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Obtain the equation of work by variable force in one dimension . |
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Answer» Solution :A force whose direction or magnitude or both change with time is a variable force . A constant force is rare .It is the variable force which is more commonly encountered . Here in figure , a plot of a varying force in one DIMENSION versus displacement `F(x)to (x) ` is shown .  If the displacement `Deltax` is small ,the force F(x) as approximately constant and the WORK done is then area under area the curve of graph `F(x) to Deltax`. Total work done is the sum of the areas of shaded rectangles ,which is written as `W = sum_(x_(i))^(x_(f))F(x) Deltax` Here ,the sum is from intial to final position . If the displacement are allowed to approach zero . then the number of TERMS in the sum increases without limit , but the sum approaches a definite value equal to the area under the curve . ` :. ` Woek done in shole path , `W = lim_(Deltal to0) sum_(x_(i))^(x_(f))F(x) Deltax` `= int_(x_(i))^(x_(f)) F(x) dx ` HENCE ,where lim stands for the limit of the sum when `Deltax` tends to zero . thus , for a varying force the work done can be EXPRESSED as a definite integral of force over displacement . |
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