7.5 Q-8

Question Statement

A particle is acted upon by two constant forces: $4 \hat{i} + \hat{j} - 3 \hat{k}$ and $3 \hat{i} - \hat{j} - \hat{k}$. The particle moves from point $A(1,2,3)$ to point $B(5,4,1)$. Find the work done.

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Background and Explanation

To calculate the work done by a force on an object, we use the formula:

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

Where:

• $\underline{F}$ is the total force vector,
• $\underline{d}$ is the displacement vector.

The displacement vector is found by subtracting the initial position vector $A$ from the final position vector $B$. Additionally, when multiple forces act on an object, the total force is the vector sum of all the individual forces.

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Solution

Step 1: Find the displacement vector $\underline{d}$

The displacement vector is calculated by subtracting the coordinates of point $A(1,2,3)$ from point $B(5,4,1)$:

$$\underline{d} = B - A = (5, 4, 1) - (1, 2, 3)$$ $$\underline{d} = (5 - 1, 4 - 2, 1 - 3) = (4, 2, -2)$$

Thus, the displacement vector is:

$$\underline{d} = 4 \hat{i} + 2 \hat{j} - 2 \hat{k}$$

Step 2: Calculate the total force vector $\underline{F}$

The total force is the sum of the individual forces $\underline{F}_1 = 4 \hat{i} + \hat{j} - 3 \hat{k}$ and $\underline{F}_2 = 3 \hat{i} - \hat{j} - \hat{k}$:

$$\underline{F} = \underline{F}_1 + \underline{F}_2 = (4 \hat{i} + \hat{j} - 3 \hat{k}) + (3 \hat{i} - \hat{j} - \hat{k})$$ $$\underline{F} = (4 + 3) \hat{i} + (1 - 1) \hat{j} + (-3 - 1) \hat{k} = 7 \hat{i} + 0 \hat{j} - 4 \hat{k}$$

Thus, the total force vector is:

$$\underline{F} = 7 \hat{i} + 0 \hat{j} - 4 \hat{k}$$

Step 3: Calculate the work done

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

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

Substitute the components of the vectors:

$$W = (7 \hat{i} + 0 \hat{j} - 4 \hat{k}) \cdot (4 \hat{i} + 2 \hat{j} - 2 \hat{k})$$

Now, compute the individual terms of the dot product:

$$W = 7(4) + 0(2) + (-4)(-2)$$ $$W = 28 + 0 + 8 = 36$$

Thus, the work done is:

$$W = 36 , \text{Joules}$$

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Key Formulas or Methods Used

• Work Done by a Force:

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

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

• Displacement Vector:

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

Where $A$ and $B$ are the initial and final position vectors.

• Total Force:

$$\underline{F} = \underline{F}_1 + \underline{F}_2$$

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

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Summary of Steps

1. Find the displacement vector: Subtract the coordinates of point $A$ from point $B$ to get the displacement vector $\underline{d}$.
2. Calculate the total force vector: Add the individual force vectors $\underline{F}_1$ and $\underline{F}_2$ to get the total force vector $\underline{F}$.
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 $36 , \text{Joules}$.