—1—WorkWhen a body moves a distancedalong a straight line as a result of being acted on by a force ofconstant magnitudeFin the direction of motion, then the workWdone by the force isW=Fd.We will generalize the definition of work to the case of variable force. Given a force functionF(x)defined and continuous at each pointxof the straight line segment[a, b], we will define the workWdone by this variable force in moving a particle along thex-axis from the pointx=ato thepointx=b.Partition the interval[a, b]intonsubintervals with the same lengthΔx. Choose an arbitrary pointckin thek’th interval[xk-1, xk]. Approximate the workΔWkdone by the force from the pointx=xk-1to the pointx=xkbyΔWk≈F(ck)Δx.We approximate the total work by summingfrom1ton, soW≈nk=1F(ck)Δx.This is a Riemann sum forF(x). WhenΔx→0, the sum approaches the definite integral ofF(x)fromatob. Therefore, we are motivated to define the workWdone by the forceF(x)in movinga particle fromatobto bebaF(x)dx .Problem.An electric elevator with a motor at the top has a multistrand cable weighing2kg/m.
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