Similarly, in the case of a linear variable force, the area under the curve from s 1 to s 2 (trapezium APQB) gives total work done W in the following figure.The work done by the non-linear variable force is represented by the area under the portion of the force-displacement graph. Method of integration is applicable if the exact way of variation in `vec"F" and vec"s"` is known and that function is integrable.The total work done can be found out by dividing the portion AB into small strips like P 1P 2P 2'P 1' and taking sum of all the areas of the strips.`"d"vec"s"` is the work done dW for this displacement. As the force is constant, the area of the strip `vec"F"`. `"d"vec"s"` is so small that the force F is practically constant for the displacement.But direction of force and displacement are same, we have.Due to this force, the body displaces through infinitesimally small-displacement ds, in the direction of the force. Let at P 1, the magnitude of force be F = P 1P 1'.For integration, we need to divide the displacement into large numbers of infinitesimal (infinitely small) displacements. In order to calculate the total work done during the displacement from s 1 to s 2, we need to use integration.Let the force vary non-linearly in magnitude between points A and B as shown in the above figure.Integral method to find work done by a variable force: One of the simplest examples of work is lifting a. The method of integration has to be applied to find the work done by a variable force. In equation form, work (joules) force (newtons) x distance (meters), where a joule is the SI unit of work.It is not applicable to viscous forces like fluid resistance as they depend upon speed and thus are often not constant with time. The formula is not applicable in several real-life situations like lifting an object through several thousand kilometers since the gravitational force is not constant.The formula for work done is applicable only if both force `vec"F"` and displacement `vec"s"` are constant and finite i.e., it cannot be applied when the force is variable.If displacement is in the direction of the force applied, θ = 0 0Ĭonditions/limitations for application of work formula:.Where θ is the angle between the applied force and displacement. Then the work done by the force is given as, Suppose a constant force `vec"F"` acting on a body produces a displacement `vec"s"` in the body along the positive X-direction.
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