Kinematics and Dynamics of Rotational Motion About a Fixed Axis

For a rigid body rotating about a FIXED axis with a CONSTANT angular acceleration $\alpha$, the angular

displacement $\theta$, angular velocity $\omega$ and angular acceleration $\alpha$ obey three equations

that are exact rotational counterparts of the familiar straight-line kinematic equations, obtained simply

by replacing every linear quantity with its angular counterpart ($s\to\theta$, $v\to\omega$, $a\to\alpha$;

see the full correspondence in Section 5.15):

where $\omega_0$ is the angular velocity at time $t=0$. These three equations are used in exactly the same

way, and under exactly the same restriction (constant angular acceleration only), as their straight-line

counterparts from earlier chapters.

Rotational dynamics answers the separate question of WHAT PRODUCES a given angular acceleration.

Exactly as Newton's second law $F = ma$ connects a net force to the linear acceleration it produces in a

body of mass $m$, its rotational counterpart connects a net external torque to the angular acceleration it

produces in a rigid body of moment of inertia $I$, about the same fixed axis:

This equation, together with the moment of inertia values from Section 5.11 and the kinematic equations

above, is what makes it possible to solve a complete rotational problem end to end -- for example, finding

how long a constant applied torque takes to spin a flywheel up from rest to some target angular velocity,

or how quickly a constant frictional torque brings a freely spinning wheel to a stop, exactly as worked out

in this chapter's numerical problems. The corresponding rotational kinetic energy of a body spinning at …