Rotational motion is all about spinning objects. We'll look at how things rotate, from merry-go-rounds to spinning tops, and learn the math behind it. It's like linear motion, but with a twist.

We'll explore torque, the rotational version of force, and how it makes things spin. We'll also dive into rotational energy, momentum, and equilibrium. These concepts help explain everything from spinning wheels to figure skaters' pirouettes.

Rotational Kinematics

Rotational vs linear kinematics

Rotational motion involves objects rotating about an axis (merry-go-round) described by angular displacement ($\theta$), angular velocity ($\omega$), and angular acceleration ($\alpha$) while linear displacement ($x$), velocity ($v$), and acceleration ($a$) describe motion along a straight line (car on a highway) Rotational kinematic equations are analogous to linear kinematic equations:

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$ describes angular displacement (total angle rotated)
• $\omega = \omega_0 + \alpha t$ describes angular velocity (rate of change of angular displacement)
• $\theta = \theta_0 + \frac{1}{2}(\omega_0 + \omega)t$ describes average angular velocity
• $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ relates angular velocity squared to angular acceleration and displacement

Linear and angular quantities are related by the radius ($r$) of the circular path:

• $x = r\theta$ converts angular displacement to linear displacement (distance traveled along the circular path)
• $v = r\omega$ converts angular velocity to linear velocity (speed along the circular path)
• $a_t = r\alpha$ converts angular acceleration to tangential acceleration (acceleration along the circular path)

Centripetal acceleration ($a_c$) is always present in rotational motion directed towards the center of the circular path (keeps object moving in a circle) and is given by $a_c = \frac{v^2}{r} = r\omega^2$

Torque and Rotational Dynamics

Torque and rotational acceleration

Torque ($\tau$) is the rotational equivalent of force causing an object to rotate about an axis (turning a doorknob) defined as $\tau = rF\sin\theta$, where $r$ is the lever arm (perpendicular distance from the axis of rotation to the line of action of the force), $F$ is the force, and $\theta$ is the angle between $r$ and $F$ measured in newton-meters (N·m) Net torque determines the rotational acceleration ($\alpha$) of an object according to $\sum \tau = I\alpha$, where $I$ is the moment of inertia (object's resistance to rotational acceleration) analogous to Newton's second law ($\sum F = ma$)

Applications of rotational dynamics

Moment of inertia ($I$) depends on the object's mass and its distribution about the axis of rotation:

• For a point mass: $I = mr^2$ (mass times distance squared from axis)
• For extended objects, use integration or the parallel-axis theorem
• For a rigid body, the moment of inertia remains constant during rotation

Rotational kinetic energy ($KE_r$) is the energy associated with rotational motion given by $KE_r = \frac{1}{2}I\omega^2$ (half moment of inertia times angular velocity squared)

Conservation of angular momentum ($L = I\omega$) applies when net external torque is zero such that $I_1\omega_1 = I_2\omega_2$ (initial equals final)

Work-energy theorem applies to rotational motion where net work done by torques equals the change in rotational kinetic energy: $W = \Delta KE_r = \frac{1}{2}I(\omega_2^2 - \omega_1^2)$

Rolling motion (wheels, gears) involves both translational and rotational motion with no slipping condition: $v_{CM} = R\omega$ (center of mass velocity equals radius times angular velocity) and total kinetic energy: $KE = \frac{1}{2}mv_{CM}^2 + \frac{1}{2}I\omega^2$ (translational plus rotational)

Advanced Rotational Concepts

Rotational Equilibrium and Precession

• Rotational equilibrium occurs when the net torque on a system is zero, resulting in no angular acceleration
• Precession is the slow rotation of the axis of rotation itself, often seen in spinning tops and gyroscopes

Conservation of Angular Momentum

• Angular momentum is conserved in the absence of external torques
• This principle explains phenomena such as the increased rotation speed of figure skaters when they pull in their arms