Problem 40 - Trigonometry

Question

Find the work performed when the given force \(\mathbf{F}\) is applied to an object, whose resulting motion is represented by the displacement vector \(d\). Assume the force is in pounds and the displacement is measured in feet. \(\mathbf{F}=-6 \mathbf{i}+19 \mathbf{j}, \mathbf{d}=8 \mathbf{i}+55 \mathbf{j}\)

Step-by-Step Solution

Answer

The work performed is 997 foot-pounds.

Step 1: Understand the Problem

The work done by a force vector as it acts on an object causing some displacement is given by the dot product of the force vector \( \mathbf{F} \) and the displacement vector \( \mathbf{d} \). We need to compute this dot product to find the work performed.

Step 2: Calculate the Dot Product

To find the dot product of two vectors \( \mathbf{F} = a \mathbf{i} + b \mathbf{j} \) and \( \mathbf{d} = c \mathbf{i} + d \mathbf{j} \), use the formula: \[ \mathbf{F} \cdot \mathbf{d} = a \cdot c + b \cdot d \]Substituting the values of the vectors, we have: \[ \mathbf{F} \cdot \mathbf{d} = (-6) \cdot 8 + 19 \cdot 55 \]

Step 3: Perform the Multiplications

Calculate each term in the dot product: 1. \((-6) \cdot 8 = -48\) 2. \(19 \cdot 55 = 1045\)

Step 4: Sum the Results

Add the calculated products to find the work done:\[ -48 + 1045 = 997 \] Thus, the work performed is 997.

Key Concepts

Vector Operations

Vectors are essential in physics and mathematics. They help us express quantities that have both magnitude and direction. Let's break down the vector operations involved in solving the given exercise.

• A vector is typically denoted as \(|\mathbf{i}+|\mathbf{j}|\)\, which shows its components along the horizontal (i) and vertical (j) axis.
• In our problem, we are given two vectors: the force vector \( \mathbf{F} = -6 \mathbf{i} + 19 \mathbf{j} \) and the displacement vector \( \mathbf{d} = 8 \mathbf{i} + 55 \mathbf{j} \).

To find out the work done by the force, we use the **dot product** of the two vectors.

• The dot product involves multiplying the corresponding components (i with i and j with j) and summing the results.
• For our vectors: \(\mathbf{F} \cdot \mathbf{d} = (-6) \cdot 8 + 19 \cdot 55 = 997\).

This operation helps us understand how much of the force vector is causing motion along the direction of the displacement vector. The result of the dot product gives us scalar values, indicating work in this context.

Work and Energy

In physics, work is done when a force acts upon an object to move it over a distance. This exercise exemplifies how to calculate work using vector operations.

• The formula for work \(W\) when a constant force is applied is: \(W = \mathbf{F} \cdot \mathbf{d}\).
• It tells us that work is the dot product of the force and displacement vectors.

Using our exercise values, the computation of the dot product resulted in **997 foot-pounds**, indicating how much work the force carried out to move the object.

Understanding work is crucial because it connects to energy transfer. Essentially, when work is done, energy is transferred from one object to another or transformed from one form to another.

Physics Concepts

Physics studies how the universe behaves, taking into account forces and energies that influence motion and changes within physical systems.

The problem primarily focuses on the concept of force and displacement among physics concepts. We are considering:

• **Force**: a vector quantity that causes an object to move or change its state of motion. It has both direction and magnitude.
• **Displacement**: a vector that represents the change in position of an object.

The goal is to compute how these two interact to produce a **simplified scalar quantity**, namely the work done.

Grasping these fundamental physics concepts enhances our ability to study more complex systems and solve real-world problems. By mastering these basics, connected problems involving forces, energies, and movements become more understandable. In our given exercise, the ability to quantify the work done helps in further understanding how much energy has been transferred or utilized during the process.