Transcript 3.3 Angular & Linear Velocity Yesterday  arc length s  r We can use it to analyze motion of a circular.

3.3 Angular & Linear Velocity
Yesterday  arc length s  r
We can use it to analyze motion of a circular path (like tires,
gears, & Ferris Wheels)
A point on the edge of a wheel will move through an angle
called angular displacement (in radians)
 To find angular displacement, multiply the rotations times
2π (1 time around in radians)
Ex 1) A gear makes 1.5 rotations about its axis. What is
angular displacement in radians of a point on the gear?
(1.5)(2π) = 3π ≈ 9.4 rad
We can calculate angular velocity 


t

angular displacement (radians)
unit of time
*We will have to be very aware of the units of our answer.
Most often we will have to convert to the correct units.
Always include the unit labels & it will be easy to see what
you need to get & cancel!! (called dimensional analysis)
Ex 2) Find the angular velocity in radians per second of a
point on a gear turning at the rate of 3.4 rpm
(rpm = revolutions per minute)

3.4 rev 2 rad 1 min (3.4)(2 ) rad
rad
 

 .36 sec
t
1 min 1 rev 60 sec
60 sec
Ex 3) What is the angular velocity in radians per second of a
notch on a wheel turning at a rate of 7600 rpm?

7600 rev 2 rad 1 min
rad
 
 795.87 sec
t
1 min
1 rev 60 sec
If you have a circle with 2 points on it at different distances
from the center
A & B will have the same angular velocity
B
but different linear velocities.
A
O
B will travel further!
To calculate linear velocity v
s r
arc length

v 

or r  r
t
t
unit of time
t
Ex 4) An ice skater moves around the edge of a circular rink
at a rate of 2 rpm. The rink has a radius of 4.1 m. What is
the skater’s velocity in meters per minute?
v  r
2 rev 2 rad

 4
1 min 1 rev
rad
v  (4.1 m)  4 min
  51.52 meters
min
rad
min
Ex 5) A unicycle has a tire with radius 10 in. It is traveling
at a speed of 5.5 mph (miles per hour)
a) Find the angular velocity of the tire in radians per second
v
 
Since v  r
r
5.5 miles 5280 ft 12 in 1 hr 1 min
in
v
 96.8 sec
1 hour 1 mile 1 ft 60 min 60 sec
96.8 in 1
rad

 9.68 sec
1 sec 10 in
b) How many revolutions per second does the tire make?
9.68 rad 1 rev
rev
 1.54 sec
1 sec 2 rad
Ex 6) Determine the linear velocity (in cm per second) of a
point on the circle 5 cm from the center that moves through
an angle of 56° in 1 min
v  r

56

14 rad
 

t 1 min 180 45 min
5 meters 14 rad 1 min 100 cm
cm
v
 8.14 sec
1
45 min 60 sec 1 meter
Homework
#303 Pg 137 #1–23 odd, 25, 27, 31, 34, 35, 41, 42, 43