Given below are two statements:
Statement I : An object moves from position $$r_{1}$$ to position $$r_{2}$$ under a conservative force field $$\overrightarrow{F}$$.
The work done by the force is W = $$\int_{r_{1}}^{r_{2}} \overrightarrow{F}.\overrightarrow{dr}.$$
Statement II: Any object moving from one location to another location can follow infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.
In the light of the above statements, choose the correct answer from the options given below :

We need to evaluate two statements about work done by conservative forces.

An object moves from position $$r_1$$ to position $$r_2$$ under a conservative force field $$\vec{F}$$. The work done by the force is $$W = \int_{r_1}^{r_2} \vec{F} \cdot d\vec{r}$$.

This is true. The work done by any force (conservative or non-conservative) on an object moving from position $$r_1$$ to $$r_2$$ is defined as the line integral:

$$W = \int_{r_1}^{r_2} \vec{F} \cdot d\vec{r}$$

This is the fundamental definition of work in mechanics.

Any object moving from one location to another can follow an infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.

This statement is false. While it is true that there are infinitely many paths between two points, the defining property of a conservative force is that the work done is path-independent. The work depends only on the initial and final positions, not on the path taken. Mathematically:

$$W = -\Delta U = -(U(r_2) - U(r_1)) = U(r_1) - U(r_2)$$

where $$U$$ is the potential energy function. Since the work depends only on the potential energy values at the two endpoints, it is the same for all paths. This is precisely what distinguishes conservative forces (like gravity, electrostatic force, spring force) from non-conservative forces (like friction).

The correct answer is Option 2: Statement I is true but Statement II is false.