Problem 61 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the work done by the force \(\mathbf{F}=6 \mathbf{i}+8 \mathbf{j}\) pounds in moving an object from \((1,0)\) to \((6,8)\), where distance is in feet.

Step-by-Step Solution

Answer

94 foot-pounds.

Step 1: Determine the Displacement Vector

The displacement vector, \( \mathbf{d} \), from point \((1,0)\) to \((6,8)\) can be found by subtracting the initial position vector from the final position vector. Thus, \( \mathbf{d} = (6-1) \mathbf{i} + (8-0) \mathbf{j} = 5\mathbf{i} + 8\mathbf{j} \).

Step 2: Use the Dot Product Formula

The work done by a force is given by the dot product of the force vector and the displacement vector: \ \( W = \mathbf{F} \cdot \mathbf{d} \). Here, \( \mathbf{F} = 6\mathbf{i} + 8\mathbf{j} \) and \( \mathbf{d} = 5\mathbf{i} + 8\mathbf{j} \). So, \( W = (6 \mathbf{i} + 8 \mathbf{j}) \cdot (5 \mathbf{i} + 8 \mathbf{j}) \).

Step 3: Calculate the Dot Product

Calculate the dot product by multiplying the corresponding components of the vectors: \ \( W = (6 \times 5) + (8 \times 8) = 30 + 64 = 94 \).

Step 4: Conclusion

The work done by the force in moving the object from point \((1,0)\) to \((6,8)\) is \(94\) foot-pounds.

Key Concepts

Displacement Vector

The concept of the displacement vector is central to understanding how an object moves from one point to another in space. It is essentially a vector that shows the change in position of an object. To determine the displacement vector, you subtract the initial position vector from the final position vector:
\[ \mathbf{d} = (x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j}\]

• The initial position is given by the coordinates \((1,0)\).
• The final position is given by the coordinates \((6,8)\).

By subtracting the initial position from the final position, the displacement vector is calculated as \(\mathbf{d} = 5\mathbf{i} + 8\mathbf{j}\).
This vector represents the path or direction in which the object has moved in the two-dimensional plane.

Dot Product

The dot product is an important operation in vector mathematics. It allows us to find the work done when a force is applied on an object over a certain displacement. To calculate the dot product between two vectors, you multiply corresponding components and sum them up:
\[ \text{If } \mathbf{A} = a_1\mathbf{i} + a_2\mathbf{j} \text{ and } \mathbf{B} = b_1\mathbf{i} + b_2\mathbf{j}, \text{ then } \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2\]
In our exercise:

• Force vector \(\mathbf{F} = 6\mathbf{i} + 8\mathbf{j}\)
• Displacement vector \(\mathbf{d} = 5\mathbf{i} + 8\mathbf{j}\)

The dot product is calculated as:
\[ W = (6 \cdot 5) + (8 \cdot 8) = 30 + 64 = 94 \]
The result gives the work done as 94 foot-pounds, indicating the effectiveness of the force applied along the path of movement.

Force Vector Calculation

The force vector is an essential part of the formula used to calculate the work done by a force. It represents both the direction and magnitude of the force applied. In this context, the force vector is given as \(\mathbf{F} = 6\mathbf{i} + 8\mathbf{j}\), measured in pounds.

• The component \(6\mathbf{i}\) represents the force in the x-direction.
• The component \(8\mathbf{j}\) represents the force in the y-direction.

Understanding the force vector is crucial because it tells us how much force is exerted in each direction. When combined with the displacement vector using the dot product, it provides a direct measure of the work performed in moving the object along its path. Each component of the force vector must be accurately used in calculations to ensure accurate results.