Angular Motion, Torque and Power

Angular displacement θ is measured in radians, angular velocity ω in rad s⁻¹ and angular acceleration α in rad s⁻². The gradient of a θ–t graph gives ω; the gradient of an ω–t graph gives α; the signed area under an ω–t graph gives angular displacement.

For constant angular acceleration, use the same algebraic structure as linear SUVAT, replacing displacement, velocity and acceleration by their angular counterparts. A complete revolution is $2\pi$ rad. Constant-acceleration equations are not valid across an interval in which net torque or inertia varies substantially.

A tangential force at radius r produces torque $T=Fr$. More generally use the perpendicular distance between the axis and force line. Sum torques with a sign convention before applying $T_{\mathrm{net}}=I\alpha$ to a body of fixed I.

A driving torque may be partly opposed by bearing or load torque. If the machine rotates at constant angular speed, acceleration is zero and the net torque is zero, even though the driving motor supplies power.

Torque has units N m, and work has units J. Although dimensionally equivalent, they describe different physical quantities: torque acts about an axis, work is energy transferred through angular displacement. For a motor at steady speed, useful output power is load torque times angular speed.

As a shaft accelerates, some input power increases stored rotational energy and some is lost to friction. When slowing under constant frictional torque, either rotational kinematics or the work–energy equation should give the same stopping result.

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.