Torque and Rotational Dynamics

Notes

Rotational kinematics

Rotation has its own versions of position, velocity and acceleration:

• Angular displacement $\Delta\theta$ (radians), angular velocity $\displaystyle \omega = \frac{\Delta\theta}{\Delta t}$, angular acceleration $\displaystyle \alpha = \frac{\Delta\omega}{\Delta t}$.
• Same equations as linear motion: $\omega = \omega_0 + \alpha t$, $\displaystyle \theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$, $\omega^2 = \omega_0^2 + 2\alpha\Delta\theta$.
• Linking to linear motion at radius $r$: $v = r\omega$, $a_t = r\alpha$.

Torque

Torque is the turning effect of a force: $\tau = rF\sin\theta$, where $r$ is the distance from the axis and $\theta$ is the angle between $\vec r$ and $\vec F$.

• Push far from the hinge and perpendicular to the door for the most torque.
• Forces through the axis make no torque.

Rotational inertia

Rotational inertia $I$ measures resistance to changes in rotation. It depends on mass and how far the mass is from the axis: $\displaystyle I = \sum mr^2$.

• A hoop has more rotational inertia than a solid disk of the same mass and radius.
• Parallel axis theorem: $I = I_{cm} + Md^2$.

Newton's second law for rotation

• Rotational equilibrium: $\tau_{net} = 0$ (and translational equilibrium: $F_{net} = 0$). Balanced seesaws and static beams.