A force acting on a particle moving in the x-y plane is given by `vecF=(2yhati+x^2hatj)N`, where x and y are in meters. The particle moves from the origin to a final position having coordinates `x=5.00m` as shown in figure. Calculate the work done by `vecF` on the particle as it moves along (a) OAC, (b) OBC, and (c) OC. (d) Is `vecF` conservative or non-conservative? Explain.

Explanation

a. For each of the paths from O to C, the work done is given by
`W=intoversetrarrF*vec(ds)`
Here `oversetrarrF=(2yhati+x^2hatj)` and `oversetrarr(ds)=(dxhati+dyhatj)`
Hence `W=int(2yhati+x^2hatj)(dxhati+dyhatj)`
`implies W=underset0overset5int2ydx+underset0overset5intx^2dy` ...(i)
The path OAC can be break up into two paths, from O to A and then from A to C.
First considering path O to A, in this path `y=0`, hence from equation (i) it is clear `W_(OA)=0`
Now consider path A to C, in this path x is constant (i.e., `dx=0`) and is equal to `5m`. From equation (i)
`W_(AC)=x^2underset0overset5intdy=(5)^2[y]_0^5=125J`
Hence total work done `W_(OAC)=0+125=125J`
b. Now consider the path OBC, this path also break up into two paths O to B and then B to C.
Considering path O to B, in this path `x=0`, hence from equation (i) it is clear `W_(OB)=0`.
Now considering path B to C, in this path y is constant (i.e. `dy=0`) from equation (i)
`W_(BC)=2yunderset0overset5intdx=2xx5xx[x]_0^5=50J`
Hence total work done in this path, `W_(OBC)=0+50=50J`
c. Motion of the particle from O to C, in this path both x and y are charging. In this path `x=yimpliesdx=dy`
From equation (i),
`W_(OC)=underset0overset5int(2x+x)^2dx=2[x^2/2]_0^5+[x^3/3]_0^5`
`=25+1/3xx125=200/3J`
d. We have calculated the work done from O to C in three different paths which is not same, hence the force F is a non-conservative force.