9.1 Rotational Motion and Angular Quantities

Angular Quantities and Rotational Motion

Rotational motion describes objects spinning around an axis. The quantities used to describe it mirror linear motion: instead of position, velocity, and acceleration along a line, you track angle, angular velocity, and angular acceleration around a circle. Mastering the parallels between linear and angular quantities makes this topic much more manageable.

Angular vs. Linear Quantities

Every linear quantity you already know has a rotational counterpart. The table below lays out the parallels:

Angular displacement ($\theta$) measures how far an object has rotated, in radians. One full revolution equals $2\pi$ radians (360°).

Angular velocity ($\omega$) is how fast the angle is changing:

$\omega_{avg} = \frac{\Delta \theta}{\Delta t}$ and instantaneously $\omega = \frac{d\theta}{dt}$

A fan blade spinning at a steady rate has constant $\omega$. The sign of $\omega$ tells you the direction of rotation (counterclockwise is typically positive).

Angular acceleration ($\alpha$) is how fast the angular velocity is changing:

$\alpha_{avg} = \frac{\Delta \omega}{\Delta t}$ and instantaneously $\alpha = \frac{d\omega}{dt}$

A merry-go-round speeding up from rest has a positive $\alpha$; one slowing to a stop has a negative $\alpha$.

Relationships in Circular Motion

The bridge between linear and angular quantities depends on the radius $r$ (the distance from the rotation axis to the point of interest):

• Arc length: $s = r\theta$ — A point on the rim of a Ferris wheel with radius 10 m that rotates through $\pi$ rad travels $s = 10 \cdot \pi \approx 31.4$ m along its circular path.
• Tangential velocity: $v = r\omega$ — Two seats on the same merry-go-round spin with the same $\omega$, but the outer seat has a larger $v$ because its $r$ is bigger.
• Tangential acceleration: $a_t = r\alpha$ — This is the component of linear acceleration along the direction of motion (tangent to the circle). It exists only when $\alpha \neq 0$.

A few useful conversions:

• $1 \text{ revolution} = 360° = 2\pi \text{ rad}$
• $\omega = 2\pi f$, where $f$ is frequency in Hz (revolutions per second)

Rotational Kinematics Equations

These are the same kinematic equations from linear motion, with angular variables swapped in. They apply whenever angular acceleration $\alpha$ is constant.

The substitution pattern is straightforward: replace $x$ with $\theta$, $v$ with $\omega$, and $a$ with $\alpha$.

Solving rotational kinematics problems:

1. List your knowns (typically three of $\omega_0$, $\omega$, $\alpha$, $\theta - \theta_0$, $t$).

2. Identify the unknown you need.

3. Pick the equation that contains your three knowns and the one unknown.

4. Solve algebraically, then substitute values. Keep angles in radians throughout.

Centripetal Acceleration

Centripetal Acceleration Concept

Any object moving in a circle is constantly changing direction, which means its velocity vector is changing even if its speed stays the same. That change in velocity requires an acceleration directed toward the center of the circle. This is centripetal acceleration.

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

A few things to note:

• Centripetal acceleration doesn't change the object's speed; it only redirects the velocity vector inward.
• In uniform circular motion (constant speed), $a_c$ is the only acceleration present. If the object is also speeding up or slowing down, there's a tangential component $a_t$ as well, and the total acceleration is the vector sum of $a_c$ and $a_t$.

Centripetal force is just Newton's second law applied to circular motion:

$F_c = ma_c = m\frac{v^2}{r} = mr\omega^2$

This isn't a new type of force. Something real must supply it: tension in a string, gravity for an orbiting planet, friction on a car rounding a curve, or the normal force on a banked road. When solving problems, always identify what is providing the centripetal force rather than just labeling it $F_c$.