Problem 41 - 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. \(\mathbf{F}=-67 \mathbf{i}+59 \mathbf{j}, \mathbf{d}=-96 \mathbf{i}-28 \mathbf{j}\)

Step-by-Step Solution

Answer

The work done is 4780 foot-pounds.

Step 1: Understand the formula for work

The work done by a force \( \mathbf{F} \) acting on an object that moves through a displacement \( \mathbf{d} \) is given by the dot product of \( \mathbf{F} \) and \( \mathbf{d} \). Mathematically, it is expressed as: \(W = \mathbf{F} \cdot \mathbf{d}.\)

Step 2: Calculate the dot product

The dot product of two vectors \( \mathbf{F} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{d} = c\mathbf{i} + d\mathbf{j} \) is calculated as \( ac + bd \). \(\For \( \mathbf{F} = -67 \mathbf{i} + 59 \mathbf{j} \) and \( \mathbf{d} = -96 \mathbf{i} - 28 \mathbf{j} \), the dot product is: (-67)(-96) + (59)(-28).\)

Step 3: Perform the multiplications

Calculate each of the products individually:\- The first product: \( -67 \times -96 = 6432 \).- The second product: \( 59 \times -28 = -1652 \).

Step 4: Find the sum of the products

Add the results from the dot product calculation:\\( 6432 + (-1652) = 4780 \).

Step 5: Interpret the result

The work done by the force vector \( \mathbf{F} \) along the displacement \( \mathbf{d} \) is \( 4780 \) foot-pounds.

Key Concepts

Work and Energy

Work and energy are fundamental concepts in physics that describe how forces interact with objects. "Work" is a measure of energy transfer when a force is applied to an object, causing it to move. To better comprehend work, it's essential to first understand that it involves both force and displacement.

In simple terms, work occurs when a force causes an object to move in the direction of the force. The amount of work done depends on both the magnitude of the force and the distance over which it acts. This relationship is formally expressed as:

• Work (\( W \)) = Force (\( \mathbf{F} \)) · Displacement (\( \mathbf{d} \))
• The unit of work is typically expressed in joules (or, in this exercise, foot-pounds, since imperial units are used)

Understanding work provides insights into energy use and transfer, highlighting the energy required for various actions and reactions. It explains how much effort is necessary to move objects and do mechanical tasks, making it critical for practical physics and engineering.

Force and Displacement

Force and displacement are two vectors that come into play in physics when discussing work. A force can be understood as a push or a pull on an object, described by both direction and magnitude, often represented in vector notation.

Displacement refers to the change in position of an object. Like force, it is a vector, so it also has magnitude and direction. When a force acts on an object and causes it to move, displacement measures how far the object moves in a particular direction, which is essential for calculating work.

• Force (\( \mathbf{F} \)) is represented as a vector: for example, \(-67 \mathbf{i}+59 \mathbf{j} \)
• Displacement (\( \mathbf{d} \)) is also expressed as a vector: \(-96 \mathbf{i}-28 \mathbf{j} \)

To determine the work done by a force, both the force and displacement vectors must align or be resolved into components that align. The concept of resolving vectors into components is crucial because it allows you to effectively calculate how much of the force contributes to moving the object.

Physics Calculations

Calculating work involves using the concept of the dot product between the force and displacement vectors. The dot product can be seen as a way to multiply two vectors to obtain a scalar, which in this context represents work.

The mathematical formula we use to calculate the dot product involves:

• Multiplying corresponding components of the force and displacement vectors
• Summing these products to get a single value, representing the work

In the example given:

• Force vector: \(-67 \mathbf{i}+59 \mathbf{j} \)
• Displacement vector: \(-96 \mathbf{i}-28 \mathbf{j} \)
• Dot product calculation is \((-67)(-96) + (59)(-28)\)

The result of this calculation tells us the total work done. Understanding these physics calculations enables students to solve real-world problems effectively, applying formulas to derive meaningful conclusions about forces and movements. This process demonstrates the power of mathematical operations to explain and predict physical phenomena.