Moment of Inertia and Flywheels

A point mass at distance r from an axis has moment of inertia $mr^2$. Add these contributions for an arrangement of particles. Mass farther from the axis contributes much more because of the square. Always identify the axis: the same object has different moments of inertia about different axes.

An extended body is modelled by summing contributions from small elements. Use the expression supplied for its shape rather than assuming every object is a thin ring. Moment of inertia has units kg m²; it is a property of the mass distribution and does not itself depend on rotational speed.

For fixed I, doubling angular speed quadruples rotational kinetic energy. Moving more mass towards the rim increases stored energy at the same speed, but real designs must withstand stress and bearing loads. The useful energy released as a flywheel slows is the difference of its initial and final energies.

A flywheel smooths changes in speed when driving torque varies over a cycle: it speeds up when supplied power exceeds demand and slows when demand exceeds supply. It is an energy store, not a source that generates energy without an input. Energy storage can support regenerative braking or intermittent production machinery.

Convert revolutions per minute to radians per second using $\omega=2\pi n/60$. The angular speed is common to a rigid body, while linear speed $v=r\omega$ increases with radius. Do not use one rim speed as the speed of every mass element.

If a rotating object also translates, account for its translational and rotational energies separately. In flywheel questions the axle is often fixed, so only the rotational term is needed. State whether bearing friction and aerodynamic losses are neglected.

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.