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Re: An Integral oF Zero



I have a question regarding what seems to be a straight forward integral
yielding null results. Suppose a ball rolls on a vertical loop-de-loop with
friction providing more than just torque (i.e., more than one point of
contact--the realistic case). Also, suppose that you wish to calculate the
velocity necessary to just make it around the loop, so that the frictional
force is zero at the top, where the normal force is zero. The equation for
friction is: f = (mu) N, where N = mv^2/r + mg cos(theta), where theta = 0
at the bottom of the loop. Now, the centripetal term can be written in
terms of potential energy of the balls release point, assuming this is how
the ball is getting its energy to make it around the loop (neglect the
friction the ball has rolling down the ramp while traveling to the loop,
since this is calculated in a straightforward manner.
To get the energy lost due to friction, an integral of the form,
Scos(theta)d(theta), from 0 to pi must be calculated but this gives zero!
If I break it up into two parts and ignore the signs, I will get a net
result which is probably correct. How can this integral be performed
without having to intervene half way through? I mean, the work integral
along the path should produce the correct result without tampering with the
process.
Thanks.

Tom McCarthy
Saint Edward's School
1895 St. Edward's Drive
Vero Beach, FL 32963
561-231-4136
Physics and Astronomy