Torque and moment of inertia, the rotational form of Newton's second law, rotational kinetic energy, angular momentum and its conservation.

What this key area is asking

The SQA wants you to use torque as the rotational equivalent of force, the moment of inertia as the rotational equivalent of mass, the relationship $\tau = I\alpha$ as the rotational form of Newton's second law, rotational kinetic energy, and angular momentum with its conservation when no external torque acts.

Torque and moment of inertia

For a single point mass at radius $r$, $I = mr^2$; for several discrete masses, $I = \sum mr^2$. Extended bodies have standard results given on the data sheet (for example a uniform rod, a solid disc, a solid sphere). The same mass has a larger moment of inertia when its material is further from the axis, which is why a hollow ring is harder to spin up than a solid disc of equal mass.

The rotational form of Newton's second law

A resultant torque produces an angular acceleration inversely proportional to the moment of inertia. Doubling the torque doubles $\alpha$; doubling $I$ halves it. Every rotational problem about getting something spinning starts from this relationship, just as linear problems start from $F = ma$.

Rotational kinetic energy

For a rolling object, total kinetic energy is $\tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$, and because $v = r\omega$ the two share the available energy. This is why a hoop rolls down a slope more slowly than a solid cylinder of the same mass: the hoop's larger $I$ takes a bigger share of the energy as rotation, leaving less for translation.

Angular momentum and its conservation

Conservation of angular momentum is the rotational analogue of conservation of linear momentum. If $I$ changes while no external torque acts, $\omega$ must change to keep $I\omega$ constant: $I_1\omega_1 = I_2\omega_2$. This single idea explains a spinning skater, a diver tucking to flip faster, and a collapsing star spinning up.

Examples in context

A figure skater pulls in their arms to reduce $I$ and spin faster, conserving angular momentum. A flywheel stores large rotational kinetic energy because of its high moment of inertia, smoothing out an engine's power delivery. A neutron star spins hundreds of times per second because the collapse of its progenitor dramatically reduced $I$. A gyroscope resists changes to its axis because of its large angular momentum, which is why it is used for stabilisation and navigation.

Try this

Q1. State the rotational equivalent of mass in dynamics. [1 mark]

• Cue. The moment of inertia, $I$.

Q2. A torque of $6.0\ \text{N m}$ acts on a body of moment of inertia $3.0\ \text{kg m}^2$. Find the angular acceleration. [2 marks]

• Cue. $\alpha = \frac{\tau}{I} = \frac{6.0}{3.0} = 2.0\ \text{rad s}^{-2}$.

Q3. State what happens to the angular velocity of an isolated spinning body when its moment of inertia is reduced. [1 mark]

• Cue. It increases, because angular momentum $I\omega$ is conserved.