Torque and rotational equilibrium

• Torque about an axis: $\tau = Fr\sin\theta$.

• Use perpendicular force component or perpendicular distance from pivot: $\tau = F r_\perp$.

• Include sense of rotation in working: choose clockwise and counterclockwise signs consistently.

• Rotational equilibrium means resultant torque = 0.

• A body can have zero resultant torque even when individual torques act.

• In exam questions, identify the pivot/axis first, then calculate torques of all forces about that point.

This diagram shows how torque depends on force, distance from the pivot, and angle of application. It is ideal for remembering why pushing farther from the hinge or more nearly perpendicular gives a larger turning effect. Source

Angular motion quantities and equations

• Angular displacement $\Delta\theta$ is measured in radians.

• Angular speed / angular velocity: $\omega = \dfrac{\Delta\theta}{\Delta t}$.

• Angular acceleration: $\alpha = \dfrac{\Delta\omega}{\Delta t}$.

• For uniform angular acceleration, use the rotational SUVAT analogies:

• $\Delta\theta = \dfrac{\omega_f+\omega_i}{2}t$

• $\omega_f = \omega_i + \alpha t$

• $\Delta\theta = \omega_i t + \dfrac{1}{2}\alpha t^2$

• $\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$

• Treat these exactly like linear kinematics, but with $\theta, \omega, \alpha$ replacing $s, v, a$.

• Check units carefully: rad, rad s$^{-1}$, rad s$^{-2}$.

Moment of inertia

• Moment of inertia $I$ is the rotational analogue of mass.

• It depends on how mass is distributed relative to the axis of rotation.

• For point masses: $I = \sum mr^2$.

• Mass farther from the axis gives a much larger moment of inertia because of the $r^2$ dependence.

• Larger $I$ means the body is harder to angularly accelerate.

• If an equation for a particular shape is needed, it will be given when necessary.

• For extended bodies in linear motion, you may treat the mass as concentrated at the centre of mass.

This figure compares the moments of inertia of standard bodies such as hoops, disks, rods, and spheres. It helps you see immediately that the same mass gives different rotational behaviour depending on how far the mass lies from the axis. Source

Newton’s second law for rotation

• Unbalanced torque causes angular acceleration.

• Rotational form of Newton’s second law: $\tau = I\alpha$.

• For the same torque, a body with larger $I$ has smaller $\alpha$.

• For the same body, a larger net torque gives a larger angular acceleration.

• This is one of the most important exam analogies: $F=ma$ becomes $\tau = I\alpha$.

• In multi-step problems, often combine force equations with torque equations.

Angular momentum and angular impulse

• Angular momentum of a rigid body: $L = I\omega$.

• If resultant torque = 0, then angular momentum is conserved.

• Conservation idea: $I_i\omega_i = I_f\omega_f$ when no external resultant torque acts.

• If $I$ decreases, then $\omega$ increases to keep $L$ constant.

• Angular impulse changes angular momentum: $\Delta L = \tau\Delta t = \Delta(I\omega)$.

• This is the rotational analogue of impulse = change in linear momentum.

• Typical applications: spinning skater, coupled rotating bodies, collisions involving rotation.

This image shows conservation of angular momentum when external torque is negligible. Pulling the arms inward reduces moment of inertia, so the skater’s angular speed increases. Source

Rotational kinetic energy

• Rotational kinetic energy: $E_k = \dfrac{1}{2}I\omega^2$.

• Also: $E_k = \dfrac{L^2}{2I}$.

• Do not confuse angular momentum conservation with kinetic energy conservation.

• In many problems, $L$ is conserved but $E_k$ changes if work is done internally or energy is lost.

• When a rotating system changes shape, check separately whether torque is zero and whether energy is conserved.

This figure shows a wheel’s free-body diagram and the key relations for rolling without slipping, including $v=r\omega$ and $a=r\alpha$. It is especially useful for linking translational motion to rotational motion in HL mechanics problems. Source

Linear–rotational analogies to memorize

• Displacement $s$ ↔ angular displacement $\theta$

• Velocity $v$ ↔ angular speed $\omega$

• Acceleration $a$ ↔ angular acceleration $\alpha$

• Mass $m$ ↔ moment of inertia $I$

• Force $F$ ↔ torque $\tau$

• Momentum $p$ ↔ angular momentum $L$

• $F=ma$ ↔ $\tau = I\alpha$

• $E_k=\dfrac{1}{2}mv^2$ ↔ $E_k=\dfrac{1}{2}I\omega^2$