Introduction
Two people on a merry-go-round have the same angular velocity (they complete the same circles per minute), but the person on the outside edge has a much higher linear velocity (they are covering more distance). Today we quantify this relationship.
Prerequisite Connection
This relies entirely on radians (Lesson 9.1). The formulas for arc length and velocity assume is in radians.
Today's Increment
We introduce (how far) and (how fast), linking curved motion to straight-line physics.
Why This Matters
This is the foundation of Related Rates in Calculus. When you differentiate with respect to time, you are essentially juggling linear and angular velocities.
Key Concepts
The Formulas
Arc Length ()
where is in radians.
Linear Velocity ()
where (omega) is rad/time.
Worked Examples
Example 1: Arc Length
Find the length of the arc intercepted by a central angle of in a circle of radius 12 cm.
Use the Formula
. Check that is in radians (it is).
Calculate
Example 2: The Trap (Degrees)
Find the arc length if the radius is 10 m and the angle is .
Convert to Radians (CRITICAL)
You CANNOT use 135 in the formula.
Calculate
Example 3: RPM to MPH
A monster truck tire with radius 2.5 feet is spinning at 150 revolutions per minute (RPM). How fast is the truck moving in feet per minute?
Convert RPM to angular velocity ()
1 revolution = radians.
Find Linear Velocity
Use .
Common Pitfalls
Using Degrees directly
The formula stops working if you use degrees. would imply an arc length of 900, which is gigantic and wrong. You must convert to first.
Real-World Application
Hard Drives and CD-ROMs
Data on a spinning disc is read at different speeds. The inner track has a small radius, so for the same RPM, the linear speed of data flying past the head is slower. The outer track moves much faster. Engineers must adjust the data read-rate or the motor speed (Constant Linear Velocity vs Constant Angular Velocity) to prevent data corruption.
Practice Quiz
Loading...