# 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.

Course: [University mechanics and waves](https://studybrix.com/en/courses/university-mechanics/index.md)
Level: university
Prerequisites: Newtonian mechanics, derivatives and energy.

## 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.

## Practice questions

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

Answer: 4 rad/s²

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

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

Answer: 25 J

Explanation: K=Iω²/2=(2)(25)/2.

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

Answer: 5 rad/s

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

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

A. Changing angular speed only
B. Renaming the axis
C. Changing the clock origin
D. Moving some mass farther from that axis

Answer: Moving some mass farther from that axis

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

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

Sample answer / marking guidance: The cylinder has translational energy mv²/2 plus rotation about its centre Iω²/2. The point-mass model includes only translation.

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

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

Answer: 27 J

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

### 7. For a fixed axis, what sets the sign of angular acceleration when I is positive?

A. The sign of elapsed time
B. The object's colour
C. The sign of net torque
D. The sign of mass

Answer: The sign of net torque

Explanation: With constant positive I, α=τnet/I.

### 8. State two assumptions behind the equation τnet=Iα in its simple scalar form.

Sample answer / marking guidance: Rotation is about the specified fixed axis and I is constant; use the torque component about that axis in an appropriate inertial treatment.

Explanation: Award credit for the axis and constant-inertia assumptions rather than applying the relation to arbitrary deforming bodies.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.