http://www.ck12.org Chapter 4. Basic Triangle Trigonometry
4.2 Circular Motion and Dimensional Analysis
Here you’ll review converting between linear and angular speeds using radians and circumference.
Converting between units is essential for mathematics and science in general. Radians are very powerful because
they provide a link between linear and angular speed. One radian is an angle that always corresponds to an arc length
of one radius. This will allow you to convert the rate at which you pedal a bike to the actual speed you can travel.
1 revolution= 2 πrThe gear near the pedals on a bike has a radius of 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?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/58061http://www.youtube.com/watch?v=sn8Y7qpYLCY James Sousa: American and Metric Conversions
Guidance
Dimensional analysis just means converting from one unit to another. Sometimes it must be done in several steps
in which case it is best to write the original amount on the left and then multiply it by all the different required
conversions. To convert 3 miles to inches you write:
3 mile 1 ·^52801 milef eet· 121 inchesf oot = 3 · 5280 · 121 inches= 190080 inches
Notice how miles and feet/foot cancel leaving the desired unit of inches. Also notice each conversion factor is the
same distance on the numerator and denominator, just written with different units.
Circular motion refers to the fact that on a spinning wheel points close to the center of the wheel actually travel
very slowly and points near the edge of the wheel actually travel much quicker. While the two points have the same
angular speed, their linear speed is very different.
Example A
Suppose Summit High School has a circular track with two lanes for running. The interior lane is 30 meters from
the center of the circle and the lane towards the exterior is 32 meters from the center of the circle. If two people run
4 laps together, how much further does the person on the outside go?
Solution:Calculate the distance each person ran separately using 1 lap to be 1 circumference and find the difference.