Arrange the four graphs in descending order of total work done; where W₁, W₂, W₃ and W₄ are the work done corresponding to figure a, b, c and d respectively.

Solution

Physics sir · 4 steps

Work is the signed area under the F–x graph, so add areas above the axis and subtract areas below it.

1. Area rule. For a force that changes with position, W = ∫ F dx. On an F–x graph that is the area between the curve and the x-axis. Area above the axis counts as positive and area below counts as negative.
2. Common first piece. In a, b and c the force rises in a straight line from -F at x = 0 to +F at x₁. So it crosses zero halfway, at x₀. The triangle below the axis and the triangle above it are the same size, and this part gives zero net work.

   Figure	Pieces of area	Net work
   a	-1/2Fx₀ + 1/2F(x₁ - x₀), two equal triangles that cancel	W₁ = 0
   b	zero from the triangles, plus a rectangle F(x₂ - x₁) above the axis	W₂ = F(x₂ - x₁) > 0
   c	the same as b, plus one more triangle above the axis after x₂	W₃ > W₂
   d	triangles from +F down to -F cancel, then a rectangle -F(x₂ - x₁) below the axis, then a small triangle above it	W₄ < 0

3. Check figure d. The last triangle in d is shorter than F and narrower than x₂ - x₁. So it is much smaller than the rectangle below the axis, and the total stays negative.
4. So. W₃ > W₂ > W₁ > W₄, with W₁ = 0 and W₄ negative. That is option (a).

The idea to keep. On an F–x graph, work is signed area. Area below the axis subtracts. A straight line from -F to +F gives equal triangles on both sides, so its net work is zero.

Answer from the NTA final answer key. Solution by PrepWiser.