6.3 Rotational Motion

Rotational motion describes how objects spin around an axis. Where linear motion uses displacement, velocity, and acceleration along a straight line, rotational motion uses angular versions of those same quantities. The math carries over almost directly, which makes learning this topic much smoother once you see the parallels.

This section covers rotational kinematics, torque and rotational dynamics, and a few advanced concepts like angular momentum conservation and rolling motion.

Rotational Kinematics

Rotational vs linear kinematics

Rotational motion describes objects rotating about an axis (like a merry-go-round), while linear motion describes objects moving along a straight line (like a car on a highway). The key insight is that every linear kinematic variable has a rotational counterpart:

Angular displacement $\theta$ is measured in radians, angular velocity $\omega$ in rad/s, and angular acceleration $\alpha$ in rad/s². The radius $r$ of the circular path connects the angular and linear quantities.

The rotational kinematic equations mirror the linear ones you already know:

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
• $\omega = \omega_0 + \alpha t$
• $\theta = \theta_0 + \frac{1}{2}(\omega_0 + \omega)t$
• $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

These work exactly like their linear counterparts ($x = x_0 + v_0 t + \frac{1}{2}at^2$, etc.), just with angular variables swapped in. You can use them the same way: identify your knowns, pick the equation that contains your unknown, and solve.

Centripetal acceleration ($a_c$) is always present during rotational motion. It points toward the center of the circular path and keeps the object moving in a circle rather than flying off in a straight line. It's given by:

$a_c = \frac{v^2}{r} = r\omega^2$

Don't confuse this with tangential acceleration ($a_t = r\alpha$). Tangential acceleration changes the object's speed along the circular path, while centripetal acceleration changes the direction of the velocity. An object can have both at the same time if it's speeding up or slowing down while moving in a circle.

Torque and Rotational Dynamics

Torque and rotational acceleration

Torque ($\tau$) is the rotational equivalent of force. Just as a force causes linear acceleration, a torque causes rotational (angular) acceleration. Think of turning a doorknob: you apply a force at some distance from the hinge, and the door rotates.

Torque is defined as:

$\tau = rF\sin\theta$

• $r$ is the distance from the axis of rotation to where the force is applied
• $F$ is the magnitude of the applied force
• $\theta$ is the angle between the $r$ vector and the $F$ vector

Torque is measured in newton-meters (N·m). Notice that $\sin\theta$ means only the component of force perpendicular to the lever arm contributes to rotation. If you push directly toward or away from the axis ($\theta = 0°$ or $180°$), you get zero torque.

The rotational version of Newton's second law ties torque to angular acceleration:

$\sum \tau = I\alpha$

This is directly analogous to $\sum F = ma$. Here, moment of inertia ($I$) plays the role of mass. It measures how much an object resists changes in its rotational motion.

Applications of rotational dynamics

Moment of inertia ($I$) depends on both the object's mass and how that mass is distributed relative to the axis of rotation. Mass farther from the axis contributes more to $I$.

• For a single point mass at distance $r$ from the axis: $I = mr^2$
• For a system of point masses: $I = \sum m_i r_i^2$
• For extended objects (disks, rods, spheres, etc.), the formulas come from integration. You'll typically be given these or expected to look them up. For example, a solid disk rotating about its center has $I = \frac{1}{2}MR^2$, while a hollow ring has $I = MR^2$.
• The parallel-axis theorem lets you find $I$ about any axis if you know $I$ about the center of mass: $I = I_{CM} + Md^2$, where $d$ is the distance between the two axes.

Rotational kinetic energy is the energy an object has due to its spin:

$KE_r = \frac{1}{2}I\omega^2$

This is analogous to $\frac{1}{2}mv^2$ for linear motion.

Angular momentum ($L$) is defined as:

$L = I\omega$

Angular momentum is conserved whenever the net external torque on a system is zero:

$I_1\omega_1 = I_2\omega_2$

This is one of the most powerful tools in rotational problems. If the moment of inertia decreases (mass moves closer to the axis), angular velocity must increase to keep $L$ constant, and vice versa.

The work-energy theorem also applies to rotation. The net work done by torques equals the change in rotational kinetic energy:

$W = \Delta KE_r = \frac{1}{2}I\omega_2^2 - \frac{1}{2}I\omega_1^2$

Rolling motion (wheels, balls on ramps) involves both translation and rotation happening simultaneously. For rolling without slipping, the center-of-mass velocity and the angular velocity are locked together:

$v_{CM} = R\omega$

The total kinetic energy of a rolling object has two parts:

$KE_{total} = \frac{1}{2}mv_{CM}^2 + \frac{1}{2}I\omega^2$

The first term is translational kinetic energy; the second is rotational. This is why a hollow cylinder rolls down a ramp more slowly than a solid cylinder of the same mass and radius. The hollow cylinder has a larger $I$, so more of its energy goes into rotation and less into translation.

Advanced Rotational Concepts

Rotational Equilibrium and Precession

Rotational equilibrium occurs when the net torque on a system is zero ($\sum \tau = 0$), meaning there's no angular acceleration. The object either isn't rotating or is rotating at a constant angular velocity. This is the rotational analog of translational equilibrium ($\sum F = 0$).

Solving rotational equilibrium problems typically involves:

1. Choose a pivot point (you can pick any point, but choosing wisely can eliminate unknown forces from the torque equation).
2. Identify all forces and where they act.
3. Calculate the torque from each force about your chosen pivot, assigning positive/negative signs based on the direction of rotation (counterclockwise is typically positive).
4. Set $\sum \tau = 0$ and solve for the unknown.

Precession is the slow rotation of the axis of a spinning object. You can see this in a spinning top that wobbles: the spin axis itself traces out a cone. Gyroscopes exhibit the same behavior. Precession occurs because gravity exerts a torque on the spinning object, and that torque changes the direction of the angular momentum vector rather than stopping the spin.

Conservation of Angular Momentum

Angular momentum conservation ($L_i = L_f$) applies whenever no net external torque acts on a system. The classic example: a figure skater spinning with arms extended has a large $I$ and a moderate $\omega$. When they pull their arms in, $I$ decreases, so $\omega$ must increase to keep $L$ constant. That's why they spin faster with arms tucked in.

This principle also shows up in orbital mechanics (planets speed up when closer to the Sun), collapsing stars (neutron stars spin extremely fast because the star's radius shrinks dramatically), and any system where mass redistributes relative to a rotation axis.