A particle acted on by constant forces 4i + j - 3k and 3i + j - k is displaced from the point i + 2j + 3k to the point 5i + 4j + k. The total work done by the force is _____.Correct answer is '40'. Can you explain this answer?

Question

Answer

Understanding the Forces
The particle is influenced by two forces:
- **Force 1**: $ \mathbf{F_1} = 4\mathbf{i} + \mathbf{j} - 3\mathbf{k} $
- **Force 2**: $ \mathbf{F_2} = 3\mathbf{i} + \mathbf{j} - \mathbf{k} $
To find the total force acting on the particle, we sum these forces:

Total Force
$$
\mathbf{F_{total}} = \mathbf{F_1} + \mathbf{F_2} = (4 + 3)\mathbf{i} + (1 + 1)\mathbf{j} + (-3 - 1)\mathbf{k} = 7\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}
$$

Calculating the Displacement
The displacement vector $ \mathbf{d} $ from the initial point $ \mathbf{r_1} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} $ to the final point $ \mathbf{r_2} = 5\mathbf{i} + 4\mathbf{j} + \mathbf{k} $ is calculated as follows:
$$
\mathbf{d} = \mathbf{r_2} - \mathbf{r_1} = (5 - 1)\mathbf{i} + (4 - 2)\mathbf{j} + (1 - 3)\mathbf{k} = 4\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}
$$

Work Done by the Forces
The work done $ W $ by the net force on the particle during the displacement is given by the formula:
$$
W = \mathbf{F_{total}} \cdot \mathbf{d}
$$
Calculating the dot product:
$$
W = (7\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}) \cdot (4\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}) = 7 \cdot 4 + 2 \cdot 2 + (-4) \cdot (-2)
$$
Calculating each term:
- $ 28 + 4 + 8 = 40 $

Conclusion
The total work done by the force is **40**.

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