Final answer:
To find the work done by the given force along the curve r = sin(2θ), we need to parameterize the curve, calculate the tangent vector, take the dot product with the force, and integrate it over one loop of the curve. The work done is -2π.
Step-by-step explanation:
To find the work done by the force along the given curve, we need to parameterize the curve and calculate the dot product of the force and the tangent vector of the curve.
Given the curve r = sin(2θ), we can parameterize it as x = sin(2θ) * cos(θ) and y = sin(2θ) * sin(θ).
The tangent vector of the curve is given by dr/dθ = (dx/dθ)i + (dy/dθ)j = 2(cos(2θ) - sin(2θ) * sin(θ))i + 2(cos(2θ) * sin(θ) - sin(2θ))j.
Now, we can calculate the work done by taking the dot product of the force and the tangent vector and integrating it over one loop of the curve:
Work = ∫(f · dr) = ∫((-2y)i + (2x)j) · ((dx/dθ)i + (dy/dθ)j)dθ.
After simplifying and performing the integration, the work done is equal to -2π.