Question

A force F = -k( yˆi + xˆj),
where k is a positive
constant, acts on a particle moving in the xy – plane. Starting from
the origin, the particle is taken along the positive x-axis to a point
(a, 0) and then parallel to the positive y-axis to a point (a, a).
Calculate the total work done by the force on the particle.

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Answer 1

Answer:

The total work done by the force on the particle can be calculated as the sum of the work done during the two stages of the particle's motion.

The work done during the first stage, when the particle moves along the positive x-axis to (a, 0), can be calculated as:

W1 = ∫ F . dr = ∫ -k(y + x) dx

from x = 0 to x = a

W1 = -ka^2

The work done during the second stage, when the particle moves parallel to the positive y-axis to (a, a), can be calculated as:

W2 = ∫ F . dr = ∫ -k(y + x) dy

from y = 0 to y = a

W2 = -ka^2

The total work done by the force is the sum of the work done during both stages:

W_total = W1 + W2 = -2ka^2