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workshop 4 problem 2

workshop 4 problem 2 - -1 Work When a body moves a distance...

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1 Work When a body moves a distance d along a straight line as a result of being acted on by a force of constant magnitude F in the direction of motion, then the work W done by the force is W = Fd . We will generalize the definition of work to the case of variable force. Given a force function F ( x ) defined and continuous at each point x of the straight line segment [ a, b ] , we will define the work W done by this variable force in moving a particle along the x -axis from the point x = a to the point x = b . Partition the interval [ a, b ] into n subintervals with the same length Δ x . Choose an arbitrary point c k in the k ’th interval [ x k - 1 , x k ] . Approximate the work Δ W k done by the force from the point x = x k - 1 to the point x = x k by Δ W k F ( c k x. We approximate the total work by summing from 1 to n , so W n k =1 F ( c k x. This is a Riemann sum for F ( x ) . When Δ x 0 , the sum approaches the definite integral of F ( x ) from a to b . Therefore, we are motivated to define the work W done by the force F ( x ) in moving a particle from a to b to be b a F ( x ) dx . Problem. An electric elevator with a motor at the top has a multistrand cable weighing 2 kg/m.
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