Question

A particle subject to a force vector F(x,y) = (yi-xj) moves clockwise along the arc of a circle with radius 4 centered at the origin, that begins at (-4, 0) and ends at (0, 4)
a) Find the work done by vector F.

Answer 1

all them question marks

Answer 2

click edit and remove them

Answer 3

there? @dan815

Answer 4

k parametrize the curve
w=int f.dr

Answer 5

r(t)=<x(t),y(t)>
dr/dt=<x'(t),y'(t)>
dr=<x'(t),y'(t)> * dt

Answer 6

f=<x,y>=<x(t),y(t)
now f.dr is all in one variable and you can integrate over some time bound

Answer 7

in more general case what u are doing here is, u wanna see how much of your force is in the tangent component of your motion, because thats the component that matters wrt to our force

Answer 8

since word=f*d

Answer 9

you could also use green's theorem by completing the path along x axis

Answer 10

so (cost, sint) for the parametrization ?

Answer 11

you have to see your bounded area though, because greens theorem will give u integral all over bounded area curve

Answer 12

you could use greens theorem still though by subtracting an area, from the main integral

Answer 13

so would the integral be from pi to pi/2?

Answer 14

subtract the work done along x axis, lets work it both ways and compare the answers

Answer 15

curlf = d/dx(-x)-(d/dy)((y)= -2, :( not path independant so u gotta do long way

Answer 16

-2*k^ to be more accurate

Answer 17

\[-\int\limits_{\pi/2}^{\pi} (sinti-costj) (-sinti+costj)dt\]

Answer 18

ok that works

Answer 19

u can just say pi to pi.2 and get rid of negative sigh no problems

Answer 20

u will see that u have -(sin^2+cos^2) so

Answer 21

you ust have 1 |pi/2 to pi

Answer 22

which is pi/2

Answer 23

wait u set p something wrong

Answer 24

it shudnt be like that becase x=4costheta,y=4sintehta

Answer 25

you will get a factor of 16 that u are missing

Answer 26

so u get pi/2 * 16 = 8pi

Answer 27

oh yeah i missed the radius 4

Answer 28

happens when u look at megan fox too long

Answer 29

im getting 16pi by greens thm

Answer 30

or rose bottom