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Question

A force $\vec{F} = (3x^2\hat{i} + 4y\hat{j})$ N acts on a particle as it moves from (0,0) to (1,1) m. Which of the following paths requires the *least* amount of work done by the force?

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: D
## CONCEPT FRAMEWORK
	**Primary Concept**: Work done by a conservative force
	**Related Topics**: Path independence, Potential energy, Line integrals
	**Prerequisites**: Vector calculus, Integration
## SOLUTION PATHWAY
**Given Information**:
	Force $\vec{F} = (3x^2\hat{i} + 4y\hat{j})$ N
	Initial position: (0,0) m
	Final position: (1,1) m
**Method & Steps**:
	1. Check for conservative force:
		Reasoning: The work done by a conservative force is path-independent and depends only on the initial and final positions.
		Formula/Rule: A force $\vec{F} = P\hat{i} + Q\hat{j}$ is conservative if $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$.
	2. Calculate partial derivatives:
		Working: $\frac{\partial P}{\partial y} = \frac{\partial (3x^2)}{\partial y} = 0$ and $\frac{\partial Q}{\partial x} = \frac{\partial (4y)}{\partial x} = 0$
		Key Point: Since $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$, the force is conservative.
	3. Conclude about work done:
		Result: The work done is the same for all paths.
		Verification:  We can calculate the work done along any path to confirm. For example, along the straight line y=x, $W = \int_{(0,0)}^{(1,1)} \vec{F} \cdot d\vec{r} = \int_0^1 (3x^2 dx + 4x dx) = \int_0^1 (3x^2 + 4x) dx = [x^3 + 2x^2]_0^1 = 1 + 2 = 3$ J. Along the x-axis and then y-axis, $W = \int_0^1 3x^2 dx + \int_0^1 4y dy = [x^3]_0^1 + [2y^2]_0^1 = 1 + 2 = 3$ J.
## OPTION ANALYSIS
[A] :x: Incorrect. While the work done along a straight line is a valid calculation, it doesn't represent the *least* work since the work is path-independent.
[B] :x: Incorrect.  The parabolic path is another valid path, but the work done will be the same as any other path.
[C] :x: Incorrect. This path also results in the same amount of work due to the conservative nature of the force.
[D] ✓: Correct.  Since the force is conservative, the work done is independent of the path taken.
## LEARNING AIDS
	**Common Mistakes**:
		1. Not recognizing a conservative force: Always check the condition $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$.
		2. Calculating work for each path: Unnecessary for conservative forces. Focus on identifying the conservative nature of the force.
	**Quick Tips**:
		1. Conservative forces are associated with potential energy.
		2. The work done by a conservative force in a closed loop is always zero.

NAGA SAI · Answer

A straight line from (0,0) to (1,1)

NAGA SAI · Answer

A parabolic path y=x^2 from (0,0) to (1,1)

NAGA SAI · Answer

A path along the x-axis from (0,0) to (1,0) followed by a path along the y-axis from (1,0) to (1,1)

NAGA SAI · Answer

The correct Option is "All paths require the same amount of work."

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