Question

A 1 kg block subjected to two simultaneous forces \((2\hat{i} + 3\hat{j} + 4\hat{k})\) N and \((3\hat{i} - \hat{j} - 2\hat{k})\) N is moved a distance of 25 m along \((3\hat{i} - 4\hat{j})\) direction. The work done in this process is ______ J.

Answer 1

Correct Answer: 35

Step 1: Understanding the Concept:
Work done by a constant force is defined as the dot product of the net force vector and the displacement vector: \(W = \vec{F}_{\text{net}} \cdot \vec{d}\).
Step 2: Key Formula or Approach:
1. Calculate the net force: \(\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2\).
2. Determine the displacement vector \(\vec{d}\). It has a magnitude of 25 m and is directed along the vector \(\vec{v} = 3\hat{i} - 4\hat{j}\).
\(\vec{d} = |\vec{d}| \hat{v}\), where \(\hat{v} = \frac{\vec{v}}{|\vec{v}|}\).
3. Compute the dot product: \(W = \vec{F}_{\text{net}} \cdot \vec{d}\).
Step 3: Detailed Explanation:
Calculate the net force:
\(\vec{F}_{\text{net}} = (2\hat{i} + 3\hat{j} + 4\hat{k}) + (3\hat{i} - \hat{j} - 2\hat{k})\)
\(\vec{F}_{\text{net}} = (2+3)\hat{i} + (3-1)\hat{j} + (4-2)\hat{k} = 5\hat{i} + 2\hat{j} + 2\hat{k} \text{ N}\).
Determine the unit vector in the direction of displacement:
\(\vec{v} = 3\hat{i} - 4\hat{j}\)
\(|\vec{v}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
\(\hat{v} = \frac{3\hat{i} - 4\hat{j}}{5}\).
Now construct the displacement vector \(\vec{d}\):
\(\vec{d} = 25 \times \hat{v} = 25 \left( \frac{3\hat{i} - 4\hat{j}}{5} \right) = 5(3\hat{i} - 4\hat{j}) = 15\hat{i} - 20\hat{j} \text{ m}\).
Calculate the work done:
\(W = \vec{F}_{\text{net}} \cdot \vec{d} = (5\hat{i} + 2\hat{j} + 2\hat{k}) \cdot (15\hat{i} - 20\hat{j} + 0\hat{k})\)
\(W = (5 \times 15) + (2 \times -20) + (2 \times 0)\)
\(W = 75 - 40 = 35 \text{ J}\).
Step 4: Final Answer:
The work done in the process is 35 J.