CK-12-Pre-Calculus Concepts

4.2. Circular Motion and Dimensional Analysis http://www.ck12.org

4 la ps

1 ·

2 π· 30 meters

1 la p ≈^754 meters

4 la ps

1 ·

2 π· 32 meters

1 la p ≈^804 meters

The person running on the outside of the track ran about 50 more meters.
Example B
Andres races on a bicycle with tires that have a 17 inch radius. When he is traveling at a speed of 30 feet per second,
how fast are the wheels spinning in revolutions per minute?
Solution:Look for ways to convert feet to revolutions and seconds to minutes.

130 secondf eet ·^601 minuteseconds·^121 inchesf oot · 21 πrevolution· 17 inches=^30 ·^602 π··^1217 revolutionsminute ≈^202.^2 minrev
Example C
When a car travels at 60 miles per hour, how fast are the tires spinning if they have 30 inch diameters?

60 miles

1 hour ·

5280 f eet

1 mile ·

12 inches

1 f oot ·

1 revolution

2 π· 15 inches·

1 hour

60 minute

=^60 · 25280 π· 15 ··^1260 revolutionsminute ≈ 672. (^3) minrev
Concept Problem Revisited
The gear near the pedals on the bike has radius 5 inches and spins once every second. It is connected by a chain to a
second gear that has a 3 inch radius. If the second wheel is connected to a tire with a 17 inch radius, how fast is the
bike moving in miles per hour?
A bike has pedals that rotate a gear at a circular speed. The gear translates this speed to a linear speed on the chain.
The chain then moves a second gear, which is a conversion to angular speed for the rear tire. This tire then converts
the angular speed back to linear speed which is how fast you are moving. Instead of doing all these calculations in
one step, it is easier to do each conversion in small pieces.
First convert the original gear into the linear speed of the chain.
11 revolutionsecond · 12 πrevolution· 5 inches= 10 πsecin
Then convert the speed of the chain into angular speed of the back gear which is the same as the angular speed of
the rear tire.
101 πsecondinches· (^12) πrevolution· 3 inches = 106 revsec
Lastly convert the angular speed of the rear tire to the linear speed of the tire in miles per hour.
10 rev
6 sec·
2 π· 17 in
1 rev ·
1 f t
12 in·
1 mile
5280 f t·
60 sec
1 min·
60 min
1 hour
=^10 ·^26 ··π 12 ·^17 · 5280 ·^60 ·^60 mileshour
≈ 10. 1 mileshour
Vocabulary
Angular speedis the ratio of revolutions that occur per unit of time.