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The force exerted by a spring when it's ...

The force exerted by a spring when it's displaced by x from its natural length is given by the equation `F(x)=-kx`, where k is a positive constant. What is the work done by a spring as it pushes out from `x=-x_(2)` to `x=-x_(1)` (where `x_(2) gt x_(1)`)?

Text Solution

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We sketch the graph of F(x)=-kx and calculate the area under the graph from `x=-x_(2)` to `x=-x_(1)`.

Here, the region is a trapezoid with area `A=(1)/(2)("base"_(1)+"base"_(2))xx"height"`, so
`W=A=(1)/(2)(kx_(2)+kx_(1))(x_(2)-x_(1))`
`=(1)/(2)k(x_(2)+x_(1))(x_(2)-x_(1))`
`=(1)/(2)k(x_(2)^(2)-x_(1)^(2))`.
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