Problem 40 - 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: Understand the Formula

The work done by a constant force along a straight line can be calculated using the dot product formula: \[ \text{Work} = \mathbf{F} \cdot \mathbf{d} \]where \(\mathbf{F}\) is the force vector and \(\mathbf{d}\) is the displacement vector.

Step 2: Find the Displacement Vector

The displacement vector can be found by subtracting the initial point coordinates from the final point coordinates. \[ \mathbf{d} = (11-2, 13-5) = (9, 8) \]

Step 3: Calculate the Dot Product

The dot product of vectors \(\mathbf{F} = \langle 2, 8 \rangle\) and \(\mathbf{d} = \langle 9, 8 \rangle\) is calculated as follows:\[ \mathbf{F} \cdot \mathbf{d} = 2 \cdot 9 + 8 \cdot 8 = 18 + 64 = 82 \]

Step 4: Apply the Dot Product to the Work Formula

Since the product we derived is the dot product, and this directly equates the work done by the force over that displacement, the work done is\[ 82 \text{ foot-pounds} \]

Key Concepts

Understanding the Dot Product

Understanding the dot product is crucial in solving physics problems that involve vectors, such as work. In simple terms, the dot product is a way to multiply two vectors together, resulting in a scalar (a single number). This is incredibly useful when we need to find the work done, since work is described as a scalar.In a mathematical sense, the dot product of two vectors \( \mathbf{A} = \langle a_1, a_2 \rangle \) and \( \mathbf{B} = \langle b_1, b_2 \rangle \) is achieved by multiplying their components and summing those products:

• \( \mathbf{A} \cdot \mathbf{B} = a_1 \cdot b_1 + a_2 \cdot b_2 \)

For example, if we apply the dot product to the vectors \( \mathbf{F} = \langle 2, 8 \rangle \) and \( \mathbf{d} = \langle 9, 8 \rangle \), the result is:

• \( 2 \cdot 9 + 8 \cdot 8 = 18 + 64 = 82 \)

This scalar tells us how much one vector "projects" onto another, which in terms of physics, is used to calculate the work done when a force is applied over a displacement.

Breaking Down the Force Vector

In physics, a force is often represented as a vector. A vector is a quantity that has both magnitude and direction, which makes it essential in describing how objects move. The force vector \( \mathbf{F} = \langle 2, 8 \rangle \) in our exercise tells us a lot.

• The first component (2) represents the force in one direction, typically the x-axis.
• The second component (8) represents the force in another direction, usually the y-axis.

Knowing the direction and magnitude of force helps in understanding how it affects an object's motion. For example, in our exercise, these components combine to give a total magnitude and direction for the force which, when applied, causes an object to move.

Calculating the Displacement Vector

A displacement vector reflects the change in position of an object. It shows us both the distance moved and the direction from the initial position to the final position. In our given problem, we need to calculate this to find work done.To find the displacement \( \mathbf{d} \), we subtract the coordinates of the initial point from the final point:

• \( \mathbf{d} = (11 - 2, 13 - 5) = (9, 8) \)

This means the object moved 9 units in one direction and 8 units in another. This vector helps us understand not only how far something has moved but also which way it went, which is vital when calculating work using the dot product.

Mastering Physics Problem Solving

Physics problem solving often involves breaking down real-world phenomena into manageable pieces, such as vectors and scalars. It's about understanding the relationships between different physical quantities like force, displacement, and work. To solve any physics problem effectively:

• Identify what is given and what needs to be found.
• Convert real-world terms into mathematical representations, such as vectors and equations.
• Apply the proper mathematical operations, like the dot product for calculating work.
• Finally, interpret the results to ensure they make sense within the context of the problem.

The solutions to problems like our exercise don't just lie in arithmetic; they require understanding how forces and movements interact in the physical world. By continuously practicing these steps, you become adept at solving more complex physics challenges.