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Math Help - Work Done By A Force

  1. #1
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    Work Done By A Force

    I've been racking my brain on this question for hours now, and I can't seem to get it. It's as follows:

    "Find the work done by the force F(x,y) = (x + 2y)i + (6y - 2x)j acting counterclockwise once along the triangle with vertices (1,1), (3,2) and (1,4)."

    A classmate thinks this should equal out to zero since it's a closed triangle and there isn't any ultimate displacement, but we cannot get it to work when we try solving it.

    Can somebody offer any help?

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by Katzenjammer View Post
    I've been racking my brain on this question for hours now, and I can't seem to get it. It's as follows:

    "Find the work done by the force F(x,y) = (x + 2y)i + (6y - 2x)j acting counterclockwise once along the triangle with vertices (1,1), (3,2) and (1,4)."

    A classmate thinks this should equal out to zero since it's a closed triangle and there isn't any ultimate displacement, but we cannot get it to work when we try solving it.

    Can somebody offer any help?

    Thanks in advance.

    Hi , Have you consider the Green's Theorem ?

     \int_{closed C} F \cdot dr = {\int\int}_D ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ) ~dydx ?

    but you should not expect that it must be a conservative field ...

    Here  P = x + 2y ~ , ~ Q= 6y-2x then

     \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}

     = -2 - 2 = -4

    so the work done is  {\int\int}_D (-4)~dydx

     = (-4) \cdot ~ ( Area ~ of ~the~ triangle)

    my answer is -12
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  3. #3
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    That does make sense.

    I think I was making it harder than it had to be. I was trying to find piecewise parametrizations of each line, C1 C2 and C3 and I wasn't getting anywhere.

    Thank you so much!
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  4. #4
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    Quote Originally Posted by Katzenjammer View Post
    That does make sense.

    I think I was making it harder than it had to be. I was trying to find piecewise parametrizations of each line, C1 C2 and C3 and I wasn't getting anywhere.

    Thank you so much!

    If the path is closed , i will think of the Green's Therorem first
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