Math 1316
3.4 Linear and Angular Speed
Angular speed is the rate at which the central angle of a spinning wheel is changing and its linear
speed is the rate at which the distance traveled by the wheel is changing.
From previous algebra courses, we know distance equals rate times time or
. The distance
traveled by a spinning wheel is represented by the letter s, therefore
. The letter r is the
linear speed of the spinning wheel which is represented by v, therefore
.
Since
, the linear speed is
(distance per unit time) and the angular speed
is
(radians per unit time). From the previous section, the arc subtended on a circle of radius r by a
central angle is
, thus
1. Suppose that a point P is on a circle with radius
angular speed
radians per sec. Find
20
, and ray
is rotating with
a) the angle generated by P in 6 seconds.
b) the distance traveled by P along the circle in 6 seconds.
c) the linear speed of P.
Solution:
a) Since
. So
, we have
b) Since
, we have
c) Since
20
20
·6
10
314
2. Find the angular velocity for each of the following:
a) the hour hand of a clock.
b) the minute hand of a clock.
c) a line from the center to the edge of a CD revolving 300 times per minute.
Solution:
a)
b)
c) 1 revolution = 2 radians, so
300 · 2
600
In problems 3 & 4, find the linear speed v.
3. The tip of the second hand of a clock, if the hand is 26 mm long.
Solution:
2
60 30
13
⁄
26
30
15
4. A point on the circumference of a tire of radius 18 cm, rotating 35 times per minute.
Solution:
1 revolution = 2 radians, so
18 70
35 · 2
70
1260
5. Suppose that a machine contains a wheel of diameter 3 feet, rotating at a rate of 1600 rpm.
a) Find the angular speed of the wheel.
b) Find the linear speed of a point on the circumference of the wheel.
Solution:
a) 1 revolution = 2 radians, so
1.5 3200
b)
1600 · 2
3200
4800
6. A typical tire for a compact car is 22 inches in diameter. If the car is traveling at a speed of
60 mph, find the number of revolutions the tire makes per minute.
Solution: must convert linear speed from miles per hour to inches per minute
60
1
5280
12
63,360
60
·
·
·
1
60
1
1
1
63,360
5760
11
and 1 revolution = 2 radians, so
917
7. Earth revolves on its axis once every 24 hours. Assuming that Earth’s radius is 6400 km, find
the following:
a) angular speed of Earth in radians per day and radians per hour.
b) linear speed at the North or South Pole.
c) linear speed at Quito, Ecuador, a city on the equator.
d) linear speed at Salem, Oregon (halfway from the equator to the North Pole)
Solution:
2
a)
b)
0·
c)
6400 ·
0
1675.5
d)
sin
4
6400
6400 sin
6400 ·
√2
2
3200√2 ·
4
3200√2
12
1184.8