<p>A particle is acted upon by constant forces \(4\hat{i}+\hat{j}-3\hat{k}\) and \(3\hat{i}+\hat{j}-\hat{k}\) which displace it from a point \(\hat{i}+2\hat{j}+3\hat{k}\) to the point \(5\hat{i}+4\hat{j}+\hat{k}\). The work done in standard units by the forces is given by</p>
Step-by-Step Solution
Key Concept: Work done by a force is calculated as W = F·s, where F is the net force and s is the displacement vector. When multiple forces act, first find the resultant force, then compute the dot product with displacement.
Step 1: Find the resultant force F_1 = 4î + ĵ - 3k̂ F_2 = 3î + ĵ - k̂ F_net = F_1 + F_2 = (4+3)î + (1+1)ĵ + (-3-1)k̂ = 7î + 2ĵ - 4k̂ Step 2: Find the displacement vector Initial position: r_1 = î + 2ĵ + 3k̂ Final position: r_2 = 5î + 4ĵ + k̂ Displacement s = r_2 - r_1 = (5-1)î + (4-2)ĵ + (1-3)k̂ = 4î + 2ĵ - 2k̂ Step 3: Calculate work done using dot product W = F_net · s = (7î + 2ĵ - 4k̂) · (4î + 2ĵ - 2k̂) W = (7)(4) + (2)(2) + (-4)(-2) W = 28 + 4 + 8 = 40 units ∴ Answer: A
Correct Answer: A