Work done by a variable force

When the force exerts to the body with force not constant. We can find the work done by integral of force F with respect to displacement x ,that a body move from point x1 to x2

For the stretched spring. To keep a spring stretched an amount x beyond its unstretched length, we have to apply a force with magnitude F at each end. If the elongation is not too great, we find that F is directly proportional to x



F = kx.........................(4)


where k is a constant called the force constant or spring constant of spring.

equation (4) is known as Hooke's law. And the total work done by this force is


W=0.5kx^2

Suppose the spring is stretched a distance x1 at the start. then the work to do it to stretch it to a greater elongation x2 is


W = 0.5kx2^2-0.5kx1^2 ................(5)


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