Consider a force F⃗ = - xî + yĵ . The work done by this force in moving a particle from point A(1, 0) to B(0, 1) along the line segment is : (all quantities are in SI units)

Solution

Physics sir · 3 steps

Split F⃗ · dr⃗ into an x-part and a y-part, and integrate each over its own limits.

1. Dot product. A small displacement is dr⃗ = dx î + dy ĵ, so the small work is dW = F⃗ · dr⃗ = -x dx + y dy. The first term has only x and the second only y, so the path between A and B does not change the result.

   W = ∫ₓ₌₁⁰ (-x) dx + ∫_(y=0)¹ y dy

2. Each part. Going from A to B, x falls from 1 to 0: ∫₁⁰ (-x) dx = [-x²/2]₁⁰ = 0 - (-1/2) = 1/2. Meanwhile y rises from 0 to 1: ∫₀¹ y dy = 1/2.
3. So. W = 1/2 + 1/2 = 1 J, which is option (c).

The idea to keep. When a force splits as Fₓ(x) î + F_y(y) ĵ, integrate each component over its own limits. Limits always run from start to end, even if upper is below lower.

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