If Wa, Wb, and Wc represent the work done in moving a particle from \(\mathrm{X}\) to \(\mathrm{Y}\) along three different path $\mathrm{a}, \mathrm{b}\(, and \)\mathrm{c}$ respectively (as shown) in the gravitational field of a point mass \(\mathrm{m}\), find the correct relation between \(\mathrm{Wa}, \mathrm{Wb}\) and \(\mathrm{Wc}\) (A) \(\mathrm{Wb}>\mathrm{Wa}>\mathrm{Wc}\) (B) \(\mathrm{Wa}<\mathrm{Wb}<\mathrm{Wc}\) (C) \(\mathrm{Wa}>\mathrm{Wb}>\mathrm{Wc}\) (D) \(\mathrm{Wa}=\mathrm{Wb}=\mathrm{Wc}\)

When dealing with gravitational forces, the force acting on an object is given by: \[F = \frac{Gm1m2}{r^2}\] where F is the force acting on the object, G is the gravitational constant, m1 and m2 are the masses being considered, and r is the distance between the centers of mass. Due to the nature of gravitational forces, they are conservative forces and the work done does not depend on the path taken. To compute the work done in moving a particle from point X to Y in the gravitational field, we will use the following formula: \(W = -\int_{x_i}^{x_f} F \cdot dr\) where W is the work done, xi and xf are the initial and final positions along the particle's path, F is the gravitational force acting on the object, and dr is the differential displacement.

Since gravitational force is a conservative force, the work done moving an object from point X to point Y will not depend on the path taken. This means that regardless of whether we choose path a, b, or c, the work done will always be the same. Let's calculate the work done for each path. For path a: \(W_a = -\int_{x_i}^{x_f} F \cdot dr\) For path b: \(W_b = -\int_{x_i}^{x_f} F \cdot dr\) For path c: \(W_c = -\int_{x_i}^{x_f} F \cdot dr\) Notice that all three work-done equations are the same, which is expected for a conservative force like the gravitational force.

Now that we've calculated the work done for each path, we can easily determine the relationship between them. Since we obtained the same equation for the work done in all three paths, it follows that: \(W_a = W_b = W_c\) This means, the correct relation between Wa, Wb, and Wc is given by option (D).