A particle of mass m is moved from the origin (0,0) to a point (d, h) in a uniform vertical gravitational field acting in the negative y -direction. The particle is taken along three distinct paths: Path 1 is a straight line connecting the two points, Path 2 is a parabolic curve, and Path 3 runs along the x -axis to (d,0) and then parallel to the y -axis to (d,h) . If W₁ , W₂ , and W₃ are the work done by gravity on

Step-by-step solution

The gravitational force is a conservative force, which means the work done by it depends only on the initial and final positions of the particle and is independent of the path taken. The force of gravity acts in the negative y -direction, so the work done by gravity is given by W = -mg(y_f - y_i) . For all three paths, the initial y -coordinate is 0 and the final y -coordinate is h . Therefore, the work done by gravity along each path is -mgh . Thus, W₁ = W₂ = W₃ = -mgh , and the ratio W₁ : W₂ : W₃ is 1 : 1 : 1 . A