Rotational Motion

Rotational Motion

Rotational motion deals with objects rotating about a fixed axis. Concepts like torque, angular velocity, and moment of inertia are analogous to force, linear velocity, and mass.

Angular Velocity and Acceleration

Angular Displacement ($\theta$): The angle through which an object rotates. Measured in radians.

Angular Velocity ($\omega$): The rate of change of angular displacement.

• $\omega = \frac{d\theta}{dt}$
• Unit: rad/s

Angular Acceleration ($\alpha$): The rate of change of angular velocity.

• $\alpha = \frac{d\omega}{dt}$
• Unit: rad/s$^2$

Relationships to Linear Quantities:

• $s = r \theta$ (Arc Length)
• $v = r \omega$ (Tangential Velocity)
• $a_t = r \alpha$ (Tangential Acceleration)
• $a_r = r \omega^2$ (Radial Acceleration)

Torque and Moment of Inertia

Torque ($\tau$)

Torque is the rotational equivalent of force, causing angular acceleration. $\tau = r F \sin \theta$ where $r$ is the distance from the pivot to the point of force application, $F$ is the force magnitude, and $\theta$ is the angle between the position vector and force vector.

Moment of Inertia ($I$)

Moment of Inertia is the rotational equivalent of mass, representing resistance to changes in rotational motion. $I = \sum m_i r_i^2 = \int r^2 \, dm$ For common shapes:

• Solid Cylinder/Disk: $I = \frac{1}{2} M R^2$
• Thin Rod (center): $I = \frac{1}{12} M L^2$
• Solid Sphere: $I = \frac{2}{5} M R^2$

Newton's Second Law for Rotation

$\sum \tau = I \alpha$

Rotational Kinetic Energy

The kinetic energy of a rotating object is: $KE_{rot} = \frac{1}{2} I \omega^2$

For an object that rolls without slipping (translating and rotating): $KE_{total} = \frac{1}{2} m v_{cm}^2 + \frac{1}{2} I \omega^2$

Angular Momentum and Conservation

Angular Momentum ($L$)

$L = I \omega$ For a point particle: $\mathbf{L} = \mathbf{r} \times \mathbf{p}$.

Conservation of Angular Momentum: If the net external torque on a system is zero, the total angular momentum is conserved. $L_i = L_f$ $I_i \omega_i = I_f \omega_f$