Linear and Angular Speeds

(3.3)

JMerrill, 2009

Linear Speed Linear speed is the speed of a car Recall the relationship between

distance, rate and time: d = rt Rate is speed so Think of Nascar: the distance is the

arc length, s. If v = speed and t = time, we have

distancespeed=

time

sv

t

Definition of Linear Speed If a point, P, moves along the

circumference of a circle at a constant speed, then the linear speed is denoted by

sv

t arclength

time

Example – Linear Speed A car travels at a constant speed

around a circular track with circumference of 2 miles. If the car records a time of 15 minutes for 9 laps, what is the linear speed of the car?

2(9 ) 18

18

15min18 60min

7215min 1

miless laps miles

lap

miv

miv mph

hr

Your Turn: Linear SpeedA car travels at a constant speed around a circular track with circumference of 3 miles. If the car records a time of 12 minutes for 7 laps, what is the linear speed in miles per hour?

Solution: t = 12 minutes, or hour and s = 7 X 3.

V = = = 105 mph.

5

1

t

s 21

0.2

Angular Speed In linear speed, we find how fast a

position along the circumference is changing. In angular speed, we find how fast the central is changing.

Angular speed is the rotation of the tires per second.

Definition of Angular Speed If a point, P, moves along the

circumference of a circle at a constant speed, then the central angle, θ, that is formed with the terminal side passing through the point P also changes over time, t, at a constant speed. The angular speed, ω (omega), is given by t

θ is in radians

Example – Angular Speed A lighthouse in the middle of a

channel rotates its light in a circular motion with a constant speed. If the beacon of light completes one rotation every 10 seconds, what is the angular speed of the beacon in radians per minute?

Example One rotation, in radians, is 2π Using we have

t

2 60sec12 / min

10sec 1min

radians ondsrad

onds ute

You Do – Angular Speed If the lighthouse is adjusted so that

the beacon rotates 1 time every 40 seconds, what is the angular speed of the beacon in radians per minute?

2 60sec3 / min

40sec 1min

radians ondsrad

onds ute

Relating Linear and Angular Speed

An important application of this is if you decide to increase the size of your tires. If you increase the size and do not upgrade the computer, your speedometer will be off-- maybe 5-10mph! Read example 3 p149.