I'm having some trouble to see whether I've taken the right approach to part of the following question.
A particle moves from point $A = (0, 0, 0)$ to point $B = (2\pi, 0, 2\pi)$, under the action of the force:
$$F = xi + yj – zk$$
Note: $i, j, k$ are the direction vectors.
ii) Find a parametric vector equation for the straight line connecting A to B,
and calculate the work done by the force F on the particle as it moves along this straight
line.
For this part, I found that $x = t, z = t$ where $0≤t≤2\pi$
therefore $r(t) = ti +tk$ so $$r'(t) = i + k$$
and $$F = ti – tk$$
the integral to find the work done is $W = \int_0^{2pi}F(r(t)).dr$, therefore $$\int_0^{2pi}F(r(t)).r'(t)dt$$
The dot product $F(r(t)).r'(t) = (ti-tk).(i+k) = t – t$ gives value $0$ which rings alarm bells for me and I think I went wrong somewhere.
I would appreciate if somebody could verify whether I've taken the correct approach.
This is also my first post so I apologise if my formatting or wording is unclear. Thanks in Advance!
Best Answer
The vanishing integrand means that the force is always directed perpendicularly to the direction of motion, so there’s no component of the force in the latter direction. The net work done by that force is by definition zero. This can certainly happen. For example, the centripetal force that keeps a mass moving at a constant speed along a circular path does no work on the mass.