A fixed-axis rotor has I=3 kg m² and net torque 12 N m. Find its angular acceleration.
Answer and explanation
4 rad/s²
Use angular acceleration = torque/I = 12/3.
Subjects
University mechanics
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Sample exercises
A rotor with I=2 kg m² spins at 5 rad/s. Find its rotational kinetic energy.
Answer and explanation
25 J
K=Iω²/2=(2)(25)/2.
A wheel of radius 0.4 m rolls without slipping at centre speed 2 m/s. Find ω.
Answer and explanation
5 rad/s
The rolling constraint gives ω=v/R=2/0.4.
Which change definitely changes the moment of inertia about an axis?
- Changing angular speed only
- Renaming the axis
- Changing the clock origin
- Moving some mass farther from that axis
Answer and explanation
Moving some mass farther from that axis
Inertia is the mass-weighted squared distance from the specified axis.
Explain why a rolling solid cylinder has more kinetic energy than a point mass of equal mass moving at the same centre speed.
Answer and explanation
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.
Using I=mr²/2 for a solid cylinder, m=4 kg and centre speed 3 m/s, find its total rolling energy.
Answer and explanation
27 J
Translation is 18 J and rotation is mv²/4=9 J.
Study notes
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.
UniversityRotational dynamics and rolling constraints · revision notes
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.
UniversityDriven oscillations, damping and resonance
A linear mass-spring oscillator obeys mx″+kx=0. Its natural angular frequency is sqrt(k/m), not k/m.
UniversityDriven oscillations, damping and resonance · revision notes
A linear mass-spring oscillator obeys mx″+kx=0. Its natural angular frequency is sqrt(k/m), not k/m.
UniversityCentral forces, angular momentum and effective potential
A central force is directed along the line joining the particle to a fixed centre and has magnitude depending only on radial distance in the usual model.
UniversityCentral forces, angular momentum and effective potential · revision notes
A central force is directed along the line joining the particle to a fixed centre and has magnitude depending only on radial distance in the usual model.
UniversityWave equations, normal modes and boundary conditions
The ideal one-dimensional wave equation is ytt=c²yxx. The parameter c is propagation speed for this nondispersive model.
UniversityWave equations, normal modes and boundary conditions · revision notes
The ideal one-dimensional wave equation is ytt=c²yxx. The parameter c is propagation speed for this nondispersive model.
UniversityRotational kinetic energy
Use moment of inertia about the stated rotation axis. Angular speed is in radians per second.
UniversityRotational kinetic energy · formula reference
Use moment of inertia about the stated rotation axis. Angular speed is in radians per second.
UniversityRolling without slipping
A uniform solid cylinder rolls without slipping, so ω = v/R; both translation and rotation contribute.
UniversityRolling without slipping · formula reference
A uniform solid cylinder rolls without slipping, so ω = v/R; both translation and rotation contribute.
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