Work as a Line Integral

In this topic:

    (1) State the relationship between a line integral and the work performed as an object moves along a smooth curve .

    (2) Find the work done by the force field on an object moving along the curve defined parametrically by for

If an object moves along a line with displacement in a constant force (vector) field the work done is We will generalize the concept of work to a variable force field and where the object moves along a smooth orientable curve Assume that is subdivided into sub-arcs and let be a point in the th sub-arc. If the length of the sub-arc is small (very small), the force will be approximately constant and we assume it has the constant value on .
Also, the direction of motion will not change much over the sub-arc, so we can assume that the object will move a distance of in the direction of the unit tangent vector for a linear displacement of Therefore, we can approximate the work performed over the sub-arc by By adding the contributions along all sub-arcs the sum work as a line integral _gr_26.gif] is an approximation to the total work performed as the object moves along in the force field As the length of the largest sub-arc tends to 0, this approximating sum approaches the value of the line integral which is considered work as a line integral.

Proposition (Work as a Line Integral) Let be a continuous force field over a domain Then the work performed as an object moves along a smooth curve in is given by the integral where is the unit tangent at each point on and is the position vector of the object moving on

Example (Work as a Line Integral) Find the work done by the force field

on an object moving along the curve defined parametrically by

    Solution. We compute,




(from ) and

Thus,

Cite this as:
Work As A Line Integral
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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