Rigid-body Rotational Motion

When an object can translate in two dimensions and can rotate only about an axis that is perpendicular to the plane, Newton’s second law can be used to show that:

$$I_O\dot{\omega }=M_O$$

where:

• $\omega$ is the angular velocity of the mass about an axis through point $O$ fixed in an inertial reference frame of reference
• $I_O$ is the mass moment of inertia of the body about point $O$. Torque $T$ is sometimes used instead of $M$.
• $M_O$ is the sum of moments applied to the body about point $O$

Energy of Rotational Motion

The work done by a moment $M$ causing a rotation through an angle $\theta$ is

$$W={\int }_0^{\theta }Md\theta$$

Multiplying both sides of $I\dot{\omega }=M$ by $\omega dt$ gives

$$I\omega d\omega =Md\theta$$

because $\omega =d\theta /dt$.

Integrating both sides gives

$${\int }_0^{\omega }I\omega d\omega =\frac{1}{2}I{\omega }^2={\int }_0^{\theta }Md\theta$$

Thus, the work done by the moment $M$ produces kinetic energy of rotation:

$$\text{KE}=\frac{1}{2}I{\omega }^2$$