Torque and Rotational Dynamics

Rotation is described with angular displacement θ (in radians), angular velocity ω and angular acceleration α, which behave just like displacement, velocity and acceleration in one dimension, with clockwise or counterclockwise chosen as positive. When α is constant you can use rotational versions of the kinematic equations, and the slopes and areas of θ, ω and α graphs connect the same way they do for linear motion.

A point a distance r from the axis travels an arc length s = rθ, has tangential speed v = rω and tangential acceleration a = rα, with angles in radians. Every point on a rigid object turns through the same angle in the same time, so points farther from the axis move faster.

Torque measures how effectively a force makes something rotate about an axis. Its size is τ = rF sin θ, where θ is the angle between the force and the line from the axis to where the force acts. You can also find it as force times lever arm (the shortest distance from the axis to the force's line of action), and you only need its size and whether it turns things clockwise or counterclockwise.

Rotational inertia I measures how hard it is to change an object's rotation and depends on how far its mass sits from the axis: a small object at distance r has I = mr², and you add these up for a group of objects. Of all parallel axes, the one through the center of mass gives the smallest I, and the parallel axis theorem, I = I_cm + Md², gives I about a parallel axis a distance d away.

If the net torque on an object is zero, its angular velocity stays constant; that's rotational equilibrium, and the object can be spinning steadily, not just sitting still. Balanced torques don't guarantee balanced forces (or the other way round), so in balanced-beam and seesaw problems you set both the net force and the net torque to zero.

When the net torque isn't zero, angular velocity changes: α = τ_net / I, so more torque means more angular acceleration and more rotational inertia means less. For problems like a pulley with mass, you often apply Newton's second law to the linear motion and its rotational form to the spinning part separately, then link them.