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Lesson 4: Angular and Linear Speed

Estimated time: 25-30 minutes

Learning Objectives

By the end of this lesson, you will be able to:

Define angular speed and linear speed
Convert between revolutions per minute and radians per second
Use the relationship v = rω to connect linear and angular speed
Solve real-world problems involving wheels, gears, and rotating objects

Angular Speed

When an object rotates, we can describe how fast it spins using angular speed.

Angular Speed (ω) — The rate at which the angle changes over time:

ω = θ / t

where θ is the angle (in radians) and t is the time. Common units: radians per second (rad/s) or radians per minute.

One full revolution = 2π radians. So an object spinning at n revolutions per minute (rpm) has angular speed:

ω = 2πn rad/min

Example 1: Finding Angular Speed

A wheel makes 150 revolutions per minute. Find the angular speed in radians per second.

Solution:

Step 1: Convert to rad/min: ω = 150 × 2π = 300π rad/min

Step 2: Convert to rad/s: 300π / 60 = 5π rad/s

Answer: ω = 5π ≈ 15.71 rad/s

Example 2: Turntable Angular Speed

A record player turntable spins at 33 1/3 rpm. Find the angular speed in rad/s.

Solution:

ω = (100/3) × 2π / 60 = 200π / 180 = 10π/9 ≈ 3.49 rad/s

Linear Speed

A point on the edge of a rotating object also moves along a circular path. The speed along that path is the linear speed.

Linear Speed (v) — The rate at which a point on a rotating object moves along its circular path:

v = s / t

where s is the arc length traveled and t is the time. Units are typically length per time (e.g., m/s, ft/min).

Example 3: Linear Speed from Arc Length

A point on the rim of a wheel travels 44 inches in 2 seconds along its circular path. Find its linear speed.

Solution:

v = s / t = 44 / 2 = 22 in/s

The Relationship Between Linear and Angular Speed

Since s = rθ, dividing both sides by t gives:

v = rω

Linear speed = radius × angular speed

This is valid when ω is in radians per unit time.

This is a powerful relationship: two points on the same rotating object have the same angular speed but different linear speeds depending on their distance from the center. Points farther from the center move faster.

Example 4: Linear Speed of a Wheel

A bicycle wheel has a radius of 13 inches and rotates at 200 rpm. Find the linear speed of a point on the rim in inches per second.

Solution:

Step 1: Angular speed: ω = 200 × 2π = 400π rad/min

Step 2: Linear speed: v = rω = 13 × 400π = 5200π in/min

Step 3: Convert: 5200π / 60 ≈ 272.3 in/s

Example 5: Comparing Points on a Wheel

A wheel rotates at 10 rad/s. Compare the linear speeds of points at r = 2 cm and r = 5 cm from the center.

Solution:

At r = 2 cm: v = 2 × 10 = 20 cm/s

At r = 5 cm: v = 5 × 10 = 50 cm/s

The point farther from the center moves 2.5 times faster, even though both complete one revolution in the same time.

Converting Between Units

Speed problems often require unit conversions. Here are some useful conversions:

1 mile = 5280 feet
1 foot = 12 inches
1 hour = 60 minutes = 3600 seconds
1 revolution = 2π radians

Example 6: Speed of a Car Tire

A car tire has a diameter of 26 inches and rotates at 480 rpm. Find the car's speed in miles per hour.

Solution:

Step 1: Radius = 13 inches

Step 2: ω = 480 × 2π = 960π rad/min

Step 3: v = rω = 13 × 960π = 12480π in/min

Step 4: Convert to mph: 12480π in/min × 60 min/hr ÷ 12 in/ft ÷ 5280 ft/mi

= 12480π × 60 / (12 × 5280) = 748800π / 63360 ≈ 37.1 mph

Applications: Gears and Pulleys

When two gears or pulleys are connected (by teeth or a belt), the linear speed at the point of contact is the same for both. This lets us relate their angular speeds.

Connected Gears/Pulleys: v₁ = v₂, so r₁ω₁ = r₂ω₂

Example 7: Gear Problem

A gear with radius 4 cm rotating at 120 rpm drives a gear with radius 6 cm. Find the angular speed of the larger gear.

Solution:

r₁ω₁ = r₂ω₂

4 × 120 = 6 × ω₂

ω₂ = 480 / 6 = 80 rpm

Problem-Solving Strategy

Identify what is given (radius, rpm, arc length, time) and what is asked for.
Convert units as needed. Make sure angular speed is in rad/time.
Apply the appropriate formula: ω = θ/t, v = s/t, or v = rω.
Convert the answer to the requested units.

Check Your Understanding

1. A fan blade rotates at 500 rpm. What is its angular speed in rad/s?

Answer: ω = 500 × 2π / 60 = 1000π/60 = 50π/3 ≈ 52.36 rad/s

2. A merry-go-round has a radius of 15 feet and angular speed of π/6 rad/s. Find the linear speed of a horse on the edge.

Answer: v = rω = 15 × π/6 = 15π/6 = 5π/2 ≈ 7.85 ft/s

3. A small gear (r = 3 cm) turns at 200 rpm and drives a large gear (r = 9 cm). What is the angular speed of the large gear?

Answer: r₁ω₁ = r₂ω₂, so 3(200) = 9ω₂, ω₂ = 600/9 = 66.67 rpm

Key Takeaways

Angular speed ω = θ/t measures how fast an angle changes (rad per unit time).
Linear speed v = s/t measures how fast a point travels along a circular path.
The key relationship: v = rω connects linear and angular speed.
All points on a rigid rotating object share the same ω, but points farther from the center have greater v.
Connected gears share the same linear speed at contact: r₁ω₁ = r₂ω₂.

Module 1 Complete!

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Practice Problems

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