Compute the work done by the force field along the curve C, where C is the helix from to .

Question

Compute the work done by the force field  along the curve C, where C is the helix  from  to  .

Quizplus · Accepted Answer

We need to evaluate the work done by the force field along the curve C, which is given by the helix from (1,0,0) to (0,1,2π). The parameterization of the curve C is given by:
r(t) = cos(t) i + sin(t) j + 2πt k, where 0 ≤ t ≤ 1.

The force field is given by F(x,y,z) = (x,y,0).

We need to evaluate the line integral of F along the curve C:
\int_C F · dr = \int_0^1 F(r(t)) · r'(t) dt
where r'(t) = -sin(t) i + cos(t) j + 2π k.

So,
F(r(t)) · r'(t) = (cos(t), sin(t), 0) · (-sin(t) i + cos(t) j + 2π k)
= -cos(t)sin(t) - sin(t)cos(t) + 2π(0)
= 0

Therefore, the work done by the force field along the curve C is zero.

Compute the Work Done by the Force Field Along