Find the work done by the vector field
in moving an object around the periphery of the rectangle with vertices , and , starting and ending at .
Ans: The required work done is
The work done by a vector field F moving a particle along the curve C is computed by the following line integral:
Hence, evaluate the line integral to evaluate the required work done.
Green's Theorem states that,
"Let R be a region in the plane with smooth boundary curve C oriented counterclockwise by for .
If a vector field is defined on R, then,
For the vector field ,
Now, first find and .
Now, use Green's Theorem (1) to evaluate the integral as follows:
Here, the boundary curve C is a rectangle with vertices ,
starting and ending at .
So the region R bounded by this rectangle is shown in the following figure.
Viewed it as an x - or y-simple region, then the region of integration will be,
Then, evaluate the integral (2) as follows:
Therefore, the required work done is .
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