Lesson 3.7

Linear vs. Angular Speed

A merry-go-round horse near the edge travels much faster than one near the center — even though they complete the same rotation in the same time. This lesson explains why.

Introduction

When an object moves in a circle, we can describe its speed in two different ways: how fast the angle is changing (angular speed) or how fast the object is physically traveling along the curved path (linear speed). These two ideas are connected by a beautifully simple formula based on the arc length equation.

Past Knowledge

Arc length is and circumference is .

Today's Goal

Calculate angular speed and linear speed , and relate them via .

Future Success

Angular speed is the gateway to physics concepts like torque, centripetal force, and orbital mechanics.

Key Concepts

Angular Speed (ω)

Angular speed measures how fast the angle is changing. It is the ratio of the angle swept to the time elapsed:

Units: radians per second (rad/s), radians per minute, etc.

For an object completing full rotations, if it makes revolutions in time , then , so:

Linear Speed (v)

Linear speed measures the actual distance traveled per unit time along the curved path:

Units: cm/s, m/s, mph, etc.

The Connection: v = rω

Since , dividing both sides by time gives:

The Bridge Formula

Linear speed = radius × angular speed

Key insight: Two points on the same rotating object share the same but have different values because their radii differ. The farther from the center, the faster the linear speed.

Worked Examples

Basic

Finding Angular Speed

Question: A wheel makes complete revolutions in seconds. Find the angular speed.

Step 1: Convert revolutions to radians.

Step 2: Divide by time.

Final Answer: rad/s

Intermediate

Finding Linear Speed from Angular Speed

Question: A Ferris wheel with radius ft rotates at rad/s. How fast is a rider traveling in ft/s?

Step 1: Apply .

Final Answer: ft/s

Advanced

Comparing Two Points on the Same Wheel

Question: A spinning disc rotates at rad/s. Point A is cm from the center and Point B is cm from the center. Find the linear speed of each point.

Point A:

Point B:

Point B travels times faster — because its radius is times larger.

Final Answer: cm/s, cm/s. Same angular speed, different linear speeds.

Common Pitfalls

Forgetting to Convert Revolutions to Radians

revolution radians, not radian. If a problem says “5 RPM,” multiply by to get the angular speed in rad/min.

Mixing Up Angular and Linear Speed

Angular speed () is in radians per time. Linear speed () is in distanceper time. If your answer for “speed” has units of radians, you gave the angular speed, not the linear speed the problem likely asked for.

Real-Life Applications

Hard Drive Platters

A traditional hard drive spins at RPM. The read/write head near the outer edge of the platter must process data at a much higher linear speed than the same head would near the center — the edge of a -inch platter is moving at roughly km/h!

Bicycle Gears

When you shift gears on a bike, you're changing the effective radius of the drive wheel. A larger gear radius increases the linear speed () for the same pedaling rate (), making you go faster but requiring more force.