Physics

John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler

Chapter 1

Introduction and Mathematical Concepts - all with Video Answers

Educators

+ 5 more educators

Chapter Questions

Problem 1

A student sees a newspaper ad for an apartment that has 1330 square feet $\left(\mathrm{ft}^{2}\right)$ of floor space. How many square meters of area are there?

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 2

Bicyclists in the Tour de France reach speeds of 34.0 miles per hour (mi/h) on fl at sections of the road. What is this speed in (a) kilometers per hour (km/h) and (b) meters per second (m/s)?

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 3

Vesna Vulovic survived the longest fall on record without a parachute when her plane exploded and she fell 6 miles, 551 yards. What is this distance in meters?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 4

Suppose a man’s scalp hair grows at a rate of 0.35 mm per day. What is this growth rate in feet per century?

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 5

Given the quantities $a=9.7 \mathrm{m}, b=4.2 \mathrm{s}, c=69 \mathrm{m} / \mathrm{s},$ what is the value of the quantity $d=a^{3} /\left(c b^{2}\right) ?$

Vishal Gupta

Numerade Educator

Problem 6

Consider the equation $v=\frac{1}{3} z x t^{2} .$ The dimensions of the variables $v$, $x,$ and $t$ are $\left.[\mathrm{L}] / \mathrm{L}^{\mathrm{T}}\right],[\mathrm{L}],$ and $[\mathrm{T}],$ respectively. The numerical factor 3 is $\mathrm{di}-$ mensionless. What must be the dimensions of the variable $z$, such that both sides of the equation have the same dimensions? Show how you determined your answer.

Vishal Gupta

Numerade Educator

Problem 7

A bottle of wine known as a magnum contains a volume of 1.5 liters. A bottle known as a jeroboam contains 0.792 U.S. gallons. How many magnums are there in one jeroboam?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 8

The CGS unit for measuring the viscosity of a liquid is the poise (P): $1 \mathrm{P}=1 \mathrm{g} /(\mathrm{s} \cdot \mathrm{cm}) .$ The SI unit for viscosity is the $\mathrm{kg} /(\mathrm{s} \cdot \mathrm{m}) .$ The viscosity of water at $0^{\circ} \mathrm{C}$ is $1.78 \times 10^{-3} \mathrm{kg} /(\mathrm{s} \cdot \mathrm{m}) .$ Express this viscosity in poise.

Vishal Gupta

Numerade Educator

Problem 9

Azelastine hydrochloride is an antihistamine nasal spray. A standard size container holds one fluid ounce (oz) of the liquid. You are searching for this medication in a European drugstore and are asked how many milliliters (mL) there are in one fluid ounce. Using the following conversion factors, determine the number of milliliters in a volume of one fluid ounce: 1 gallon $($ gal $)=128$ oz, $3.785 \times 10^{-3}$ cubic meters $\left(\mathrm{m}^{3}\right)=1$ gal, and $1 \mathrm{mL}=10^{-6} \mathrm{m}^{3}.$

Anand Jangid

Numerade Educator

Problem 10

A partly full paint can has 0.67 U.S. gallons of paint left in it. (a) What is the volume of the paint in cubic meters? (b) If all the remaining paint is used to coat a wall evenly (wall area $=13 \mathrm{m}^{2}$ ), how thick is the layer of wet paint? Give your answer in meters.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 11

A spring is hanging down from the ceiling, and an object of mass $m$ is attached to the free end. The object is pulled down, thereby stretching the spring, and then released. The object oscillates up and down, and the time $T$ required for one complete up-and-down oscillation is given by the equation $T=2 \pi \sqrt{m / k},$ where $k$ is known as the spring constant. What must be the dimension of $k$ for this equation to be dimensionally correct?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 12

You are driving into St. Louis, Missouri, and in the distance you see the famous Gateway to the West arch. This monument rises to a height of $192 \mathrm{m}$. You estimate your line of sight with the top of the arch to be $2.0^{\circ}$ above the horizontal. Approximately how far (in kilometers) are you from the base of the arch?

