THE OUTPATIENT CENTER DOOR
The outpatient center at the hospital where I work has a very large revolving door to allow stretchers and wheelchairs to be brought through. Walking through it one day, I began to wonder how my velocity would have to vary if I walked through it along a chord so that I would always be on the same rotating radius on which I entered. Realizing that this description of the problem is a bit fuzzy, the picture below is my attempt to illustrate the situation. I enter the door from the right and walk to the left along the cord; the door is rotating counter-clockwise with constant velocity. My goal is to walk so that at any point along my journey I am still on the rotating radius on which I entered.
The next illustration attaches mathematical interpretations to the situation. The chord is distance D from the parallel diameter. 
At any point P along the way, my position defines a right triangle with height D forming angle A with the parallel diameter. The hypotenuse of this triangle connects the center of rotation O with the chord of travel, lies along "my" radius and is of length D/SIN(A).
The lower leg of the triangle would be COS(A)*(D/SIN(A)), which is equivalent to D*(COS(A)/SIN(A)), or D*COT(A).
Given that the door rotates with constant speed, I take angle A as my measure of time. My position P on the horizontal chord at time A is D*COT(A). My velocity along the chord should then be dP/dA, which is equivalent to
d(D*COT(A))/dA =
-D*CSC(A)^2
according to the inter-active derivative page at http://www.webmath.com/diff.html.
However, when I use the same site to find the derivative of my original lower leg expression, COS(A)*(D*SIN(A)), it gives an answer of -D. If the expressions are equal, shouldn't they have the same derivative? If you have an answer, please email me at pavel314@comcast.net
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