rotational kinematics angular velocity
When a rigid body such as a wheel or a disk turns about an axis, every point on it turns through the same angle θ in the same time. That is why rotation is far simpler to describe with a single angle than with the position of each point. The x·v·a of straight-line motion carry over to rotation as angular position θ, angular velocity ω, and angular acceleration α, with only the names changed, and the same relations of differentiation and integration hold. The bridge between the two worlds is the radius r alone: multiply angular velocity by the radius and you get the linear speed at which that point actually moves (v=rω). This lesson sets up the three quantities that describe rotation, one after another, and crosses that bridge.
Drag the angle θ with the slider and the disk turns; every point away from the axis sweeps through the same angle. The inner blue dot and the outer gold dot turn through the same angle, but the outer one travels a longer arc. A point at radius r covers an arc of length s = rθ, with θ measured in radians. A radian is arc length divided by radius, and thanks to that definition the arc-length formula needs no extra constant. The arc length in the readout belongs to the outer gold dot.
Angular velocity ω says how fast the angle changes and is written ω = dθ/dt. The graph is a θ-t graph, time on the horizontal axis and angle on the vertical, and at constant angular velocity it is a straight green line. The height of the dashed step divided by its width, the slope of the line, is ω. Increase ω with the slider and the line steepens, so the disk turns through a larger angle in the same time. It is exactly the structure of straight-line motion, where velocity was the slope of the position-time graph.
Angular acceleration α says how fast the angular velocity changes and is defined as α = dω/dt. This graph is an ω-t graph, with angular velocity on the vertical axis, and the slope of the blue line is α. Set α positive with the slider and the line climbs to the right, so the rotation speeds up, as it always does when α>0. With α<0 the line falls and the rotation slows, and at the instant it crosses the horizontal axis the angular velocity changes sign and the body starts turning the other way. The initial angular velocity is held fixed, so the line starts at the same point and only its slope changes.
Every point on the disk turns with the same angular velocity, yet the outer points move faster, because the tangential speed of a point at radius r is v = rω. The disk carries an inner blue dot and an orange dot on the rim, at the positions r=1 and r=2. Drag ω with the slider and both tangential-speed arrows grow together, the outer one always twice as long. It is the same reason the outer seats of a ride feel more thrilling and the edge of a record moves faster than its center.
With constant angular acceleration, the constant-acceleration formulas of straight-line motion carry straight over to rotation. Drag α with the slider and the blue line of the ω-t graph tilts, while the area of the shaded trapezoid under it is the angle Δθ swept so far, just as the area under the velocity graph was the displacement in straight-line motion. The formulas only change letters: ω=ω₀+αt, Δθ=ω₀t+½αt², ω²=ω₀²+2αΔθ. In the readout you can see the final angular velocity and the swept angle grow together as α increases.