Rashmi Sinha

Numerade Educator

Problem 13

A highway is to be built between two towns, one of which lies $35.0 \mathrm{km}$ south and $72.0 \mathrm{km}$ west of the other. What is the shortest length of highway that can be built between the two towns, and at what angle would this highway be directed with respect to due west?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 14

A hill that has a $12.0 \%$ grade is one that rises 12.0 m vertically for every $100.0 \mathrm{m}$ of distance in the horizontal direction. At what angle is such a hill inclined above the horizontal?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 15

The corners of a square lie on a circle of diameter $D=0.35 \mathrm{m}$ Each side of the square has a length $L$. Find $L$.

Prabhat Tyagi

Numerade Educator

Problem 16

which an antenna is mounted. The horizontal distance between the person's eyes and the building is $85.0 \mathrm{m} .$ In part $a$ the person is looking at the base of the antenna, and his line of sight makes an angle of $35.0^{\circ}$ with the horizontal. In part $b$ the person is looking at the top of the antenna, and his line of sight makes an angle of $38.0^{\circ}$ with the horizontal. How tall is the antenna?

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 17

The two hot-air balloons in the drawing are 48.2 and $61.0 \mathrm{m}$ above the ground. A person in the left balloon observes that the right balloon is $13.3^{\circ}$ above the horizontal. What is the horizontal distance $x$ between the two balloons?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 18

Available on WileyPLUS.

Manish Jain

Numerade Educator

Problem 19

The drawing shows sodium and chloride ions positioned at the corners of a cube that is part of the crystal structure of sodium chloride (common table salt). The edges of the cube are each $0.281 \mathrm{nm}(1 \mathrm{nm}=1$ nanometer $=$ $\left.10^{-9} \mathrm{m}\right)$ in length. What is the value of the angle $\theta$ in the drawing?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 20

A person is standing at the edge of the water and looking out at the ocean (see the drawing). The height of the person's eyes above the water is $h=1.6 \mathrm{m},$ and the radius of the earth is $R=6.38 \times 10^{6} \mathrm{m}$ (a) How far is it to the horizon? In other words, what is the distance $d$ from the person's eyes to the horizon? (Note: At the horizon the angle between the line of sight and the radius of the earth is $90^{\circ} .$ ) (b) Express this distance in miles.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 21

Three deer, $\mathrm{A}, \mathrm{B},$ and $\mathrm{C},$ are grazing in a field. Deer $\mathrm{B}$ is located $62 \mathrm{m}$ from deer $\mathrm{A}$ at an angle of $51^{\circ}$ north of west. Deer $\mathrm{C}$ is located $77^{\circ}$ north of east relative to deer A. The distance between deer $\mathrm{B}$ and $\mathrm{C}$ is $95 \mathrm{m}$. What is the distance between deer A and C? (Hint: Consider the law of cosines given in Appendix E.)

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 22

An aerialist on a high platform holds on to a trapeze attached to a support by an $8.0-\mathrm{m}$ cord. (See the drawing.) Just before he jumps off the platform, the cord makes an angle of 41 $^{\circ}$ with the vertical. He jumps, swings down, then back up, releasing the trapeze at the instant it is $0.75 \mathrm{m}$ below its initial height. Calculate the angle $\theta$ that the trapeze cord makes with the vertical at this instant.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 23

(a) Two workers are trying to move a heavy crate. One pushes on the crate with a force $\overrightarrow{\mathbf{A}},$ which has a magnitude of 445 newtons and is directed due west. The other pushes with a force $\overrightarrow{\mathbf{B}}$, which has a magnitude of 325 newtons and is directed due north. What are the magnitude and direction of the resultant force $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ applied to the crate? (b) Suppose that the second worker applies a force $-\overrightarrow{\mathbf{B}}$ instead of $\overrightarrow{\mathbf{B}}$. What then are the magnitude and direction of the resultant force $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ applied to the crate? In both cases express the direction relative to due west.

