# Finding work done in spherical coordinates

1. Dec 13, 2011

### HeisenbergW

1. Find the work done by the force F=r3*cos2$\varphi$*sin$\varphi$*$\hat{r}$ + r3*cos$\varphi$*cos(2$\varphi$) $\hat{\varphi}$
from the point (0,0,0) to (2,0,0)

2. Relevant equations
Work=$\int$ F*dr
where dr= dr$\hat{r}$ + rd$\varphi$$\hat{\varphi}$

3. The attempt at a solution

When muliplying the line element, dr, by the force, F, I come up with
$\int$ r3*cos2$\varphi$*sin$\varphi$ dr +$\int$ r4*cos$\varphi$*cos(2$\varphi$) d$\varphi$

I believe the r goes from 0 to 2, and there is no change in $\varphi$

I end up with 4*cos$^{2}$$\varphi$*sin$\varphi$
but then when I plug in 0 for $\varphi$, the answer ends up being zero, which I have a hard time believing since it moves from 0 to 2.
Any help is greatly appreciated

Last edited: Dec 13, 2011
2. Dec 13, 2011

### Simon Bridge

What is the force along $\varphi = 0$?
(This should simplify your line integral.)

3. Dec 13, 2011

### vb2341

Check the force along the path you are given (because it's really a line integral through a vector field), and it should become fairly simple to see why that is. Notice that your psi component didn't really change in the integral that mattered.

4. Dec 14, 2011

### HeisenbergW

Thanks for the replies
I believe you are saying that the force along $\varphi$=0 is just zero along the r component, which is the only component that matters, since there is no motion in the other two coordinates. Since my force is actually zero at $\varphi$=0, it doesn't matter that I went from (0,0,0) to (2,0,0), since no force means no work.