Rotational Kinematics and Torque

Rotational Kinematics

Door handles and wrenches exploit one core idea: rotational effect equals force times lever arm. Rotation has a full parallel dictionary to linear motion where $\theta, \omega, \alpha$ mirror $x, v, a$, torque mirrors force, and moment of inertia mirrors mass.

For constant angular acceleration, use these direct kinematic equations:

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

Torque & Moment of Inertia

Torque is the rotational analogue of force, defined as $\vec{\tau} = \vec{r}\times\vec{F}$ with magnitude $|\tau| = rF\sin\theta$. Newton's second law for rotation is $\tau_{\text{net}} = I\alpha$, where $I = \sum m_i r_i^2$ is the moment of inertia.

Common shapes and their inertia

The parallel axis theorem ($I = I_{\text{cm}} + Md^2$) shifts any axis from the centre of mass.

Common pitfall: $I$ is not just mass, but where mass sits. Rim mass counts far more than hub mass because of $r^2$ weighting.