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

To analyze the two statements given in the question, we need to understand the concept of conservative forces and work done by these forces.

Statement I Analysis: 

Statement I states: "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 \(W = -\int_{\vec{r}_1}^{\vec{r}_2} \vec{F} \cdot d\vec{r}\).

A conservative force is one for which the work done does not depend on the path taken but only on the initial and final positions. This type of force has a potential energy associated with it, and the work done by the force on a closed path is zero.

The work done by a conservative force is given by the negative of the change in potential energy, which is represented mathematically as:

\(W = -\int_{\vec{r}_1}^{\vec{r}_2} \vec{F} \cdot d\vec{r}\)

This equation correctly describes the work done by a conservative force. Hence, Statement I is true.

Statement II Analysis:

Statement II states: "Any object moving from one location to another location 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 incorrectly applies the concept of conservative forces. By definition, for conservative forces, the work done does not depend on the path followed but is solely dependent on the initial and final positions. Therefore, even though an object may take an infinite number of paths, the work done for a conservative force will be the same for all paths. This contradicts the assertion made in Statement II. Thus, Statement II is false.

Conclusion:

After analyzing both statements, we conclude:

• Statement I is true but Statement II is false.