Topic 5.2 Notes – Connecting Linear and Rotational Motion

1. Linear and Angular Quantities Describe the Same Motion

Picture a disk rotating about a fixed axis. Every point on that disk moves in a circle, whether it is near the center (at radius $r_1$) or near the rim (at radius $r_2$, where $r_2 > r_1$).

There are two ways to describe what’s happening:

Angular quantities (describe the whole object)

• Angular position $\theta$ in radians
• Angular velocity $\omega$ in rad/s
• Angular acceleration $\alpha$ in rad/s²

If the disk turns counterclockwise, every pointshares the same $\theta$, $\omega$, and $\alpha$. On the AP exam, direction is just clockwise (CW) or counterclockwise (CCW) relative to a stated axis.

Linear (tangential) quantities (describe one specific point)

• Arc length $s$ (meters)
• Linear speed $v$ (m/s)
• Tangential acceleration $a_t$ (m/s²)

These depend on how far the point is from the axis. A point near the rim travels farther and faster than a point near the center, even though both complete one rotation together.

That “distance from the axis” is the radius $r$. It’s the bridge between angular and linear descriptions.

2. The Three Core Equations That Connect Linear and Angular Motion

All three equations follow the same pattern. Linear quantity equals radius times angular quantity.

Arc length

$$s = r\theta$$

• $s$ is the distance along the circle.
• $\theta$ must be in radians.
• Larger $r$ means larger distance for the same angle.

Quick example: if a point 0.4 m from the center rotates through $2$ radians, $s = (0.4)(2) = 0.8$ m.

If you accidentally use degrees, your answer will be wrong by a factor of $\pi/180$. That mistake shows up a lot on quizzes.

Linear velocity

$$v = r\omega$$

• $\omega$ is the same for every point in a rigid object.
• $v$ increases as $r$ increases.

If a wheel spins at $6 \text{ rad/s}$, a point at 0.5 m has $v = (0.5)(6) = 3 \text{ m/s}$. A point at 0.2 m has $v = (0.2)(6) = 1.2 \text{ m/s}$.

Same $\omega$. Different $v$.

Tangential acceleration

$$a_t = r\alpha$$

If the object speeds up or slows down, points farther out have larger tangential acceleration.

If $\alpha = 4 \text{ rad/s}^2$ and $r = 0.3 \text{ m}$, $a_t = (0.3)(4) = 1.2 \text{ m/s}^2$.

Seeing the pattern together

It’s always “multiply by $r$.”

3. What Makes a System Rigid

A rigid system keeps its shape and size while rotating.

That gives you three powerful statements:

• All points have the same angular displacement $\theta$.
• All points have the same angular velocity $\omega$.
• All points have the same angular acceleration $\alpha$.

Every point completes one full rotation in the same time. If the object’s angular speed increases, it increases for every point at the same rate.

But the linear quantities are different because $v = r\omega$ and $a_t = r\alpha$.

On FRQs, you may need to explain this in words. A strong sentence sounds like this: “All points share the same angular velocity because the object is rigid, but the linear speed is greater at larger radii since $v$ is proportional to $r$.”

That proportional reasoning is often what earns the point.

4. Moving Between Linear and Rotational Descriptions

AP questions love switching perspectives. You might be given $v$ and asked for $\omega$, or given $\alpha$ and asked about tangential acceleration at a certain radius.

When you see a problem:

1. Identify whether the given quantity is angular or linear.
2. Find the radius of the specific point.
3. Use the matching equation.

If you’re describing direction, just state CW or CCW relative to the axis provided. Nothing more complicated is expected in this course.

The key mental model is simple. Angular quantities describe the rotation of the whole object. Linear quantities describe the motion of one point. The radius connects them.