Question

Consider a particle conned to the xy-plane under the inuence of the force given by:

Fx = -ky Fy = kx

where k is a constant and x & y are the coordinates of the particle. Assume the particle is initially

at the origin. We wish to move the particle in a closed counter-clockwise loop, consisting of four straight segments:

Segment A - { [0,0] to [a,0] }

B - { [a,0] to [a,b] }

C = { [a,b] to [0,b] }

D { [0,b] to [0,0] }

a. Determine the work done for each segment, as well as the total amount of work done to go

around the loop.

b. Is this force conservative or not?

c.

Now assume the particle is rst moved in the previously specied counter-clockwise loop,

and then back along the same path in the clock-wise direction. Determine the total

amount of work done.

Answer 1

Consider a particle conned to the xy-plane under the inuence of the force given by: Fx = -ky Fy = kx where k is a constant and x & y are the coordinates of the particle. Assume the particle is initially at the origin. We wish to move the particle in a closed counter-clockwise loop, consisting of four straight segments: Y (a,b) (a,b) D B A Segment A - { [0,0] to (a,0] } B - { [a,O] to [a,b] } C = { [a,b] to (0,b] } D{ [0,b] to [0,0] } a. Determine the work done for each segment, as well as the total amount of work done to go around the loop. b. Is this force conservative or not? (0 ) t ول.مرحبا (ه C. Now assume the particle is rst moved in the previously specied counter-clockwise loop, and then back along the same path in the clock-wise direction. Determine the total amount of work done. a Wer (-kytkx *x)/åx _K(0) + Kx² , z - Ky x + Kx? - -KI(-a) -ka² Jawa: Ex +Fy).dx Jawa= fky + Kx) dx 0 - Kỹ Jelos= (fx + Fy) dy (ky tkxfly = -Ky24• **** - Kba-ka² b Z wo Sfky tkx) dy .b =-Ky? 1 + Kurdi b =k6² WB = -K16² + Kab

- 2 (4) b) For conservative force work done in closed loobco on + 8 + +“ - жа* -К* 4Ka) + K54 - Ка“ + къ° 12 y not conservative force. д) 4-5» - 254, C-ya:- 46 =\ Со го