Given below are two statements : Statement I : An object moves from position r₁ to position r₂ under a conservative force field F⃗. The work done by the force is W = − ∫(from r₁ to r₂) F⃗ · d r⃗. 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 :

Given below are two statements : Statement I An object moves from position r1 to position r2 under a conservative force field

Q. Given below are two statements :

Statement I : 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 $$ W = - \int_{r_1}^{r_2} \vec F \cdot d\vec r. $$
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 :

(A) Statement I is true but Statement II is false

(B) Statement I is false but Statement II is true

(D) Both Statement I and Statement II are true

Correct Answer: Both Statement I and Statement II are false

Explanation

For a force field, the work done by the force when a particle moves from position $r_1$ to $r_2$ is defined as

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

The negative sign appears only when work done is expressed in terms of potential energy change, because for a conservative force,

$$ W = -\Delta U $$

Therefore, writing the work done by the force itself as

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

is incorrect. Hence, Statement I is false.

A key property of a conservative force is that the work done between two fixed points depends only on the initial and final positions and is independent of the path followed.

Although an object can indeed follow infinitely many paths between two points, for a conservative force the work done remains the same for all possible paths.

Statement II incorrectly concludes that work changes with path for a conservative force, which contradicts the defining property of conservative forces.

Hence, Statement II is also false.

Therefore, the correct option is:

Both Statement I and Statement II are false

Explanation

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