Torque and Rotational Dynamics

Rotation is described by angular position θ (in radians), angular velocity $\omega = \frac{d\theta}{dt}$ and angular acceleration $\alpha = \frac{d\omega}{dt}$. They behave just like x, v and a: slopes and areas of graphs connect them, and when α is constant you can use rotational kinematic equations such as $\omega = \omega_0 + \alpha t$.

A point a distance r from the axis travels an arc length $s = r\theta$, so its speed is $v = r\omega$ and its tangential acceleration is $a_t = r\alpha$. Every point on a rigid object has the same ω and α, but points farther from the axis move faster.

Torque is a force's turning effect about an axis. Only the part of the force perpendicular to the line from the axis counts, so $\tau = rF\sin\theta$. That's the same as the force times the lever arm, the perpendicular distance from the axis to the force's line of action. As a vector, torque is the cross product $\vec{\tau} = \vec{r} \times \vec{F}$, but you'll usually describe its direction as clockwise or counterclockwise.

Rotational inertia measures how hard it is to change an object's rotation, and it depends on how far the mass sits from the axis: a point mass has $I = mr^2$, you add the pieces of a system, and for a solid object you integrate, $I = \int r^2\,dm$. The parallel axis theorem, $I = I_{\text{cm}} + Md^2$, gives I about any axis parallel to one through the center of mass.

An object is in rotational equilibrium when the net torque on it is zero, and then its angular velocity stays constant (Newton's first law for rotation). In statics problems you set both the net force and the net torque to zero, often choosing a pivot point that takes an unknown force out of the torque equation.

When the net torque isn't zero, the angular acceleration is $\alpha = \frac{\tau_{\text{net}}}{I}$: more torque spins an object up faster, and more rotational inertia makes it harder. Problems like a block hanging from a pulley with mass need Newton's second law for both the linear motion and the rotation, linked by $a = r\alpha$.