7.5 Q-7

Question Statement

Find the work done when a constant force $\underline{F} = 4 \hat{i} + 3 \hat{j} + 5 \hat{k}$ is applied to an object, which moves from point $P_1(3,1,-2)$ to point $P_2(2,4,6)$.

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Background and Explanation

In physics, the work done by a constant force is calculated using the formula:

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

Where:

• $\underline{F}$ is the force vector,
• $\underline{d}$ is the displacement vector (the change in position).

To calculate displacement $\underline{d}$, subtract the initial position vector from the final position vector:

$$\underline{d} = P_2 - P_1$$

Then, work is the dot product of the force and displacement vectors.

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Solution

We are given the force vector $\underline{F} = 4 \hat{i} + 3 \hat{j} + 5 \hat{k}$, and the points $P_1(3,1,-2)$ and $P_2(2,4,6)$.

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

The displacement vector is calculated by subtracting the coordinates of point $P_1$ from those of point $P_2$:

$$\underline{d} = P_2 - P_1 = (2, 4, 6) - (3, 1, -2)$$ $$\underline{d} = (2-3, 4-1, 6+2) = (-1, 3, 8)$$

Thus, the displacement vector is:

$$\underline{d} = -\hat{i} + 3 \hat{j} + 8 \hat{k}$$

Step 2: Compute the dot product of $\underline{F}$ and $\underline{d}$

Now we calculate the work done, which 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:

$$W = (4 \hat{i} + 3 \hat{j} + 5 \hat{k}) \cdot (-\hat{i} + 3 \hat{j} + 8 \hat{k})$$

Now, compute the individual terms of the dot product:

$$W = 4(-1) + 3(3) + 5(8)$$ $$W = -4 + 9 + 40$$

Thus, the total work done is:

$$W = 45 , \text{Joules}$$

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Key Formulas or Methods Used

• Work Done by a Constant Force:

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

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

• Displacement Vector:

$$\underline{d} = P_2 - P_1$$

Where $P_1$ and $P_2$ are the initial and final position vectors.

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Summary of Steps

1. Find the displacement vector: Subtract the initial position vector $P_1$ from the final position vector $P_2$.
2. Compute the dot product: Multiply corresponding components of the force and displacement vectors and sum the results.
3. Calculate the work done: The result from the dot product gives the work done, in this case, $45 , \text{Joules}$.