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Chapter 1: Problem 43

If a vector \(\vec{A}\) has the following components, use trigonometry to find its magnitude and the counterclockwise angle it makes with the \(+x\) axis: (a) \(A_{x}=8.0\) lb, \(A_{y}=6.0\) lb (b) \(A_{x}=-24 \frac{m}{s}, A_{y}=-31 \frac{m}{s}\) (c) \(A_{x}=-1500\) km, \(A_{y}=2000\) km (d) \(A_{x}=71.3\) N, \(A_{y}=-54.7\) N

Short Answer

Expert verified
Calculate each vector's magnitude and angle using the Pythagorean theorem and arctangent function.

Step by step solution

01

Understand the components

Identify the given components of the vector \(\vec{A}\) for each part of the question. The components are given as \(A_x\) and \(A_y\), which are the projections of the vector along the \(x\)- and \(y\)-axes, respectively.
02

Calculate Magnitude

The magnitude \( |\vec{A}| \) of the vector is calculated using the Pythagorean theorem: \[ |\vec{A}| = \sqrt{A_x^2 + A_y^2} \]. Perform this calculation for each part to find the vector's magnitude.
03

Find the Angle with the +x-axis

To determine the angle \( \theta \) that \(\vec{A}\) makes with the positive x-axis, use the arctangent function: \[ \theta = \tan^{-1}\left(\frac{A_y}{A_x}\right) \]. Be sure to consider the signs of \(A_x\) and \(A_y\) to determine the correct quadrant for the angle. Adjust \(\theta\) based on which quadrant the vector lies in.
04

Apply to Each Scenario

Repeat Steps 2 and 3 for parts (a), (b), (c), and (d) of the exercise. Calculate the magnitude and angle for each given set of components.

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Key Concepts

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

Vector Magnitude
The magnitude of a vector is a crucial concept in physics and mathematics. It represents the length or size of the vector, irrespective of its direction. To find the magnitude of a vector with components
  • \( A_x \) along the x-axis,
  • and \( A_y \) along the y-axis,
we use the Pythagorean theorem. The formula is:\[ |\vec{A}| = \sqrt{A_x^2 + A_y^2} \] This formula helps in treating the components as sides of a right triangle.By plugging the given values of \( A_x \) and \( A_y \) into this formula, one can calculate the vector's magnitude for any scenario.
Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. In vector analysis, it is particularly helpful in determining the direction or angle of a vector. When a vector is described by its components, the angle it makes with the x-axis can be found using trigonometric functions. These functions relate the ratios of a right triangle's sides to its angles. A right triangle is naturally formed with the vector as the hypotenuse and its components as the adjacent and opposite sides.
Arctangent Function
The arctangent function, denoted as \( \tan^{-1} \) or \( \atan \), is used to find an angle when the opposite and adjacent side lengths are known. For a vector with components \( A_x \) and \( A_y \), it provides an efficient way to calculate the angle \( \theta \) that the vector makes with the positive x-axis.The formula:\[ \theta = \tan^{-1}\left(\frac{A_y}{A_x}\right) \] requires consideration of the sign of components. Since angles can differ by more than 90 degrees depending on the vector's quadrant, care must be taken during the calculation.
Quadrant Determination
Determining the quadrant in which a vector lies is essential for accurately finding the angle it makes with the x-axis. Vectors are represented in the coordinate plane divided into four quadrants:
  • Quadrant I: Both \( A_x \) and \( A_y \) are positive.
  • Quadrant II: \( A_x \) is negative, \( A_y \) is positive.
  • Quadrant III: Both \( A_x \) and \( A_y \) are negative.
  • Quadrant IV: \( A_x \) is positive, \( A_y \) is negative.
The signs of \( A_x \) and \( A_y \) determine the vector's respective quadrant. This identification adjusts the calculated angle by adding (or subtracting) appropriate values like 180 degrees or 360 degrees to get the actual direction.

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Most popular questions from this chapter

As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of 8.50 \(\mathrm{cm}\) and a thickness of 0.050 \(\mathrm{cm} .\) Find (a) the volume of a single cookie and (b) the ratio of the diameter to the thickness, and express both in the proper number of significant figures.

Space station. You are designing a space station and want to get some idea how large it should be to provide adequate air for the astronauts. Normally, the air is replenished, but for security, you decide that there should be enough to last for two weeks in case of a malfunction. (a) Estimate how many cubic meters of air an average person breathes in two weeks. A typical human breathes about 1\(/ 2 \mathrm{L}\) per breath. (b) If the space station is to be spherical, what should be its diameter to contain all the air you calculated in part (a)?

\(\bullet\) While driving in an exotic foreign land, you see a speed-limit sign on a highway that reads \(180,000\) furlongs per fort-night. How many miles per hour is this? (One furlong is \(\frac{1}{8}\) mile, and a fortnight is 14 days. A furlong originally referred to the length of a plowed furrow.)

A ladybug starts at the center of a 12 -in.-diameter turntable and crawls in a straight radial line to the edge. While this is happening, the turntable turns through a \(45^{\circ}\) angle. (a) Draw a sketch showing the bug's path and the displacement vector for the bug's progress. (b) Find the magnitude and direction of the ladybug's displacement vector.

\(\bullet\) White dwarfs and neutron stars. Recall that density is mass divided by volume, and consult Chapter 0 and Appendix \(E\) as needed. (a) Calculate the average density of the earth in \(g / c m^{3},\) assuming our planet to be a perfect sphere. (b) In about 5 billion years, at the end of its lifetime, our sun will end up as a white dwarf, having about the same mass as it does now, but reduced to about \(15,000 \mathrm{km}\) in diameter. What will be its density at that stage? (c) A neutron star is the remnant left after certain supernovae (explosions of giant stars). Typically, neutron stars are about 20 \(\mathrm{km}\) in diameter and have around the same mass as our sun. What is a typical neutron star density in \(\mathrm{g} / \mathrm{cm}^{3} ?\)
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