Problem 46 - Algebra and Trigonometry

Question

Work A constant force \(\mathbf{F}=\langle 2,8\rangle\) moves an object along a straight line from the point \((2,5)\) to the point \((11,13) .\) Find the work done if the distance is measured in feet and the force is measured in pounds.

Step-by-Step Solution

Answer

The work done is 82 foot-pounds.

Step 1: Determine the Displacement Vector

The displacement vector \( \mathbf{d} \) from point \((2,5)\) to \((11,13)\) is found by subtracting the coordinates: \( \mathbf{d} = \langle 11-2, 13-5 \rangle = \langle 9, 8 \rangle \).

Step 2: Express the Work Done Formula

The work done by a constant force is calculated using the dot product of the force vector \( \mathbf{F} \) and the displacement vector \( \mathbf{d} \): \( W = \mathbf{F} \cdot \mathbf{d} \).

Step 3: Calculate the Dot Product

Calculate the dot product \( W = \langle 2,8 \rangle \cdot \langle 9,8 \rangle \). This is done by multiplying corresponding components and adding them: \( W = 2 \times 9 + 8 \times 8 = 18 + 64 = 82 \).

Step 4: Interpret the Result

The calculated dot product \( 82 \) gives the work done in foot-pounds. There are no additional calculations needed, as both the force and distance were in consistent units.

Key Concepts

Dot Product

The dot product is a mathematical operation used in the context of vectors, which are quantities characterized by both magnitude and direction. It's especially important when dealing with physics concepts like work done, as it helps to connect force and displacement in an effective way. For two vectors, say \( \mathbf{A} = \langle a_1, a_2 \rangle \) and \( \mathbf{B} = \langle b_1, b_2 \rangle \), the dot product is calculated by multiplying their respective components and summing the results. In formula form, this is written as \( \mathbf{A} \cdot \mathbf{B} = a_1 \cdot b_1 + a_2 \cdot b_2 \).
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\)In the realm of physics, the dot product is often used to find work done by a force acting over a certain distance. It gives you a scalar value which incorporates both the magnitude of the vectors and their direction. Only the component of force that acts in the direction of the displacement does the work. So, understanding dot product is crucial for accurately calculating work, like in our example where we found \( 82 \) foot-pounds.

Displacement Vector

The displacement vector represents the change in position of an object. In our example, it’s the vector from point \((2,5)\) to point \((11,13)\). To find this vector, we subtract the initial position coordinates from the final position coordinates, resulting in the vector \( \mathbf{d} = \langle 9, 8 \rangle \).

This vector not only tells us how far the object has moved but also in which direction. In physics, this is critical because the force's effectiveness (in terms of work done) depends heavily on the direction relative to this displacement. The displacement vector is a key component when performing calculations like the dot product, which is necessary for determining work done.

Force Vector

In physics, the force vector is a representation of a force with both magnitude and direction. In our problem, the force vector is provided as \( \mathbf{F} = \langle 2, 8 \rangle \), given in terms of its components along the x and y axes. This means that the total force is divided into 2 pounds of force in the x-direction and 8 pounds of force in the y-direction.
Understanding this concept helps in comprehending how forces are applied in real-life scenarios and their effects on different objects. For calculating work done, as demonstrated in the exercise, the force vector must be accurately portrayed since it significantly impacts the outcome of the dot product with the displacement vector. This mathematical operation then gives us the total work performed, which is crucial in physics to understand energy transfer.