SOLUTION: A car is moving at a rate of 50 miles per hour with a wheel radius of 16 inches, find the Angular speed in radians per minute and the number of revolutions per minute

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Question 1208203: A car is moving at a rate of 50 miles per hour with a wheel radius of 16 inches, find the Angular speed in radians per minute and the number of revolutions per minute
Found 3 solutions by mccravyedwin, ikleyn, math_tutor2020:
Answer by mccravyedwin(406) About Me  (Show Source):
You can put this solution on YOUR website!
A car is moving at a rate of 50 miles per hour with a wheel radius of 16 inches,
find the Angular speed in radians per minute and the number of revolutions per
minute.

16 inches = 16%2F12=4%2F3 ft. So the circumference of the wheel is
2%2Api%2A%284%2F3%29=expr%288%2F3%29pi feet.
   
In 1 hour it has 'rolled out' 50 miles or 50x5280 ft or 264000 feet.
So in 1 minute it has 'rolled out' 264000/60= 4400 ft.

So we find how many circumferences it 'rolls out' in 1 minute.

4400%2F%28expr%288%2F3%29pi%29=+1650%2Fpi

When the wheel has rolled out 1 circumference, that's 1 revolution.

So 1650%2Fpi revolutions per minute or about 525.2 rpms.

That's the answer to the second part.

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Each revolution is 2pi radians so we divide 1650%2Fpi by 2pi

and get %281650%2Fpi%29%2F%282pi%29%22%22=%22%22825%2Fpi%5E2 radians per minute

or about 83.6 radians per minute.

That's the answer to the first part.

Edwin

Answer by ikleyn(52771) About Me  (Show Source):
You can put this solution on YOUR website!
.


        Correction to Edwin' solution in the last part of his post. 


Edwin correctly determined that the number of revolutions per minute is  1650%2Fpi,

but after that, to get angular velocity, he mistakenly divided it by  2pi,

although, to be correct,  1650%2Fpi  must be  highlight%28highlight%28MULTIPLIED%29%29  by 2pi.


So, the angular velocity in this problem is  %281650%2Fpi%29%2A%282pi%29 = 1650*2 = 3300  radians per minute.



Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Tutor Edwin arrived at the correct revolutions per minute (rpm) value, which was 1650%2Fpi

However, he made an error when computing the "83.6 radians per minute" value.
Instead of dividing, multiply 1650%2Fpi with 2pi

You may be asking "how do I know whether to multiply or divide by 2pi?"
One way to remember is to set something up like this

The "revolutions" units cancel out. So do the pi terms.

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Here is how I would compute the revolutions per minute (rpm) value

Each conversion factor is based on these facts
  • 1 hour = 60 min
  • 5280 feet = 1 mile
  • 12 inches = 1 foot
  • 1 revolution = 32pi inches of circumference (since circumference = 2*pi*radius = 2*pi*16 = 32pi)
Nearly every unit cancels except for the "revolutions" up top and "minutes" down below.

1650%2Fpi+=+525.211312 approximately
I used the calculator's stored version of pi to get the most accuracy possible.

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If you wanted to use a similar template to find the radians per minute value, then,

The pi terms cancel, along with nearly every unit except for the "radians" up top and "minutes" down below.
The stuff in green is the calculation in the previous section.

Side note: it's unfortunate that both "revolutions" and "radians" start with R.
For new students it might be easy to mistake "rpm" to mean "radians per minute"

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Answers:
Radians per minute = 3300
Revolutions per minute (rpm) = 1650/pi = 525.211312 (approximate)