/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Consider the following four forc... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 22

Consider the following four force vectors: $$ \begin{array}{l} \overrightarrow{\mathbf{F}}_{1}=50.0 \text { newtons, due east } \\\ \overrightarrow{\mathbf{F}_{2}}=10.0 \text { newtons, due east } \\\ \overrightarrow{\mathbf{F}_{3}}=40.0 \text { newtons, due west } \\\ \overrightarrow{\mathbf{F}}_{4}=30.0 \text { newtons, due west } \end{array} $$ Which two vectors add together to give a resultant with the smallest magnitude, and which two vectors add to give a resultant with the largest magnitude? In each case specify the magnitude and direction of the resultant.

Short Answer

Expert verified
Smallest resultant: 30.0 N west (\(\overrightarrow{\mathbf{F}}_{2}\) + \(\overrightarrow{\mathbf{F}}_{3}\)). Largest resultant: 70.0 N west (\(\overrightarrow{\mathbf{F}}_{3}\) + \(\overrightarrow{\mathbf{F}}_{4}\)).

Step by step solution

01

Identifying Directions

First, understand that forces in opposite directions will affect the resultant magnitude. Here, \( \overrightarrow{\mathbf{F}}_{1} \) and \( \overrightarrow{\mathbf{F}}_{2} \) are directed east, while \( \overrightarrow{\mathbf{F}}_{3} \) and \( \overrightarrow{\mathbf{F}}_{4} \) are directed west.
02

Adding Vectors for Smallest Magnitude

To find the vectors whose sum has the smallest magnitude, look for one east and one west vector, as they will partially cancel each other. Try adding \( \overrightarrow{\mathbf{F}}_{2} = 10.0 \) N (east) and \( \overrightarrow{\mathbf{F}}_{3} = 40.0 \) N (west). This gives: \[ \overrightarrow{\mathbf{F}}_{2} + \overrightarrow{\mathbf{F}}_{3} = 10.0 \, \text{N (east)} - 40.0 \, \text{N (west)} = -30.0 \, \text{N} \]Since west is negative, the resultant is 30.0 N west.
03

Adding Vectors for Largest Magnitude

To achieve the largest resultant, add two vectors in the same direction. For east, add \( \overrightarrow{\mathbf{F}}_{1} = 50.0 \) N and \( \overrightarrow{\mathbf{F}}_{2} = 10.0 \) N (both east). This results in:\[ \overrightarrow{\mathbf{F}}_{1} + \overrightarrow{\mathbf{F}}_{2} = 50.0 \, \text{N} + 10.0 \, \text{N} = 60.0 \, \text{N (east)} \]Alternatively, adding west-directed vectors: \( \overrightarrow{\mathbf{F}}_{3} = 40.0 \) N and \( \overrightarrow{\mathbf{F}}_{4} = 30.0 \) N gives:\[ \overrightarrow{\mathbf{F}}_{3} + \overrightarrow{\mathbf{F}}_{4} = 40.0 \, \text{N} + 30.0 \, \text{N} = 70.0 \, \text{N (west)} \]So, the largest magnitude is 70.0 N west.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Resultant Force
When you combine two or more forces, the overall force produced is called the resultant force. This resultant is what influences the motion of an object. In our exercise, we are dealing with vector addition, where direction plays a crucial role.
  • Forces in the same direction add up, increasing the total magnitude.
  • Forces in opposite directions subtract from each other, reducing the magnitude.
In the case of the given vectors: - The smallest resultant is achieved by pairing an eastward and a westward force. Adding a smaller east force with a larger west force ( 10 N east and 40 N west) results in a force of 30 N directed west. - The largest resultant comes from adding the vectors going the same way. Either 50 N and 10 N both east for 60 N east or 40 N and 30 N both west for 70 N west. Remember: The direction of the resultant is determined by the direction of the larger of the combining vectors.
Magnitude and Direction
The magnitude of a force tells you how strong the force is. Direction, on the other hand, tells you where the force is headed. Both of these aspects are fundamental in vector calculations.
  • Magnitude can be thought of as the length or size of a vector, expressed in units like newtons (N) in physics.
  • Direction might be described in terms of compass points (east, west, etc.) or angles.
When solving the exercise: - The smallest magnitude of 30 N west results when choosing forces that reduce each other's effect. - The largest magnitude is 70 N west when both forces added together point in the same direction. Understanding these two components helps predict how objects will move and interact under these forces.
Physics Problem Solving
Solving physics problems often involves breaking down a problem into manageable steps. Here, we applied this method to the vector addition exercise involving forces. Let's take a closer look:
  • Step 1: Identify directions - Recognize which vectors point east and which point west. This helps in determining how they will interact.
  • Step 2: Calculate the resultant of opposing forces - To find the smallest resultant, combine forces pointing in opposite directions.
  • Step 3: Add forces in the same direction - To calculate the largest magnitude, combine vectors with the same direction.
By following these steps, you simplify the process, making it easier to predict outcomes and enhance understanding. Each of these skills is fundamental in physics problem solving, allowing you to tackle complex situations with ease.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In wandering, a grizzly bear makes a displacement of \(1563 \mathrm{~m}\) due west, followed by a displacement of \(3348 \mathrm{~m}\) in a direction \(32.0^{\circ}\) north of west. What are (a) the magnitude and (b) the direction of the displacement needed for the bear to return to its starting point? Specify the direction relative to due east.

Vector \(\overrightarrow{\mathrm{A}}\) has a magnitude of 6.00 units and points due east. Vector \(\overrightarrow{\mathrm{B}}\) points due north. (a) What is the magnitude of \(\overrightarrow{\mathbf{B}},\) if the vector \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) points \(60.0^{\circ}\) north of east? (b) Find the magnitude of \(\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}\).

The corners of a square lie on a circle of diameter \(D\). Each side of the square has a length \(L\). Is \(L\) smaller or larger than \(D ?\) Explain your reasoning using the Pythagorean theorem. In a \(1330-\mathrm{ft}^{2}\) apartment, how many square meters of area are there? Be sure that your answer is consistent with your answers to the Concept Questions. The diameter \(D\) of the circle is \(0.35 \mathrm{~m}\). Each side of the square has a length \(L\). Find \(L\). Be sure that your answer is consistent with your answer to the Concept Question.

The \(x\) vector component of a displacement vector \(\overrightarrow{\mathbf{r}}\) has a magnitude of \(125 \mathrm{~m}\) and points along the negative \(x\) axis. The \(y\) vector component has a magnitude of \(184 \mathrm{~m}\) and points along the negative \(y\) axis. Find the magnitude and direction of \(\overrightarrow{\mathbf{r}}\). Specify the direction with respect to the negative \(x\) axis.

Vector \(\overrightarrow{\text { A }}\) points along the \(+y\) axis and has a magnitude of 100.0 units. Vector \(\overrightarrow{\mathbf{B}}\) points at an angle of \(60.0^{\circ}\) above the \(+x\) axis and has a magnitude of 200.0 units. Vector \(\overrightarrow{\mathrm{C}}\) points along the \(+x\) axis and has a magnitude of 150.0 units. Which vector has (a) the largest \(x\) component and (b) the largest \(y\) component?
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Rotational Dynamics

Read Explanation

Physical Quantities And Units

Read Explanation

Astrophysics

Read Explanation

Electrostatics

Read Explanation

Solid State Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.