Question

check my work

Answer 1

Find the work done in moving a particle once round the ellipse \[\frac{x^2}{25}+\frac{y^2}{16}=1\], z=0 under the field of force given by
\[\vec F=(2x-y+z)\hat i+(x+y-z^2)\hat j+(3x-2y+4z)\hat k\]
z=0 so we have
\[\vec F=(2x-y)\hat i+(x+y)\hat j+(3x-2y)\hat k\]
Parametric equation of the ellipse,
\[x=5\cos(\theta)\]\[y=4\sin(\theta)\]
\[z=0\]
\[\therefore dx=-5\sin(\theta)d\theta, dy=4\cos(\theta)d\theta, dz=0\]
\[\therefore d\vec r=dx \hat i+dy \hat j +dz \hat k=-5\sin(\theta)d\theta \hat i+4\cos(\theta)d\theta \hat j\]
\[\vec F.d \vec r=(2x-y)dx+(x+y)dy=2xdx-ydx+xdy+ydy\]
\[\vec F. d \vec r=-25\sin(2\theta)d\theta+20\sin^2(\theta)d\theta+20\cos^2(\theta)d\theta+8\sin(2\theta)d\theta\]\[\vec F.d \vec r=(20-16\sin(2\theta))d\theta\]
For an ellipse
\[0\le \theta \le 2\pi\]
therefore
\[W=\int\limits_{C}(\vec F . d \vec r)=\int\limits_{0}^{2\pi}(20-16\sin(2\theta))d\theta\]
\[W=[20\theta+8\cos(2\theta)]_{0}^{2\pi}=[(40\pi+8)-(0+8)]=40\pi+8-8=40\pi\]

Answer 2

sorry should be
\[\oint_{C} (\vec F . d \vec r)=\oint_{0}^{2\pi}(20-16\sin(2\theta)d\theta\]
Since ellipse is a closed curve :)

Answer 3

doesnt work require displacement? if you end up where you started ... my physics is rusty.

Answer 4

Hmmm that's true but since there's a field present it will affect the particle's motion as it moves around the ellipse, kinda like how a car moves around an elliptical road, friction present at every point will cause it to do some work. I think that is one way to think about it

Answer 5

Also I think this field is non-conservative
\[\vec \nabla \times \vec F \neq \vec 0\]
So taking a particle to some point and bringing it back in the form of an elliptical path will create a difference in going to the point and return trip

Answer 6

2x dx

2 (5cos) (-5sin) .... -25 (2 sin cos), which is simplified to a half angle ...

it seems to me that all your substitutions are valid.

Answer 7

@amistre64 thx!!

Answer 8

if you had any concern about it, where would your concern be focused on?

your formulas appear fine, and your substitutions are good to me.

Answer 9

the pun is nice to: 'check my work' and its a problem that involves finding the work done :)

Answer 10

Oh, I didn't see that!! I guess that deserves 2 medals, 1 for maths and 1 for english!! :P