If $W_1$, $W_2$ and $W_3$ represent the work done in moving a particle from $A$ to $B$ along three different paths, $1$, $2$ and $3$ respectively (as shown) in the gravitational field of a point mass $m$, find the correct relation between $W_1$, $W_2$ and $W_3$.

Step 1: Identify the nature of the gravitational force.

The gravitational force exerted by a point mass $m$ on a particle of mass $m_0$ is a central and conservative force.

$$\vec{F} = -\frac{G m m_0}{r^2} \hat{r}$$

Step 2: Apply the property of conservative forces.

The work done by a conservative force depends only on the initial and final positions, and is independent of the path taken.

$$W = \int_A^B \vec{F} \cdot d\vec{r} = -\Delta U$$

Step 3: Relate the work done along the three paths.

Integrating the force from radial distance $r_A$ to $r_B$ shows that the work depends only on the coordinates of the endpoints $A$ and $B$.

$$W = G m m_0 \left(\frac{1}{r_B} - \frac{1}{r_A}\right)$$

Since the endpoints are identical for all three paths, the work done along each path is equal.

$$W_1 = W_2 = W_3$$

Answer: (2) $W_1 = W_2 = W_3$