Diameter of 32 inches. 12 revolutions per second. What is the speed (in miles per hours)

Question

I can't seem to set it up

Answer 1

“Knowing how to set up” this problem involves more than just calculating an answer (whether correct or not). That’s why teachers often write, “Show your work;” they want to know whether you are following the desired process.

In addition to following a desired process, it is important to know definitions of terms and the unit equivalences in order to do “conversions” required by dimensional analysis.

One important value is called The Multiplicative Identity (that is, 1). Any number multiplied by 1 remains the same. Thus, we can change units without changing the value by multiplying by the “1” that allows units to cancel – that’s why units are so very important. For example, 1 = (12 inches) / (1 foot), so 3 feet is also (3 feet)*( (12 inches)/(1 foot) ) = 36 inches. Now, for practice, do the conversion from 36 inches to feet to yards.

The value of π (that is, pi) is defined as the Circumference/diameter of a circle. Therefore, C = πd.

In this problem, a wheel travels 12 revolutions (that is, 12 Circumferences) per second. To convert that to miles per hour, multiply by 1:
12 revolution per second
( (12 rev)/(1 sec) ) * 1
( (12 rev)/(1 sec))*((1 circumference)/(1 revolution))
(12 circumferences)/(1 second)                [revs cancel]
((12 circs)/(1 sec))*((πd)/(1 circs)) = ((12 circs)/(1 sec))*((32π inches)/(1 circ))
((12*32π)) inches/sec             [note: circs cancel, leaving inches/sec]
((12*32π inches)/(1 sec))*((1 ft)/(12 inches))*((1 mile)/(5280 ft))
[        How convenient that the 12’s cancel; inches and ft also cancel]
( (32π miles)/(5280 seconds) ) * ((60 seconds)/(1 minute)) * ((60 minutes)/(1 hr))
((32π)*60*60) miles) / (5280 hour)
68.5 miles per hour

Now, sometimes it is helpful to string the values together in one line like this:
(A rev/sec)*(B circ/rev)*(C in/circ)*(D ft/in)*(E mi/ft)*(F sec/min)*(G min/hr)

     So that you can check that the result has units of mi/hr.

 

[Note that the constants do not need to be calculated, and may even cancel, until all the units are resolved. This usually helps to get a more exact answer.]

Answer 2

In each revolution it travels the distance of the

circumference of the "wheel" ...

c = πd = (3.1416)(32) = 100.531 inches/revolution

 

at 12 revolutions per second that is

12(100.531) = 1206.3716 inches per second

 

(1206.3716 in/sec)(60 sec/min) = 72382.29474 in/min

(72382.29474 in/min)(1/12 ft/in) = 6031.857895 ft/min

(6031.857895 ft/min)(1/5280 mile/ft) = 1.142397 mile/min

(1.142397 mi/min)(60 min/hr) = 68.5439 mph

 

approximately 68.5 miles per hour