centre Iω²/2. The point-mass model includes only translation.

    Credit both terms and use of the same centre speed; do not claim that all rolling bodies have the same I.

  4. 27 J

    Translation is 18 J and rotation is mv²/4=9 J.

Further reading and review references

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