Final answer:
To determine the work done by the force as the object is displaced from (2i + 3j) m to (4i + 6j) m, we compute the dot product of the force vector (which is position dependent) and the displacement vector, integrate each component over the displacement, and sum the results.
Step-by-step explanation:
To calculate the work done by the force F as the object moves from position vector r1 = (2i + 3j) m to r2 = (4i + 6j) m, we use the concept of the dot product. The work done by a force during a displacement is the dot product of the force vector and the displacement vector. Given that the force F is a function of position, F = (3x2i + 2yj) N, we need to evaluate it at each point along the path of displacement.
First, calculate the displacement Δr = r2 - r1 = (4i + 6j) - (2i + 3j) = (2i + 3j) m.
Since the force depends on x and y, we need to consider its value at the initial and final points for each component separately: Fxi at x = 2 m, and x = 4 m, and Fyj at y = 3 m, and y = 6 m.
Finally, the work done W is given by:
W = Fx · Δx + Fy · Δy
This can be broken into two parts, one for each component of the force.
For the x-component: Fx = 3x2, where x changes from 2 m to 4 m.
For the y-component: Fy = 2y, where y changes from 3 m to 6 m.
To find the work done by each component, we need to integrate the corresponding force component over the respective displacement range:
Wx = ∫ (3x2) dx from x=2 to x=4
Wy = ∫ (2y) dy from y=3 to y=6
Following integration and calculation, we sum the work done by each component to find the total work done by the force.