10.1 Angular Acceleration

Angular acceleration describes how quickly an object's rotation speed changes over time. It's the rotational equivalent of linear acceleration, and you'll need it whenever a spinning object speeds up, slows down, or changes direction of rotation.

Circular Motion and Angular Acceleration

Uniform vs non-uniform circular motion

These two types of circular motion differ in one key way: whether the speed is changing.

• Uniform circular motion means the object moves with constant angular velocity ($\omega$) along a circular path. The only acceleration present is centripetal acceleration ($a_c$), which points toward the center and keeps the object on its curved path. A planet in a roughly circular orbit is a good example.
• Non-uniform circular motion means the angular velocity is changing over time. This introduces two additional quantities: angular acceleration ($\alpha$), which describes the rate of that change, and tangential acceleration ($a_t$), which acts along the direction of motion to speed the object up or slow it down. A figure skater pulling their arms in to spin faster is a classic example.

In non-uniform motion, the object also undergoes angular displacement ($\theta$), which tracks how far it has rotated from its starting position.

Calculation of angular acceleration

Angular acceleration ($\alpha$) is defined as the change in angular velocity divided by the time it takes for that change:

$\alpha = \frac{\Delta \omega}{\Delta t}$

The units are radians per second squared (rad/s²). A positive $\alpha$ means the object is spinning faster; a negative $\alpha$ means it's slowing down.

When angular acceleration is constant, you can use three kinematic equations that mirror the linear kinematic equations you already know:

1. $\omega_f = \omega_i + \alpha t$
2. $\theta = \omega_i t + \frac{1}{2} \alpha t^2$
3. $\omega_f^2 = \omega_i^2 + 2 \alpha \theta$

Each equation is useful depending on which variable you're missing. For example, if you don't know the time, equation 3 lets you relate angular velocities directly to angular displacement.

Quick example: A wheel starts from rest ($\omega_i = 0$) and accelerates at $\alpha = 2 \text{ rad/s}^2$ for 5 seconds. Using equation 1: $\omega_f = 0 + (2)(5) = 10 \text{ rad/s}$. Using equation 2: $\theta = 0 + \frac{1}{2}(2)(5^2) = 25 \text{ rad}$.

Linear and angular acceleration relationship

Tangential acceleration ($a_t$) and angular acceleration ($\alpha$) are connected through the radius of the circular path:

$a_t = r \alpha$

This tells you that points farther from the center of rotation experience greater tangential (linear) acceleration, even though every point on a rigid object shares the same angular acceleration. Think of a merry-go-round: a rider near the edge feels a much stronger push than one near the center.

The total acceleration of a point in non-uniform circular motion combines two perpendicular components:

• Tangential acceleration ($a_t$): acts along the direction of motion, changing the speed
• Centripetal acceleration ($a_c$): acts toward the center, changing the direction

Because these two are perpendicular, you find the total acceleration with:

$a_{total} = \sqrt{a_t^2 + a_c^2}$

Note: "radial acceleration" and "centripetal acceleration" refer to the same thing here. Both point inward along the radius toward the center.

Applications of angular acceleration

Angular acceleration shows up in many everyday situations:

• A figure skater pulling their arms inward spins faster (angular velocity increases, so $\alpha > 0$)
• A car braking causes its wheels to undergo negative angular acceleration
• A pendulum swinging back and forth continuously changes its angular velocity as it moves through its arc

What causes angular acceleration? Torque ($\tau$). The relationship is:

$\tau = I \alpha$

This is the rotational version of Newton's second law ($F = ma$). Here, $I$ is the moment of inertia, which measures how much an object resists changes in its rotation. A larger moment of inertia means you need more torque to achieve the same angular acceleration.

Moment of inertia depends on both mass and how that mass is distributed relative to the axis of rotation:

• Point mass at distance $r$: $I = mr^2$
• Solid cylinder or disk (rotating about its center): $I = \frac{1}{2}mr^2$

Mass concentrated farther from the axis means a larger $I$ and more resistance to angular acceleration.

Rotational dynamics and energy

Angular momentum ($L$) ties together moment of inertia and angular velocity:

$L = I\omega$

Angular momentum is conserved when no net external torque acts on a system. This is why the figure skater example works: pulling arms inward decreases $I$, so $\omega$ must increase to keep $L$ constant. This conservation law is the rotational analog of conservation of linear momentum.