Vishal Gupta

Numerade Educator

Problem 24

A force vector $\overrightarrow{\mathbf{F}}_{1}$ points due east and has a magnitude of 200 newtons. A second force $\overrightarrow{\mathbf{F}}_{2}$ is added to $\overrightarrow{\mathbf{F}}_{1} .$ The resultant of the two vectors has a magnitude of 400 newtons and points along the east/west line. Find the magnitude and direction of $\overrightarrow{\mathbf{F}}_{2} .$ Note that there are two answers.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 25

Consider the following four force vectors:
$$\begin{array}{l}\overrightarrow{\mathbf{F}}_{1}=50.0 \text { newtons, due east } \\\overrightarrow{\mathbf{F}}_{2}=10.0 \text { newtons, due east } \\\overrightarrow{\mathbf{F}}_{3}=40.0 \text { newtons, due west } \\\overrightarrow{\mathbf{F}}_{4}=30.0 \text { newtons, due west }\end{array}$$
Which two vectors add together to give a resultant with the smallest magnitude, and which two vectors add to give a resultant with the largest magnitude? In each case specify the magnitude and direction of the resultant.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 26

Vector $\overrightarrow{\mathbf{A}}$ has a magnitude of 63 units and points due west, while vector $\overrightarrow{\mathbf{B}}$ has the same magnitude and points due south. Find the magnitude and direction of (a) $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ and (b) $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}} .$ Specify the directions relative to due west.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 27

Two bicyclists, starting at the same place, are riding toward the same campground by two different routes. One cyclist rides $1080 \mathrm{m}$ due east and then turns due north and travels another $1430 \mathrm{m}$ before reaching the campground. The second cyclist starts out by heading due north for $1950 \mathrm{m}$ and then turns and heads directly toward the campground. (a) At the turning point, how far is the second cyclist from the campground? (b) In what direction (measured relative to due east) must the second cyclist head during the last part of the trip?

Massimo Antonelli

Massimo Antonelli

Numerade Educator

Problem 28

The drawing shows a triple jump on a checkerboard, starting at the center of square $A$ and ending on the center of square $B .$ Each side of a square measures $4.0 \mathrm{cm} .$ What is the magnitude of the displacement of the colored checker during the triple jump?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 29

Given the vectors $\overrightarrow{\mathbf{P}}$ and $\overrightarrow{\mathbf{Q}}$ shown on the grid, sketch and calculate the magnitudes of the vectors (a) $\overrightarrow{\mathbf{M}}=\overrightarrow{\mathbf{P}}+\overrightarrow{\mathbf{Q}}$ and (b) $\overrightarrow{\mathbf{K}}=2 \overrightarrow{\mathbf{P}}-\overrightarrow{\mathbf{Q}}$ Use the tail-to-head method and express the magnitudes in centimeters with the aid of the grid scale shown in the drawing.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 30

Vector $\overrightarrow{\mathbf{A}}$ has a magnitude of 12.3 units and points due west. Vector $\overrightarrow{\mathbf{B}}$ points due north. (a) What is the magnitude of $\overrightarrow{\mathbf{B}}$ if $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ has a magnitude of 15.0 units? (b) What is the direction of $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ relative to due west? (c) What is the magnitude of $\overrightarrow{\mathbf{B}}$ if $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ has a magnitude of 15.0 units? (d) What is the direction of $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ relative to due west?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 31

A car is being pulled out of the mud by two forces that are applied by the two ropes shown in the drawing. The dashed line in the drawing bisects the $30.0^{\circ}$ angle. The magnitude of the force applied by each rope is 2900 newtons. Arrange the force vectors tail to head and use the graphical technique to answer the following questions. (a) How much force would a single rope need to apply to accomplish the same effect as the two forces added together? (b) How would the single rope be directed relative to the dashed line?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 32

