Question

A truck with 0.420-m-radius tires travels at 32.0 m/s. What is the angular velocity of the rotating tires in radians per second? What is this in rev/min?

Question by OpenStax is licensed under CC BY 4.0
Final Answer

76.2 rad/s76.2 \textrm{ rad/s}, 728 rpm728 \textrm{ rpm}

Solution video

OpenStax College Physics, Chapter 6, Problem 7 (Problems & Exercises)

OpenStax College Physics, Chapter 6, Problem 7 (PE) video thumbnail

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Calculator Screenshots

  • OpenStax College Physics, Chapter 6, Problem 7 (PE) calculator screenshot 1
Video Transcript
This is College Physics Answers with Shaun Dychko. The truck wheel radius is 0.42 meters and its linear velocity is 32.0 meters per second. We know that linear velocity is the wheel radius times its angular velocity and so we can solve for angular velocity omega by dividing both sides by r and then switching the sides around. We have omega is v divided by r. So that's 32 meters per second divided by 0.42 meters which gives 76.2 radians per second. Then the question asks us to convert this into revolutions per minute and so we multiply the radians per second by one revolution for every two pi radians. So the radians cancel leaving us with revolutions on the numerator of our units. Then we multiply by 60 seconds for every minute leaving us with minutes in our denominator. So this is revolutions per minute. For some strange reason, revolutions per minute is abbreviated rpm instead of writing revolutions per minute as a fraction. That's just the tradition. So we have 728 rpm.

Comments

Why is it not 76.2/2pi*60 (why is pi multiplied by the 60 not the 2

Hi goblinking88, thank you for the question. Do you mean in the calculator screenshot? This is just the way I tend to enter things into the calculator. The 76.2 is getting divided by both the 2 and by the pi, so I type it in as 76.2/2/pi. Then it gets multiplied by 60. All together it looks like 76.2/2/pi*60. The "pi" is not multiplied by 60, but rather the result of 76.2/2/pi is multiplied by 60.
Hope this helps,
Shaun

Hi benjamin, thank you for your comment. The answer shown in correct. I think there is confusion about the concise way I'm typing into the calculator. Ans/2/pi*60 is the same as the more verbose Ans2π×60\dfrac{Ans}{2 \pi} \times 60. Since the calculator evaluates from left to right, it's the result of Ans/2/pi which is multiplied by 60. Multiplying by 60 seconds per minute is necessary since, without doing so, the units would be "revolutions per second", whereas the goal is to have units of "revolutions per minute" - rpm, in other words.
All the best,
Shaun

I should add that the more verbose way of typing this into the calculator would be by including brackets, as in (Ans / 2 / pi) * 60, which is equivalent to what is shown above in the screenshot.