Torque and Angular Acceleration

$F_t$ will cause a torque $\tau = F_{t}r = mr^2\alpha$.

The total torque about an axis for a rigid body composed of discrete masses:

$$\tau = \Sigma \tau_{i} = \Sigma (mr^2)_{i}\alpha$$ thus $\tau = I\alpha$. This is the angular analogue of Newton's Second Law.

Newton's Second Law for Rotation

For any object rotating about an axis, the sum of torques gives the moment of inertia times the angular acceleration.

$$\Sigma \vec{\tau} = I\vec{\alpha}$$

Parallel Axis Theorem

If the rotational inertia $I_{cm}$ about an axis through the center of mass of a body is known, the parallel axis theorem allows the calculation of rotational inertia through any axis parallel thereto.

$$ $$ Where:

• $I_{cm}$ is the rotational inertia about an an axis through the center of mass
• $d$ is the distance from the center of mass axis to the parallel axis
• $M$ is the total mass of the object