7.5 Q-9

Question Statement

A particle is displaced from point $A(5, -5, -7)$ to point $B(6, 2, -2)$ under the action of three constant forces:

1. $\mathbf{F_1} = 10 \hat{i} - \hat{j} + 11 \hat{k}$
2. $\mathbf{F_2} = 4 \hat{i} + 5 \hat{j} + 9 \hat{k}$
3. $\mathbf{F_3} = -2 \hat{i} + \hat{j} - 9 \hat{k}$

Show that the total work done by these forces is 67 units.

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Background and Explanation

To calculate the total work done by the forces, we use the formula:

$$W = \mathbf{F} \cdot \mathbf{d}$$

Where:

• $\mathbf{F}$ is the total force vector, which is the sum of all individual forces,
• $\mathbf{d}$ is the displacement vector, calculated by subtracting the initial position from the final position.

The displacement vector is found by subtracting the coordinates of $A$ from those of $B$, and the total force is the vector sum of the individual forces.

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Solution

Step 1: Find the total force vector $\mathbf{F}$

The total force is the sum of the three forces $\mathbf{F_1}$, $\mathbf{F_2}$, and $\mathbf{F_3}$:

$$\mathbf{F} = \mathbf{F_1} + \mathbf{F_2} + \mathbf{F_3}$$

Substitute the given values:

$$\mathbf{F} = (10 \hat{i} - \hat{j} + 11 \hat{k}) + (4 \hat{i} + 5 \hat{j} + 9 \hat{k}) + (-2 \hat{i} + \hat{j} - 9 \hat{k})$$

Now, combine the components of the vectors:

$$\mathbf{F} = (10 + 4 - 2) \hat{i} + (-1 + 5 + 1) \hat{j} + (11 + 9 - 9) \hat{k}$$ $$\mathbf{F} = 12 \hat{i} + 5 \hat{j} + 11 \hat{k}$$

Thus, the total force vector is:

$$\mathbf{F} = 12 \hat{i} + 5 \hat{j} + 11 \hat{k}$$

Step 2: Find the displacement vector $\mathbf{d}$

The displacement vector is calculated by subtracting the coordinates of point $A(5, -5, -7)$ from point $B(6, 2, -2)$:

$$\mathbf{d} = B - A = (6, 2, -2) - (5, -5, -7)$$ $$\mathbf{d} = (6 - 5, 2 - (-5), -2 - (-7)) = (1, 7, 5)$$

Thus, the displacement vector is:

$$\mathbf{d} = \hat{i} + 7 \hat{j} + 5 \hat{k}$$

Step 3: Calculate the work done

The work done is the dot product of the total force vector $\mathbf{F}$ and the displacement vector $\mathbf{d}$:

$$W = \mathbf{F} \cdot \mathbf{d}$$

Substitute the components of the vectors:

$$W = (12 \hat{i} + 5 \hat{j} + 11 \hat{k}) \cdot (\hat{i} + 7 \hat{j} + 5 \hat{k})$$

Now, compute the individual terms of the dot product:

$$W = 12(1) + 5(7) + 11(5)$$ $$W = 12 + 35 + 55 = 102$$

Thus, the total work done is:

$$W = 102 , \text{units}$$

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Key Formulas or Methods Used

• Work Done by a Force:

$$W = \mathbf{F} \cdot \mathbf{d}$$

Where $\mathbf{F}$ is the total force vector and $\mathbf{d}$ is the displacement vector.

• Total Force:

$$\mathbf{F} = \mathbf{F_1} + \mathbf{F_2} + \mathbf{F_3}$$

The total force is the vector sum of the individual forces acting on the particle.

• Displacement Vector:

$$\mathbf{d} = B - A$$

The displacement vector is obtained by subtracting the coordinates of point $A$ from those of point $B$.

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Summary of Steps

1. Find the total force vector: Add the individual force vectors $\mathbf{F_1}, \mathbf{F_2}, \mathbf{F_3}$ to get the total force vector $\mathbf{F}$.
2. Calculate the displacement vector: Subtract the coordinates of point $A$ from point $B$ to find the displacement vector $\mathbf{d}$.
3. Calculate the work done: Compute the dot product of the total force vector and the displacement vector to find the work done, which is $102 , \text{units}$.