Linear and Angular Speed

Linear speed is how fast a point moves along a circular path: v = \frac{s}{t}, the arc length per unit of time. Angular speed is how fast the angle changes: \omega = \frac{\theta}{t}, in radians per unit of time. The two are linked by v = r\omega, where r is the radius.

Before this lesson: You should know Radians, Arc Length and Unit Rates.

Two Ways to Measure Speed on a Circle

Picture two children on a merry-go-round, one near the center and one at the edge. Both go around once in the same time, so they turn through the same angle. They have the same angular speed.

The child at the edge travels a much bigger circle in that time, so she moves faster along her path. She has a greater linear speed. The farther a point is from the center, the faster it moves for the same turning rate.

Formulas for Linear and Angular Speed

The Greek letter \omega is omega. The connection comes from arc length in radians, s = r\theta. Divide both sides by t to get \frac{s}{t} = r \cdot \frac{\theta}{t}, which is v = r\omega.

Speeds are often given in revolutions per minute (rpm). One revolution is 2\pi radians, so multiply rpm by 2\pi to get radians per minute.

Worked Examples

Example 1: Find an angular speed

A wheel turns through 12 radians in 3 seconds. Find its angular speed.

\omega = \frac{\theta}{t}

\omega = \frac{12}{3}

\omega = 4

Answer: 4 radians per second

Example 2: Convert revolutions per minute

A ceiling fan spins at 300 revolutions per minute. Find its angular speed in radians per minute.

\omega = 300 \cdot 2\pi

\omega = 600\pi

600\pi \approx 1884.96

Answer: 600\pi \approx 1884.96 radians per minute

Example 3: Linear speed from angular speed

A carousel horse is 4 meters from the center. The carousel turns at 0.5 radian per second. How fast is the horse moving?

v = r\omega

v = 4 \cdot 0.5

v = 2

Answer: 2 meters per second. A horse 2 meters from the center would move only 1 meter per second.

Example 4: A bike wheel in miles per hour

A bike wheel has a diameter of 26 inches and turns at 180 revolutions per minute. How fast is the bike going, in miles per hour? (There are 63,360 inches in a mile.)

r = \frac{26}{2} = 13

\omega = 180 \cdot 2\pi = 360\pi

v = 13 \cdot 360\pi

v = 4680\pi \approx 14{,}702.65

14{,}702.65 \cdot 60 = 882{,}159

\frac{882{,}159}{63{,}360} \approx 13.92

Answer: about 13.92 miles per hour

Example 5: Word problem with Earth’s rotation

Earth’s radius is about 3960 miles, and it turns once every 24 hours. How fast is a person on the equator moving because of this rotation?

\omega = \frac{2\pi}{24}

\omega = \frac{\pi}{12}

v = 3960 \cdot \frac{\pi}{12}

v = 330\pi

330\pi \approx 1036.73

Answer: about 1037 miles per hour, even though it feels like you are standing still.

Common Mistakes

Related Methods

Solve for angular speed

Divide both sides of v = r\omega by r to get \omega = \frac{v}{r}. If a point 4 feet from the center moves at 20 feet per second, \omega = \frac{20}{4} = 5 radians per second.

Use the circumference

For whole revolutions, you can also multiply the number of revolutions by the circumference, 2\pi r. At 180 rpm, the bike wheel in Example 4 travels 180 \cdot 26\pi = 4680\pi inches per minute, the same answer.

Linear and angular velocity

In physics, velocity also includes a direction. Angular velocity is positive for counterclockwise turning and negative for clockwise turning, like the angles in standard position.

Practice Problems

1. A point travels 30 centimeters along a circle in 6 seconds. Find its linear speed.
2. An angle of 5\pi radians is swept out in 10 seconds. Find the angular speed.
3. A record turns at 45 revolutions per minute. Find its angular speed in radians per minute.
4. A point is 8 centimeters from the center of a wheel turning at 3 radians per second. Find its linear speed.
5. A point 4 feet from the center moves at 20 feet per second. Find the angular speed.
6. A Ferris wheel with a radius of 25 meters makes one revolution every 2 minutes. Find the linear speed of a seat.
7. The second hand of a clock is 6 centimeters long. Find the linear speed of its tip in centimeters per second.
8. A car wheel with a radius of 0.3 meter spins at 50 radians per second. How fast is the car moving?
9. A bike moves at 220 inches per second, and its wheels have a radius of 14 inches. Find the wheels’ angular speed in radians per second and in revolutions per minute.
10. A lawn mower blade is 0.25 meter long from the center to the tip and spins at 3000 rpm. Find the linear speed of the tip in meters per second.

v = \frac{s}{t}

v = \frac{30}{6}

v = 5

\omega = \frac{\theta}{t}

\omega = \frac{5\pi}{10}

\omega = \frac{\pi}{2} \approx 1.57

\omega = 45 \cdot 2\pi

\omega = 90\pi \approx 282.74

v = r\omega

v = 8 \cdot 3

v = 24

\omega = \frac{v}{r}

\omega = \frac{20}{4}

\omega = 5

\omega = \frac{2\pi}{2} = \pi

v = 25 \cdot \pi

25\pi \approx 78.54

\omega = \frac{2\pi}{60} = \frac{\pi}{30}

v = 6 \cdot \frac{\pi}{30}

v = \frac{\pi}{5} \approx 0.63

v = r\omega

v = 0.3 \cdot 50

v = 15

\omega = \frac{v}{r} = \frac{220}{14}

\omega = \frac{110}{7} \approx 15.71

\frac{110}{7} \cdot 60 = \frac{6600}{7} \approx 942.86

\frac{942.86}{2\pi} \approx 150.06

\omega = 3000 \cdot 2\pi = 6000\pi

\omega = \frac{6000\pi}{60} = 100\pi

v = 0.25 \cdot 100\pi = 25\pi

25\pi \approx 78.54

FAQ

What is the difference between linear speed and angular speed?

Linear speed is the distance a point travels along its circular path per unit of time. Angular speed is the angle it turns through per unit of time.

What is the formula that connects linear and angular speed?

It is v = r\omega, where r is the radius and \omega is the angular speed in radians per unit of time.

How do you convert rpm to radians per second?

Multiply by 2\pi to get radians per minute, then divide by 60. For example, 60 \text{ rpm} = \frac{60 \cdot 2\pi}{60} = 2\pi radians per second.

Why must angular speed be in radians?

The formula v = r\omega comes from s = r\theta, which only works with radians. With degrees, you would need an extra conversion factor.

Summary

• Linear speed: v = \frac{s}{t}, distance along the arc per unit of time.
• Angular speed: \omega = \frac{\theta}{t}, radians per unit of time.
• They are connected by v = r\omega.
• One revolution is 2\pi radians, so multiply rpm by 2\pi.
• Points farther from the center move faster for the same angular speed.

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