Question

Given below are two statements:
Statement I: An object moves from position \( \vec{r}_1 \) to position \( \vec{r}_2 \) under a conservative force field \( \vec{F} \). The work done by the force is \[ W = -\int_{\vec{r}_1}^{\vec{r}_2} \vec{F} \cdot d\vec{r}. \]
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
• C. Both Statement I and Statement II are true
• D. Both Statement I and Statement II are false

Hint:

For conservative forces, work done is path independent and depends only on the initial and final positions.

Answer 1

The Correct Option is A

Step 1: Analyze Statement I.
For a conservative force field, the work done in moving a particle from position \( \vec{r}_1 \) to \( \vec{r}_2 \) is equal to the negative change in potential energy. Mathematically, \[ W = -\int_{\vec{r}_1}^{\vec{r}_2} \vec{F} \cdot d\vec{r}. \] This expression correctly represents the work done by a conservative force. Hence, Statement I is true.

Step 2: Analyze Statement II.
Although a particle can move between two points along infinitely many paths, the defining property of a conservative force is that the work done depends only on the initial and final positions and not on the path followed. Therefore, the work done by a conservative force does not change with the path. Hence, Statement II is false.

Step 3: Final conclusion.
Statement I is true but Statement II is false.

Final Answer: \[ \boxed{\text{Statement I is true but Statement II is false}} \]