Rotational Kinematics and Torque

Everything from door handles to wrenches exploits one idea: rotational effect = force × lever arm. Rotation has a full parallel dictionary to linear motion — $\theta, \omega, \alpha$ mirror $x, v, a$; torque mirrors force; moment of inertia mirrors mass. Learn the dictionary once and every linear result translates.

Rotational motion has direct analogues to every translational quantity.

Angular quantities

Constant angular acceleration equations

$$\omega = \omega_0 + \alpha t, \qquad \theta = \theta_0 + \omega_0 t + \tfrac{1}{2}\alpha t^2$$

Torque is the rotational analogue of force:

$$\vec{\tau} = \vec{r}\times\vec{F}, \qquad |\tau| = rF\sin\theta$$

Newton's second law for rotation

$$\tau_{\text{net}} = I\alpha$$

where $I = \sum m_i r_i^{2}$ is the moment of inertia.

Common moments of inertia

• Solid cylinder: $I = \tfrac{1}{2}MR^{2}$
• Solid sphere: $I = \tfrac{2}{5}MR^{2}$
• Thin rod (centre): $I = \tfrac{1}{12}ML^{2}$
• Thin hoop: $I = MR^{2}$

Physics link: The parallel axis theorem $I = I_{\text{cm}} + Md^{2}$ lets you compute $I$ about any axis from the centre-of-mass value.

Common pitfall: Moment of inertia is not just "how much mass" but "where the mass sits": $I = \sum m r^{2}$. A hoop and a disk of equal mass and radius have different $I$ — the $r^{2}$ weighting makes rim mass count far more than hub mass.