Expert: Steve Holleran - 3/5/2007

Question
Steve,
      Stuck on this questiona and would appreciate any and all help.  

I have a piece of thread being pulled off of a sppol at the rate of 59.4cm per sec.  How do I go about finding the radius of the spool if it makes 152 revolutions per min?

Thank you in advance.

Arthur

Answer
Hi Arthur,

Just some definitions first:  the rotation of any point on the spool is its ANGULAR velocity.  The speed of the string at its edge is the LINEAR velocity.

To work this out, we need the formula for angular velocity related to linear velocity.

If we let v = the linear velocity of the thread
         r = the radius of the spool
         w = the angular velocity of the spool

Then the formula is :    v = r * w

We've also got to watch units:

w must be expressed in radians per unit time, so we have this series of conversions for the 152 rpm:

152 rev / 1 min * 2pi rad / 1 rev * 1 min / 60 sec will give us w = 15.9 rad / sec.  So the equation is

                v = r * w

               59.4 cm / sec  = r * 15.9 rad / sec

(Notice how the time units have to match)

Dividing, we get r = 3.74 cm.

If you have to work one of these again, just make sure that the length units of the radius and linear velocity and time units on the linear velocity and angular velocity match.  If they don't just convert.

I hope this is ok and helps you out.
Steve Holleran