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Chapter 3: Problem 24

Vector \(\vec{A}\), which is directed along an \(x\) axis, is to be added to vector \(\vec{B}\), which has a magnitude of \(7.0 \mathrm{~m}\). The sum is a third vector that is directed along the \(y\) axis, with a magnitude that is \(3.0\) times that of \(\vec{A}\). What is that magnitude of \(\vec{A}\) ?

Short Answer

Expert verified
The magnitude of \( \vec{A} \) is approximately \( 2.33 \) m.

Step by step solution

01

Understand the Problem

We need to find the magnitude of vector \( \vec{A} \) when added to vector \( \vec{B} \) results in a vector that points solely in the \( y \)-direction. The resulting vector's magnitude is given as three times the magnitude of \( \vec{A} \). Vector \( \vec{B} \) has a magnitude of \( 7.0 \) m.
02

Identify Vector Components

Since \( \vec{A} \) is directed along the \( x \)-axis, it has components \( A_x = A \) and \( A_y = 0 \). Vector \( \vec{B} \), which is unspecified in direction but part of the sum is directed along the \( y \)-axis, must have a \( B_y \) component that counterbalances \( \vec{A} \). We'll assume it's fully in the y-direction: \( B_x = 0 \), \( B_y = 7.0 \) m.
03

Write the Equation for Vector Addition

The resulting vector \( \vec{C} \) is said to be directed along the \( y \)-axis. This implies the \( y \)-component of \( \vec{C} \) is \( C_y = B_y \) while the \( x \)-component \( C_x = 0 \). The equation becomes \[ C_y = B_y = 3A \].
04

Solve for \( A \)

Since we know \( C_y = 3A = 7.0 \) m (because \( C_y = B_y \)), we can solve:\[ 3A = 7.0 \mathrm{~m} \]Divide both sides by 3:\[ A = \frac{7.0 \mathrm{~m}}{3} \approx 2.33 \mathrm{~m} \]
05

Verify the Solution

Verify the calculated magnitude of \( \vec{A} \) produces a resulting vector \( \vec{C} \) directed entirely in the \( y \)-axis with a magnitude of \( 3 \times A = 7.0 \) m.

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Key Concepts

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

Understanding Vector Components
Vectors are quantities that have both a magnitude and a direction. When dealing with vectors in physics, it is essential to understand their components. A vector aligned with a particular axis, such as the x or y axis, primarily has non-zero components only in that direction. For example, in our exercise, vector \( \vec{A} \) is directed along the x-axis.
  • The components of \( \vec{A} \) are \( A_x = A \) and \( A_y = 0 \).
This means \( \vec{A} \) has its full magnitude in the x-direction. Similarly, vector \( \vec{B} \) is assumed to be entirely in the y-direction for simplicity, with components:
  • \( B_x = 0 \)
  • \( B_y = 7.0 \) m
Understanding vector components is crucial because it allows us to perform vector addition and solve more complex physics problems efficiently. By breaking down vectors into their components, we can easily analyze their impact in different directions.
Finding the Resultant Vector
A resultant vector is what you get when two or more vectors are added together. It is crucial because it represents the combined effect of these vectors. In this exercise, our goal is to find a resultant vector \( \vec{C} \), directed along the y-axis. This tells us that all force or movement is in the y direction, with nothing left in the x direction.

We can express the summation of vectors in component form. Since \( \vec{A} \) and the resultant vector \( \vec{C} \) are directed explicitly along the axis:
  • For the x-direction: \( C_x = A_x = A \)
  • For the y-direction: \( C_y = B_y \)
The challenge is ensuring that the sum of these vectors in the y-dimension matches the problem's constraints. The problem also gives the condition: \( C_y = 3A \). This provides a direct relationship to solve for \( A \), noting that \( B_y = 7.0 \) m. Our task was to confirm that these two vectors summed to the conditions provided.
Approaches in Physics Problem Solving
Tackling physics problems often involves systematically breaking down and understanding the problem at hand. This is true for our vector addition problem. Keeping these steps in mind can help:
  • **Understand the Problem**: Identify what is asked and what information is given. We immediately saw that we needed to find the magnitude of \( \vec{A} \).
  • **Identify and Use Concepts**: Knowing that the sum vectors would result in a vector along the y-direction pointed us towards vector components. We utilized \( B_y = 7.0\) m and \( C_y = 3A \).
  • **Solve the Equations**: Set up the mathematical equations and solve them. Here we derived \( 3A = 7.0 \) m, which led to our solution \( A = 2.33 \) m.
  • **Verify**: Always check that the solution makes sense and meets the problem's conditions.
With these strategies, we can effectively approach and solve physics problems, making even the most complicated vectors manageable.

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Most popular questions from this chapter

A car is driven east for a distance of \(50 \mathrm{~km}\), then north for 30 \(\mathrm{km}\), and then in a direction \(30^{\circ}\) east of north for \(25 \mathrm{~km}\). Sketch the vector diagram and determine (a) the magnitude and (b) the angle of the car's total displacement from its starting point.

An explorer is caught in a whiteout (in which the snowfall is so thick that the ground cannot be distinguished from the sky) while returning to base camp. He was supposed to travel due north for \(5.6 \mathrm{~km}\), but when the snow clears, he discovers that he actually traveled \(7.8 \mathrm{~km}\) at \(50^{\circ}\) north of due east. (a) How far and (b) in what direction must he now travel to reach base camp?

A fire ant, searching for hot sauce in a picnic area, goes through three displacements along level ground: \(\vec{d}_{1}\) for \(0.40 \mathrm{~m}\) southwest (that is, at \(45^{\circ}\) from directly south and from directly west), \(\vec{d}_{2}\) for \(0.50 \mathrm{~m}\) due east, \(\vec{d}_{3}\) for \(0.60 \mathrm{~m}\) at \(60^{\circ}\) north of east. Let the positive \(x\) direction be east and the positive \(y\) direction be north. What are (a) the \(x\) component and (b) the \(y\) component of \(\vec{d}_{1} ?\) Next, what are (c) the \(x\) component and (d) the \(y\) component of \(\vec{d}_{2}\) ? Also, what are (e) the \(x\) component and (f) the \(y\) component of \(\vec{d}_{3}\) ?

A man goes for a walk, starting from the origin of an \(x y z\) coordinate system, with the \(x y\) plane horizontal and the \(x\) axis eastward. Carrying a bad penny, he walks \(1300 \mathrm{~m}\) east, \(2200 \mathrm{~m}\) north, and then drops the penny from a cliff \(410 \mathrm{~m}\) high. (a) In unit-vector notation, what is the displacement of the penny from start to its landing point? (b) When the man returns to the origin, what is the magnitude of his displacement for the return trip?

In a game of lawn chess, where pieces are moved between the centers of squares that are each \(1.00 \mathrm{~m}\) on edge, a knight is moved in the following way: (1) two squares forward, one square rightward; (2) two squares leftward, one square forward; (3) two squares forward, one square leftward. What are (a) the magnitude and (b) the angle (relative to "forward") of the knight's overall displacement for the series of three moves?
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