Question

If C is the curve given by r(t)=(1+2sint)i+(1+3sin2t)j+(1+1sin3t)k, 0≤t≤π2 and F is the radial vector field F(x,y,z)=xi+yj+zk, compute the work done by F on a particle moving along C.

Answer 1

The work done by vector field F on a particle moving along curve C can be computed using the line integral ∫F.dr. To compute the line integral, we need to parameterize curve C and evaluate the dot product of the vector field and the parameterized curve.

Step-by-step explanation:

To compute the work done by a vector field F on a particle moving along a curve C, we can use the line integral. The line integral of a vector field F along a curve C can be computed using the formula: ∫F.dr. In this case, F(x, y, z) = xi + yj + zk and C is given by r(t) = (1 + 2sin(t))i + (1 + 3sin(2t))j + (1 + sin(3t))k. We need to parameterize C to compute the line integral. Let's rewrite r(t) as:

r(t) = i + 2sin(t)i + j + 3sin(2t)j + k + sin(3t)k

We can then calculate the line integral using the given parameterization. Substituting r(t) into F(x, y, z), we get:

F(r(t)) = (1 + 2sin(t))i + (1 + 3sin(2t))j + (1 + sin(3t))k

Now, we can compute the line integral by evaluating ∫F(r(t)).dr over the given interval 0 ≤ t ≤ π/2. This involves evaluating the dot product of F(r(t)) and r'(t). The work done by F on the particle moving along C is the value of the line integral.