5 Miscellaneous Exercises

The initial force vector is given as:

$$\vec{F}_0 = (-2\hat{i} + 4\hat{j} + 2\hat{k})\, \text{N}$$

The initial position of the point of application is:

$$P_1 = (3, -5, 1)$$

The final position of the point of application is:

$$P_2 = (5, 9, 7)$$

The total displacement vector $\vec{d}_{total}$ is found by subtracting the initial position vector from the final position vector:

$$\vec{d}_{total} = P_2 - P_1 = (5-3)\hat{i} + (9-(-5))\hat{j} + (7-1)\hat{k}$$ $$\vec{d}_{total} = (2\hat{i} + 14\hat{j} + 6\hat{k})\, \text{m}$$

The problem states that the force is halved when the point of application has moved half the distance. This means the total displacement is divided into two equal halves.
Displacement for the first half:

$$\vec{d}_1 = \frac{1}{2}\vec{d}_{total} = \frac{1}{2}(2\hat{i} + 14\hat{j} + 6\hat{k}) = (1\hat{i} + 7\hat{j} + 3\hat{k})\, \text{m}$$

Displacement for the second half:

$$\vec{d}_2 = \frac{1}{2}\vec{d}_{total} = \frac{1}{2}(2\hat{i} + 14\hat{j} + 6\hat{k}) = (1\hat{i} + 7\hat{j} + 3\hat{k})\, \text{m}$$

During the first half, the force is $\vec{F}_0$. The work done $W_1$ is the dot product of $\vec{F}_0$ and $\vec{d}_1$:

$$W_1 = \vec{F}_0 \cdot \vec{d}_1 = (-2\hat{i} + 4\hat{j} + 2\hat{k}) \cdot (1\hat{i} + 7\hat{j} + 3\hat{k})$$ $$W_1 = (-2)(1) + (4)(7) + (2)(3)$$ $$W_1 = -2 + 28 + 6$$ $$W_1 = 32\, \text{J}$$

During the second half, the force is halved, so $\vec{F}_1 = \frac{1}{2}\vec{F}_0$:

$$\vec{F}_1 = \frac{1}{2}(-2\hat{i} + 4\hat{j} + 2\hat{k}) = (-1\hat{i} + 2\hat{j} + 1\hat{k})\, \text{N}$$

The work done $W_2$ is the dot product of $\vec{F}_1$ and $\vec{d}_2$:

$$W_2 = \vec{F}_1 \cdot \vec{d}_2 = (-1\hat{i} + 2\hat{j} + 1\hat{k}) \cdot (1\hat{i} + 7\hat{j} + 3\hat{k})$$ $$W_2 = (-1)(1) + (2)(7) + (1)(3)$$ $$W_2 = -1 + 14 + 3$$ $$W_2 = 16\, \text{J}$$

The total work done $W_{total}$ is the sum of the work done in both halves:

$$W_{total} = W_1 + W_2$$ $$W_{total} = 32\, \text{J} + 16\, \text{J}$$ $$W_{total} = 48\, \text{J}$$

The final answer is $\text{48 J}$.