91Ó°ÊÓ

The components of vector \(\overrightarrow{\mathbf{A}}\) are \(A_{x}\) and \(A_{v}\) (both positive), and the angle that it makes with respect to the positive \(x\) axis is \(\theta\). (a) Does increasing the component \(A_{x}\) (while holding \(A_{y}\) constant) increase or decrease the angle \(\theta\) ? (b) Does increasing the component \(A_{y}\) (while holding \(A_{x}\) constant) increase or decrease the angle \(\theta\) ? Account for your answers. The components of displacement vector \(\overrightarrow{\mathrm{A}}\) are \(A_{x}=12 \mathrm{~m}\) and \(A_{y}=12 \mathrm{~m}\). Find \(\theta\). (b) The components of displacement vector \(\overrightarrow{\mathrm{A}}\) are \(A_{x}=17 \mathrm{~m}\) and \(A_{y}=12 \mathrm{~m}\). Find \(\theta\). (c) The components of displacement vector \(\overrightarrow{\mathbf{A}}\) are \(A_{x}=12 \mathrm{~m}\) and \(A_{y}=17 \mathrm{~m}\). Find \(\theta\). Be sure that your answers are consistent with your answers to the Concept

Step by step solution

01

Analyze the Effect of Increasing Ax on θ

The angle θ can be calculated using the formula \( \theta = \tan^{-1}\left(\frac{A_{y}}{A_{x}}\right) \). If we increase \( A_{x} \) and hold \( A_{y} \) constant, \( \frac{A_{y}}{A_{x}} \) decreases, resulting in a smaller angle \( \theta \). Therefore, increasing \( A_{x} \) decreases \( \theta \).

02

Analyze the Effect of Increasing Ay on θ

Using the same formula, \( \theta = \tan^{-1}\left(\frac{A_{y}}{A_{x}}\right) \), if \( A_{y} \) is increased while \( A_{x} \) is held constant, \( \frac{A_{y}}{A_{x}} \) increases, leading to a larger angle \( \theta \). Therefore, increasing \( A_{y} \) increases \( \theta \).

03

Calculate θ with Ax = 12 m and Ay = 12 m

Substitute \( A_{x} = 12 \) m and \( A_{y} = 12 \) m into the formula \( \theta = \tan^{-1}\left(\frac{A_{y}}{A_{x}}\right) \). Thus, \( \theta = \tan^{-1}\left(\frac{12}{12}\right) = \tan^{-1}(1) = 45^{\circ} \).

04

Calculate θ with Ax = 17 m and Ay = 12 m

Substitute \( A_{x} = 17 \) m and \( A_{y} = 12 \) m into the formula \( \theta = \tan^{-1}\left(\frac{A_{y}}{A_{x}}\right) \). Therefore, \( \theta = \tan^{-1}\left(\frac{12}{17}\right) \approx 35.3^{\circ} \).

05

Calculate θ with Ax = 12 m and Ay = 17 m

Substitute \( A_{x} = 12 \) m and \( A_{y} = 17 \) m into the formula \( \theta = \tan^{-1}\left(\frac{A_{y}}{A_{x}}\right) \). Hence, \( \theta = \tan^{-1}\left(\frac{17}{12}\right) \approx 54.7^{\circ} \).

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Key Concepts

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

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. It's essential for understanding vector components in physics.
The core concept involves functions - sine, cosine, and tangent, which are fundamental in calculating angles and lengths. These functions relate to right-angled triangles, where the tangent particularly helps find angles within the triangle using the opposite and adjacent sides.

• The sine function ( \( \sin \theta \) ) involves the opposite side over the hypotenuse.
• The cosine function ( \( \cos \theta \) ) involves the adjacent side over the hypotenuse.
• The tangent function ( \( \tan \theta \) ) is the opposite side over the adjacent side.

Understanding these functions assists in converting angles and vector components effectively, which is required in various scientific and engineering tasks.

Angle Calculation

Angle calculation using vector components is vital in determining the orientation of a vector. The angle, typically denoted as \( \theta \), gives the direction of a vector with respect to a reference axis, often the positive x-axis.
To find \( \theta \), the inverse tangent function (\( \tan^{-1} \)) is utilized, which converts a ratio of the vector's vertical over horizontal components ( \( \frac{A_{y}}{A_{x}} \) ) into an angle:\[ \theta = \tan^{-1} \left( \frac{A_{y}}{A_{x}} \right) \]This formula means if the vertical component \( A_y \) is greater than \( A_x \), the angle \( \theta \) will increase, showing more orientation towards the vertical axis.
Conversely, larger \( A_x \) will decrease \( \theta \), making the vector more horizontal.

Displacement Vector

A displacement vector provides both the distance and direction from one point to another. It's an essential concept in physics to understand movements and forces.
This vector is composed of two components in a two-dimensional plane: the x-component and the y-component. These can be visually represented by plotting on a coordinate axis, showing the movement from origin to a point based on x and y distances.

• x-component ( \( A_x \) ) measures the displacement along the horizontal axis.
• y-component ( \( A_y \) ) measures the displacement along the vertical axis.

The combined measurement of these two components forms the vector, and the angle of this vector signifies its directional orientation.

Tangent Function

The tangent function ( \( \tan \theta \) ) is a crucial trigonometric tool used to determine angles by relating two sides of a right triangle. Specifically, the tangent function compares the length of the side opposite the angle to the side adjacent to it.
This function is used to find the angle \( \theta \), when given the displacement vector components:
\[ \theta = \tan^{-1} \left( \frac{A_y}{A_x} \right) \]When vector components change, the tangent function helps adjust the angle \( \theta \) accordingly by evaluating the ratio of vertical to horizontal displacements.

• If the opposite side ( \( A_y \) ) increases and the adjacent ( \( A_x \) ) holds, \( \theta \) increases.
• If the adjacent increases while opposite stays, \( \theta \) decreases.

This application ensures precise direction finding concerning the vector orientation in various fields, such as navigation and engineering.