Two forces \(\vec{F}_1=4\hat{\imath}+\hat{\jmath}-3\hat{k}\) and \(\vec{F}_2=3\hat{\imath}+\hat{\jmath}-\hat{k}\) act together on a particle and move it from the point A(1, 2, 3) to the point B(5, 4, 1). The work done by the two forces is (in units of work)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 40
The work done by a force \(\vec{F}\) over a displacement \(\vec{d}\) is the scalar product \(\vec{F}\cdot\vec{d}\). Here the force is the resultant of the two forces:
\(\vec{F} = \vec{F}_1+\vec{F}_2 = 7\hat{\imath}+2\hat{\jmath}-4\hat{k}\)
The displacement is
\(\vec{d} = \overrightarrow{AB} = 4\hat{\imath}+2\hat{\jmath}-2\hat{k}\)
\(\vec{F}\cdot\vec{d} = (7)(4)+(2)(2)+(-4)(-2)\)
\( = 28+4+8 = 40\)
Why the other options are incorrect
Option (2) 39 — Incorrect. This uses the position vector of B, \(5\hat{\imath}+4\hat{\jmath}+\hat{k}\), instead of the displacement \(\overrightarrow{AB}\).
Option (3) 24 — Incorrect. This is the work done by \(\vec{F}_1\) alone.
Option (4) 16 — Incorrect. This is the work done by \(\vec{F}_2\) alone.