Problem 38 - 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. \(\mathbf{F}=22 \mathbf{i}+9 \mathbf{j}, \mathbf{d}=30 \mathbf{i}+4 \mathbf{j}\)

Step-by-Step Solution

Answer

The work performed is 696 pound-feet.

Step 1: Understanding the Problem

Given two vectors, a force vector \(\mathbf{F}\) and a displacement vector \(\mathbf{d}\), we are asked to find the work done. The work is calculated using the dot product of these two vectors.

Step 2: Formula for Work with Vectors

The work done \(W\) by a constant force \(\mathbf{F}\) when an object is displaced by a vector \(\mathbf{d}\) is given by the dot product formula:\[W = \mathbf{F} \cdot \mathbf{d}\]

Step 3: Calculate the Dot Product

The dot product \(\mathbf{F} \cdot \mathbf{d}\) is calculated as follows:\[\mathbf{F} \cdot \mathbf{d} = (22 \mathbf{i} + 9 \mathbf{j}) \cdot (30 \mathbf{i} + 4 \mathbf{j})\]Expand this expression:\[= 22 \times 30 + 9 \times 4\]Simplify the calculations:\[= 660 + 36 = 696\]

Step 4: Interpret the Result

The calculated dot product value is the work done by the force. In this case, the work done is 696 pound-feet, as both the force and displacement are expressed in consistent units.

Key Concepts

Dot Product

The dot product is a way to multiply two vectors together to get a scalar quantity. This is important in physics calculations, especially when determining work. When dealing with two vectors, such as a force vector \(\mathbf{F}\) and a displacement vector \(\mathbf{d}\), the dot product helps us find out how much of the force is effective in causing movement in the direction of the displacement.

To calculate the dot product, you take the components of each vector and multiply them together, then sum those products. For two-dimensional vectors \(a \mathbf{i} + b \mathbf{j}\) and \(c \mathbf{i} + d \mathbf{j} \), the dot product is calculated as:

• \(a \times c\)
• \(b \times d\)

Add these results to get the dot product: \(a \times c + b \times d\).

In terms of work, if the dot product is zero, there is no work done because the force is perpendicular to the displacement. If the dot product is positive, the force does effective work in moving the object along the displacement. As a concept, it's tangible and simple once visualized. Think of it as measuring how much of a push or pull is actually helping move the object along the path.

Vector Mathematics

Vector mathematics allows us to efficiently work with quantities that have both a magnitude and a direction, such as force and displacement. In physics and engineering, vectors are vital for describing various physical phenomena.

Vectors in two dimensions can be represented as \(\mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors along the x and y-axes, respectively. This means you can break down any vector into its horizontal and vertical components, simplifying complex problems.

For calculations, operations on vectors include addition, subtraction, and multiplication (dot and cross products). With the dot product, you work with projections of one vector onto another, as discussed. Understanding vector operations is crucial because they offer a way to quantify and compute physical effects, like work or force, in different directions.

Vector mathematics is not just a fundamental part of the equations but serves as a bridge to understanding how forces impact movement, especially when those forces act in multiple directions.

Force and Displacement

Force and displacement are core components in calculating work. The fundamental concept behind work is understanding how a force causes an object to move over a distance.

A force vector \(\mathbf{F}\) represents how strong a push or pull is, and in which direction this is being applied. In our example, \(\mathbf{F} = 22 \mathbf{i} + 9 \mathbf{j}\) indicates a force applied with a significant horizontal and a smaller vertical component.

Displacement vector \(\mathbf{d}\) indicates the change in position from the start to the end of the movement, symbolized as \(\mathbf{d} = 30 \mathbf{i} + 4 \mathbf{j}\) in our case. It tells us how far and in what direction the object has moved.

When a force acts upon an object, causing it to move in the direction of the force, work is performed. Calculating this work with the dot product ensures we account for only the portion of the force vector that aligns with the displacement. This concept allows us to separate effective force from the components which do not result in movement.

Remember, no movement means no work, even if a force is being applied. Understanding how force and displacement vectors interact is key to mastering work calculations in physics.