Pre ap calculus homework

Question

A bicycle with 24-inch diameter wheels is traveling at 15 mi/h. Find the angular speed of the wheels in rad/ min. How many revolutions per minute do the wheels make?

 

Answer 1

Hi Nick.

 

This is a problem that deals more with understanding how to manipulate the units, than the numbers.  Because if you can set up the units right, you will know where all of the numbers go.   Here is what I mean.

 

You start with a circle with a 24 INCH diameter.   It travels  15 MILES/HOUR.   Need angular spped in RADIANS/MINUTE  and you need to know how many REVOLUTIONS/MINUTE the wheels make.  It is all about converting all of your units.

 

 

To start with, a circle with a 24-inch diameter has a circumference of 24(pi).  This is the distance that the wheel will travel in 1 revolution.   So lets find the revolutions per minute first.  

 

The bicycle travels:  Notice below the "inches" cancel and the "feet" cancel leaving miles/revolution

 

24(pi) inches                 1 foot               1 mile            24(pi)  miles

--------------     *    ---------------  *  -------------  =  -----------

1 revolution                 12 inches         5280 feet         (12)(5280) revolution

 

 

 

Traveling at :   (Notice the "hours" cancel as do the "miles", leaving revolutions per minute = what we want!)

 

15 miles          1 hour               (12)(5280)  revolutions          (15)(12)(5280)  revolutions

----------  * -------------     *  ------------------------------  =  ---------------------------------       

  1 hour           60 minutes                24(pi) miles                          (60)(24)(pi)  minutes

 

 

Plug those numbers into your calculator and you will have the number of revolutions per minute that you need.

 

 

Tackling "angular speed"  requires that we know it is in RADIANS/MINUTE.

 

So we will start with our number in revolutions/minute, and convert it to radians/minute.

 

(15)(12)(5280) revolutions             2(pi) radians                        (15)(12)(5280)(2)(pi) radians
---------------------------------  * ------------------------   =   ------------------------------------------
 (60)(24)(pi) minutes                          1 revolution                               (60)(24)(pi)   minutes

 

(notice the "revolutions" cancelled above, leaving radians/minute which is what we wanted.)

 

So put all of those numbers into your calculator  and you will find the angular speed in radians/minute.

 

 

I get  the angular speed = 1,320 radians per minute.

 

I get   660/pi  which is approx 210.0845 revolutions per minute.