Problem 76 - Fundamentals of Physics

Question

A vector \(\vec{B}\), with a magnitude of \(8.0 \mathrm{~m}\), is added to a vector \(\vec{A}\), which lies along an \(x\) axis. The sum of these two vectors is a third vector that lies along the \(y\) axis and has a magnitude that is twice the magnitude of \(\vec{A}\). What is the magnitude of \(\vec{A}\) ?

Step-by-Step Solution

Answer

The magnitude of \( \vec{A} \) is 4.0 m.

Step 1: Understanding the Variables

We have two vectors: \( \vec{A} \) whose magnitude we need to find, and which lies along the \( x \) axis.\( \vec{B} \) with a magnitude of \( 8.0 \mathrm{~m} \). Given the sum of \( \vec{A} \) and \( \vec{B} \) results in a vector that is along the \( y \) axis and has a magnitude twice that of \( \vec{A} \). We need to find the magnitude of \( \vec{A} \).

Step 2: Setting up the Equation

The sum of the vectors \( \vec{A} \) and \( \vec{B} \), written as \( \vec{A} + \vec{B} = \vec{C} \), must satisfy the condition that the resulting vector \( \vec{C} \) lies entirely on the \( y \) axis.This implies that the \( x \)-component of \( \vec{C} \) must be zero.

Step 3: Components of the Vectors

Vector \( \vec{A} \) lies on the \( x \)-axis, so it can be expressed as \( \vec{A} = A \hat{i} \) where \( A \) is its magnitude and \( \hat{i} \) is the unit vector along the \( x \)-axis.Vector \( \vec{B} \) must be primarily along the \( y \) axis to ensure the sum results in a \( y \)-axis vector.Thus, \( \vec{B} = B \hat{j} \), where \( B = 8.0 \text{ m} \) and \( \hat{j} \) is the unit vector along the \( y \)-axis.

Step 4: Resultant Vector Equation

The resulting vector \( \vec{C} \) can be expressed as: \( \vec{C} = A \hat{i} + B \hat{j} \) where the \( x \)-component must be zero and the \( y \)-component equals the magnitude of \( \vec{C} = 2A \).

Step 5: Solving for the Magnitude of \( \vec{A} \)

Since the resultant vector \( \vec{C} \) is entirely on the \( y \) axis, \( A = 0 \) for the \( x \)-component. For the \( y \)-component, we have:- \( B = 2A \)- Since \( B = 8.0 \text{ m} \), we have \( 8.0 \text{ m} = 2A \).Solve this equation to find \( A = \frac{8.0}{2} = 4.0 \text{ m} \).

Key Concepts

Component Vectors

To fully grasp the concept of vector addition, we first need to understand what component vectors are. In any vector, you can break it down into components that align with specific axes, typically the x-axis and y-axis. This helps simplify calculations when adding vectors together.

For instance, when a vector lies along the x-axis, it only has an x-component and no y-component. This can be depicted as \( \vec{A} = A \hat{i} \), where \( A \) is the magnitude and \( \hat{i} \) is the unit vector. Likewise, if a vector lies along the y-axis, it can be presented as \( \vec{B} = B \hat{j} \). Here, \( B \) represents the magnitude and \( \hat{j} \) is the unit vector for the y-axis.

With component vectors, each vector's contribution to the overall direction and magnitude can be isolated and analyzed, which is crucial for calculating the resultant vector.

Resultant Vector

The concept of a resultant vector arises when two or more vectors are combined. This resultant vector represents the overall effect of combining vectors. In vector addition, the goal is to determine the resulting vector from the sum of given vectors.

When we add vectors \( \vec{A} \) and \( \vec{B} \), the resultant vector \( \vec{C} \) can be expressed as \( \vec{A} + \vec{B} = \vec{C} \). Each component (x and y) must be added separately. In the exercise, because \( \vec{C} \) is entirely on the y-axis, its x-component must be zero. Therefore, the x-component of \( \vec{A} \) and \( \vec{B} \) both contribute, but only the y-component of \( \vec{B} \) is non-zero in the final vector, as \( A = 0 \) for x-component.

This efficiently shows how vectors influence one another based on their components and orientation in different directions.

Magnitude Calculation

Calculating the magnitude of a vector is a fundamental part of understanding vector addition. Magnitude provides the length or size of a vector and is independent of direction.

In the given problem, we had to find the magnitude of vector \( \vec{A} \) based on the magnitudes of the other vectors involved. We know:

• Vector \( \vec{B} \) has a magnitude of 8.0 m.
• The resultant vector \( \vec{C} \) is twice the magnitude of \( \vec{A} \).

Hence, if \( \vec{C} \) lies solely along the y-axis, with no x-component, the equation simplifies to show that \( B = 2A \). Plugging in the given values, \( 8.0 \ \text{m} = 2A \), we divide both sides by 2 to solve for \( A \). Consequently, \( A = 4.0 \ \text{m} \). This straightforward calculation is crucial for comprehending vector relationships and ensuring that all parts of the problem fit together logically.