So far, we've talked about motion in a straight line: a car driving down a road, a ball falling through the air. This is called linear or translational motion. But what happens when an object spins, turns, or orbits? Think of a spinning top, a planet orbiting the sun, or a wheel on a bicycle. This is rotational motion.

Instead of tracking how far an object travels in meters, we track how much it turns in angles. We're shifting our perspective from a straight path to a circular one.

To describe rotation, we need a new set of tools. Let's start with the basics. Imagine a single point on the edge of a spinning record. How can we describe its position?

We can use an angle, $\theta$, measured from a reference line (like the positive x-axis). As the record spins, this angle changes. The change in the angle is called angular displacement.

Here, $\theta_f$ is the final angle and $\theta_i$ is the initial angle. While we often measure angles in degrees, in physics it's much more useful to use radians.

Just as linear velocity describes how quickly position changes, angular velocity ($\omega$) describes how quickly the angle changes. It's the rate of rotation.

Angular velocity is measured in radians per second (rad/s). A positive value usually means counter-clockwise rotation, while a negative value means clockwise.

And if the angular velocity changes? That's angular acceleration ($\alpha$). It tells us how quickly the rotation is speeding up or slowing down.

Angular acceleration is measured in radians per second squared (rad/s²).

Every point on a spinning object is also moving linearly. Think about a person on a carousel. They are rotating around the center, but they also have a linear speed. How do these two types of motion relate?

The answer depends on the radius ($r$), the distance from the axis of rotation. The length of the arc ($s$) a point travels is related to the angular displacement ($\,\theta\,$) and the radius.

This equation only works when $\theta$ is in radians.

From this, we can find the relationship between linear speed ($v$) and angular speed ($\omega$). The linear speed of a point on a rotating object is often called its tangential velocity because its direction is always tangent to the circular path.

If we divide the arc length equation by time, we get the relationship between tangential speed and angular speed.

This tells us that points farther from the center of rotation move faster. That's why the outer edge of a record travels a greater distance and has a higher linear speed than a point near the center, even though they both have the same angular velocity.

Similarly, tangential acceleration ($a_t$) is related to angular acceleration ($\alpha$).

This is the acceleration that speeds up or slows down the rotation. But there's another kind of acceleration to consider. For an object to move in a circle, it must constantly be pulled toward the center. This is caused by centripetal acceleration ($a_c$), which is always directed radially inward.

The great thing about these new rotational quantities is that they behave just like their linear counterparts. The equations we used for linear motion with constant acceleration have direct equivalents in rotational motion. We just swap the variables.

Displacement ($x$) becomes angular displacement ($\theta$). Velocity ($v$) becomes angular velocity ($\omega$). Acceleration ($a$) becomes angular acceleration ($\alpha$).

This parallel makes solving problems involving constant angular acceleration very familiar. If you can solve a linear kinematics problem, you can solve a rotational one.

Now you have the basic language to describe rotational motion. We can track how an object turns, how fast it's turning, and how that rate of turning changes, all using angles and radians.