Problem 23 - College Physics

Question

For each of the given vectors, give a vector that, when added to it, yields a null vector (a vector with a magnitude of zero). Express the vector in the form other than that in which it is given (component or magnitudeangle): (a) \(\overrightarrow{\mathrm{A}}=4.5 \mathrm{~cm}, 40^{\circ}\) above the \(+x\) -axis; (b) \(\overrightarrow{\mathbf{B}}=(2.0 \mathrm{~cm}) \hat{\mathbf{x}}-(4.0 \mathrm{~cm}) \hat{\mathbf{y}} ;(\mathrm{c}) \overrightarrow{\mathrm{C}}=8.0 \mathrm{~cm}\) at an angle of \(60^{\circ}\) above the \(-x\) -axis.

Step-by-Step Solution

Answer

Opposite vectors are: (a) \((-4.5 \cos(40^{\circ})) \hat{i} + (-4.5 \sin(40^{\circ})) \hat{j}\), (b) \(4.47 \mathrm{~cm}, 116.57^{\circ}\), and (c) \((8.0 \cos(60^{\circ})) \hat{i} - (8.0 \sin(60^{\circ})) \hat{j}\)."

Step 1: Understanding the Problem

For each vector provided, we need to find another vector such that their sum results in a null vector, meaning a vector with zero magnitude. The given vectors are in different forms, so we will express the result in a form different from the given.

Step 2: Finding the Opposite of Vector A

Vector A is given in polar form as \(\overrightarrow{\mathrm{A}}=4.5 \mathrm{~cm}, 40^{\circ}\) above the \(+x\)-axis. - To find its opposite vector, we calculate it in component form: - \( A_x = 4.5 \cos(40^{\circ}) \) - \( A_y = 4.5 \sin(40^{\circ}) \)- Therefore, the opposite vector (\(\overrightarrow{A'}\)) is: - \( A'_x = -A_x = -4.5 \cos(40^{\circ}) \) - \( A'_y = -A_y = -4.5 \sin(40^{\circ}) \)- The components are \( \overrightarrow{A'} = (-4.5 \cos(40^{\circ})) \hat{i} + (-4.5 \sin(40^{\circ})) \hat{j} \).

Step 3: Finding the Opposite of Vector B

Vector B is given in component form as \( \overrightarrow{\mathbf{B}}=(2.0 \mathrm{~cm}) \hat{\mathbf{x}}-(4.0 \mathrm{~cm}) \hat{\mathbf{y}} \).- The opposite vector (\(\overrightarrow{B'}\)) is: - \( B'_x = -2.0 \mathrm{~cm} \) - \( B'_y = 4.0 \mathrm{~cm} \)- Express \(\overrightarrow{B'}\) in polar form using magnitude and direction angle: - Magnitude \( B' = \sqrt{(-2.0)^2 + (4.0)^2} = \sqrt{4 + 16} = \sqrt{20} = 4.47 \mathrm{~cm} \) - Angle \( \theta = \tan^{-1}\left(\frac{4.0}{-2.0}\right) = \tan^{-1}(-2) \approx 153.43^{\circ} \) (but need to adjust for complete grid)- \(\overrightarrow{B'} = 4.47 \mathrm{~cm}, 116.57^{\circ} \), noting that the angle should finalize due to the negative x-component.

Step 4: Finding the Opposite of Vector C

Vector C is in polar form \(\overrightarrow{\mathrm{C}}=8.0 \mathrm{~cm}\) at an angle of \(60^{\circ}\) above the \(-x\)-axis.- Convert vector C into components: - \( C_x = -8.0 \cos(60^{\circ}) \) - \( C_y = 8.0 \sin(60^{\circ}) \)- The opposite vector (\(\overrightarrow{C'}\)) is: - \( C'_x = 8.0 \cos(60^{\circ}) \) - \( C'_y = -8.0 \sin(60^{\circ}) \)- Express \(\overrightarrow{C'}\) in component form: - \(\overrightarrow{C'} = (8.0 \cos(60^{\circ})) \hat{i} - (8.0 \sin(60^{\circ})) \hat{j} \).

Key Concepts

Vector Addition

When adding two vectors, each vector is broken down into its components along the x and y axes. This allows for straightforward calculation of their sum. The result is a new vector that starts at the tail of the first and points to the tip of the second. The process of adding vectors is called vector addition. In mathematics and physics, this is a fundamental process used in analyzing forces, motion, and more.

• To add two vectors, sum up their respective x-components and y-components separately.
• The sum of these components gives the x and y coordinates of the resultant vector.
• Geometrically, vector addition can be visualized using the "head-to-tail" method, where one vector's head is placed at another's tail.

Understanding how vector addition works is key in determining the overall effect, like finding a null vector when vectors cancel each other out by precise opposition.

Polar and Component Forms

Vectors can be expressed in two common forms: polar and component. Each has its own advantages depending on what you're trying to solve.

• Polar form of a vector provides its magnitude and direction angle with respect to a reference direction, usually the positive x-axis.
• Component form of a vector breaks it down into perpendicular elements along the x and y axes, specified using components like \(x\hat{i} + y\hat{j}\).

Polar to component conversion involves using trigonometric functions:- Given a vector's magnitude \(r\) and angle \(\theta\), its components are found as: - \(x = r\cos(\theta)\) - \(y = r\sin(\theta)\)Component to polar conversion utilizes the inverse functions and Pythagorean theorem:- The magnitude is \(r = \sqrt{x^2 + y^2}\), and the angle is \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).Understanding these conversions is critical for solving vectors in various forms, as witnessed in the original exercise.

Magnitude and Direction

Magnitude refers to the size or length of a vector, while direction indicates where it points. Together, these describe a vector completely and are key aspects of its representation in polar form.

• The magnitude of a vector is a measure of its length and can be thought of as the distance from its start point to its endpoint.
• The direction is often denoted by an angle measured from a reference direction, typically the positive x-axis. It specifies the vector's orientation in space.

When given component form, the vector's magnitude and direction can be calculated:- Use the Pythagorean theorem to find the magnitude: \[r = \sqrt{x^2 + y^2}\]- Determine the direction using the tangent function: \[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]A comprehensive understanding of magnitude and direction allows us to solve more complex physics problems, such as determining resultant forces.

Vectors in Physics

In physics, vectors are used to express quantities that have both magnitude and direction, such as velocity, force, and displacement. This dual property makes vectors an indispensable tool in physics for representing the complexities of the real world.

• Force vectors express the intensity and direction of physical forces acting upon an object.
• Velocity vectors provide information on both speed and direction of motion.
• Displacement vectors describe the change in position of an object, considering both distance and direction.

By employing vectors, physicists can model and solve real-world phenomena. This includes the ability to determine net forces and movements through vector addition. Understanding vectors in physics informs much of classical mechanics and is foundational in advanced study, making vector exercises like finding null vectors crucial in building analytical skills.