- #1

Ebby

- 41

- 14

- Homework Statement
- Show that the force is non-conservative

- Relevant Equations
- F = kx

x^2 + y^2 = R^2

I have to show that the force is non-conservative, i.e. that work done for the round trip ##\neq 0##.

Rearranging the circle equation, I can say that:$$x = \sqrt {R^2 - y^2}$$$$y = \sqrt {R^2 - x^2}$$

Then I have:$$F_x = -\sqrt {R^2 - x^2}$$$$F_y = \sqrt {R^2 - y^2}$$

Now, as I understand it, I must integrate these forces thus:

$$W_x = \int_a^b -\sqrt {R^2 - x^2} \, dx$$$$W_y = \int_a^b \sqrt {R^2 - y^2} \, dy$$

So now I'm not sure what my limits of integration ##a## and ##b## should be. Taking the expression for ##W_x##, for example, I imagine the ##x## coordinate starting at a value of ##R##, going to ##-R## and then back to ##R## again. But this will just mean ##W_x## is zero, which is not what I want.

Should I change the integraton to be over an angle rather than over a displacement to reflect the fact that the path is circular, and use limits of ##0## and ##2\pi##?