Work Done by a General Variable Force — Question 5

Question 5

Identical endpoints can give different work for a variable field

A planar force is $\vec F=(2y\,\hat i)\,$N, with y in meters. Move a particle from O=(0,0) to B=(3,2). Route A goes horizontally first; route B goes vertically first.

1. Find work along route A.

2. Find work along route B.

3. Find work around the loop that follows A outward and reverses B back to O.

Question 5 - Solution

Identical endpoints can give different work for a variable field

(a) Keep the height on the horizontal segment.

The governing relation is $$\boxed{W_C=\int_C2y\,dx}$$ On A, the horizontal segment lies at y=0, so the force vanishes. The vertical segment has $dx=0$. Hence $W_A=0$.

(b) Use the different horizontal force.

On B, the vertical segment does zero work; the horizontal segment at y=2 has $$F_x=4\ \mathrm N$$ through 3 m. Thus $$W_B=12\ \mathrm J.$$

(c) Reverse the displacement on the return route.

The reverse of B contributes $-12\ \mathrm J$, so loop work is $-12\ \mathrm J$. These explicit integrals show that endpoint coordinates alone do not specify work for this force field.