Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

A two-dimensional force field is given by the equation $$f(x,y)=cxy\textbf{i}+x^6y^2\textbf{j}$$, 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 $b>0$

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

I started by expressing the parametric equations of the path namely: $x(t)=t$ and $y(t)=at^b$

Then we have: $\int\limits_{0}^1cxydx+x^6y^2dy=\int\limits_{0}^1(a^3bt^{3b+5}+act^{b+1})dt=\frac{a^3b}{3b+6}+\frac{ac}{b+2} $

From here I am not sure how to continue.

Please help

share|improve this question
    
@julien can you help me ? –  Carpediem Apr 27 '13 at 15:36

1 Answer 1

up vote 3 down vote accepted

Let $g(b)=\frac{a^3b}{3b+6}+\frac{ac}{b+2}$. If $g$ is independent of $b$, then $g(b)$ must be constant. Thus, \begin{equation} 0=g'(b)=\frac{2a^3-3ac}{3(b+2)^2} \end{equation} so that $a=\sqrt{\frac{3c}{2}}$.

share|improve this answer
    
Thank you very much –  Carpediem Apr 27 '13 at 16:14

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.