catagory: Mechanics
question: A man and a bicycle have a total mass of 100 kg.
He is cycling at a constant 8m/s up a hill that makes a
4 degree angle with the horizontal direction. What power must the rider
expend against gravitational forces?



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The idea here is to utilize the formula P = F*v where P is the power 
output, F is the force, v is the velocity and * is the so called "dot"
or "inner product" of the vectors F and v.  In other words 

(1)    P = |F||v|cos(a)

Where a is the angle between F and v.  In this case the angle is 94
degrees because F is down (gravity force) and the incline is 4 degrees
above horizontal. OK so let plug all this stuff in.  We get

       P = |mg||v| cos(94)
         
         =100kg X 9.8m/s^2 X 8m/s X cos(94) =  -546.9 J/s
         
Note: The minus sign means that energy is being expended by the cyclist.
         
             ***********************
             *                     *
             *  P = -546.9 W       *              
             *                     * 
             ***********************          

P.S. I traveled in N.Z. in 1981 and loved every minute of it! A great
country and I was treated wonderfully by the citizens! :-)