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 is the dot product of the net force acting on the object and the displacement vector. If multiple forces act simultaneously, we first find their vector sum to get the resultant force.

Step 2: Key Formula or Approach:
1. \(\vec{F}_{net} = \vec{F}_1 + \vec{F}_2\)
2. \(\vec{d} = d \hat{u}\), where \(\hat{u}\) is the unit vector in the given direction.
3. \(W = \vec{F}_{net} \cdot \vec{d}\)

Step 3: Detailed Explanation:
1. Calculate \(\vec{F}_{net}\):
\[ \vec{F}_{net} = (2+3)\hat{i} + (3-1)\hat{j} + (4-2)\hat{k} = 5\hat{i} + 2\hat{j} + 2\hat{k} \] 2. Find the displacement vector \(\vec{d}\):
Direction vector is \(\vec{A} = 3\hat{i} - 4\hat{j}\). Magnitude \(|\vec{A}| = \sqrt{3^2 + (-4)^2} = 5\). Unit vector \(\hat{u} = \frac{3\hat{i} - 4\hat{j}}{5}\).
Displacement \(\vec{d} = 25 \left( \frac{3\hat{i} - 4\hat{j}}{5} \right) = 5(3\hat{i} - 4\hat{j}) = 15\hat{i} - 20\hat{j}\). 3. Calculate Work:
\[ W = (5\hat{i} + 2\hat{j} + 2\hat{k}) \cdot (15\hat{i} - 20\hat{j} + 0\hat{k}) \] \[ W = (5 \times 15) + (2 \times -20) + (2 \times 0) = 75 - 40 = 35 \text{ J} \]

Step 4: Final Answer:
The work done in this process is 35 J.