Vector Algebra
Work done using dot product
Grade 12
Question:
<p>A particle acted on by constant forces \(4\hat{i}+\hat{j}-3\hat{k}\) and \(3\hat{i}+\hat{j}-\hat{k}\) is displaced from the point \(\hat{i}+2\hat{j}+3\hat{k}\) to the point \(5\hat{i}+4\hat{j}+\hat{k}\). The total work done by the forces is</p>
<p>20 units</p>
<p>30 units</p>
<p>40 units</p>
<p>50 units</p>
Step-by-Step Solution
Key Concept: Work done is the dot product of net force with displacement vector. First, add the force vectors, then compute the dot product with the displacement vector (final position minus initial position).
Step 1: Find the net force by adding the two constant forces. 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 from initial position to final position. Initial position: r_1 = î + 2ĵ + 3k̂ Final position: r_2 = 5î + 4ĵ + k̂ Displacement: d = r_2 - r_1 = (5-1)î + (4-2)ĵ + (1-3)k̂ = 4î + 2ĵ - 2k̂ Step 3: Calculate work done using W = F_net · d. W = (7î + 2ĵ - 4k̂) · (4î + 2ĵ - 2k̂) W = (7)(4) + (2)(2) + (-4)(-2) W = 28 + 4 + 8 W = 40 units ∴ Answer: C
Correct Answer: C