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Line integral problems in Apostol calculus

  1. May 27, 2017 #1
    1. The problem statement, all variables and given/known data
    A two dimensional force field f is give by the equation f(x,y)=cxyi+x^6 y^2j, where c is a positive constant. This force acts on a particle which must move from (0,0) to the line x=1 along a curve of the form y=ax^b where a>0 and b>0

    2. Relevant equations
    Find a value of a(in terms of c) such that the work done by this force is independent of b

    3. The attempt at a solution
    I pluck the information x=1 into y=ax^b which gives y=a,so i believe the curve move from (0,0) to (1,a)
    then i parametrize the curve as r(t)=ti+at^bj which give r'(t)=i+abt^(b-1)j
    With plucking x=t, y=at^b into f(x,y) with the upper and lower limit in the integral, the solution i got is
    ac/(b+2)+a^(3)b/(b+18)

    However, the solution from the book is a=(3c/2)^(1/2)

    May i know which of my steps are correct and wrong, and teach me the right way of doing this question?
     
  2. jcsd
  3. May 27, 2017 #2

    pasmith

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    Homework Helper

    I get the line integral as [tex]
    W = \frac{ac}{b+2} + \frac{a^3b}{6 + 3b}.[/tex] I can only assume that you did not multiply the [itex]\mathbf{j}[/itex] components correctly or did not correctly integrate the result; as you haven't actually shown that working I can't help you.

    You have yet to finish the question: how do arrange that [itex]W[/itex] (which is a quadratic in [itex]b[/itex] whose coefficients are functions of [itex]a[/itex] and [itex]c[/itex] divided by a quadratic in [itex]b[/itex] whose coefficients are known constants) is independent of [itex]b[/itex]?
     
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