A jogger travels a route that has two parts. The first is a displacement $\overrightarrow{\mathbf{A}}$ of $2.50 \mathrm{km}$ due south, and the second involves a displacement $\overrightarrow{\mathbf{B}}$ that points due east. (a) The resultant displacement $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ has a magnitude of $3.75 \mathrm{km} .$ What is the magnitude of $\overrightarrow{\mathbf{B}},$ and what is the direction of $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ relative to due south? (b) Suppose that $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ had a magnitude of $3.75 \mathrm{km}$. What then would be the magnitude of $\overrightarrow{\mathbf{B}},$ and what is the direction of $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ relative to due south?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 33

water at a beach ball from three directions. As a result, three forces act on the ball, $\overrightarrow{\mathbf{F}}_{1}, \overrightarrow{\mathbf{F}}_{2},$ and $\overrightarrow{\mathbf{F}}_{3}$ (see the drawing). The magnitudes of $\overrightarrow{\mathbf{F}}_{1}$ and $\overrightarrow{\mathbf{F}}_{2}$ are $F_{1}=50.0$ newtons and $F_{2}=90.0$ newtons. Using a scale drawing and the graphical technique, determine (a) the magnitude of $\overrightarrow{\mathbf{F}}_{3}$ and (b) the angle $\theta$ such that the resultant force acting on the ball is zero.

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 34

A force vector has a magnitude of 575 newtons and points at an angle of $36.0^{\circ}$ below the positive $x$ axis. What are (a) the $x$ scalar component and (b) the $y$ scalar component of the vector?

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 35

Vector $\overrightarrow{\mathbf{A}}$ points along the $+y$ axis and has a magnitude of 100.0 units. Vector $\overrightarrow{\mathbf{B}}$ points at an angle of $60.0^{\circ}$ above the $+x$ axis and has a magnitude of 200.0 units. Vector $\overrightarrow{\mathbf{C}}$ points along the $+x$ axis and has a magnitude of 150.0 units. Which vector has (a) the largest $x$ component and (b) the largest y component?

Nishant Kumar

Numerade Educator

Problem 36

Soccer player #1 is $8.6 \mathrm{m}$ from the goal (see the drawing). If she kicks the ball directly into the net, the ball has a displacement labeled $\overrightarrow{\mathbf{A}}$. If, on the other hand, she first kicks it to player $\# 2,$ who then kicks it into the net, the ball undergoes two successive displacements, $\overrightarrow{\mathbf{A}}_{y}$ and $\overrightarrow{\mathbf{A}}_{x} .$ What are the magnitudes and directions of $\overrightarrow{\mathbf{A}}_{x}$ and $\overrightarrow{\mathbf{A}}_{y} ?$

Ajay Singhal

Numerade Educator

Problem 37

The components of vector $\overrightarrow{\mathbf{A}}$ are $A_{x}$ and $A,$ (both positive), and the angle that it makes with respect to the positive $x$ axis is $\theta$. Find the angle $\theta$ if the components of the displacement vector $\overrightarrow{\mathbf{A}}$ are (a) $A_{x}=12 \mathrm{m}$ and $A_{y}=12 \mathrm{m},$ (b) $A_{x}=17 \mathrm{m}$ and $A_{y}=12 \mathrm{m},$ and (c) $A_{x}=12 \mathrm{m}$ and $A_{y}=17 \mathrm{m}$.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 38

During takeoff, an airplane climbs with a speed of $180 \mathrm{m} / \mathrm{s}$ at an angle of $34^{\circ}$ above the horizontal. The speed and direction of the airplane constitute a vector quantity known as the velocity. The sun is shining directly overhead. How fast is the shadow of the plane moving along the ground? (That is, what is the magnitude of the horizontal component of the plane's velocity?)

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 39

The $x$ vector component of a displacement vector $\overrightarrow{\mathbf{r}}$ has a magnitude of $125 \mathrm{m}$ and points along the negative $x$ axis. The $y$ vector component has a magnitude of $184 \mathrm{m}$ and points along the negative $y$ axis. Find the magnitude and direction of $\overrightarrow{\mathbf{r}}$. Specify the direction with respect to the negative ic

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 40

Your friend has slipped and fallen. To help her up, you pull with a force $\overrightarrow{\mathbf{F}}$, as the drawing shows. The vertical component of this force is 130 newtons, and the horizontal component is 150 newtons. Find (a) the magnitude of $\overrightarrow{\mathbf{F}}$ and $(\mathrm{b})$ the angle $\theta.$

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 41

Available on WileyPLUS.

Manish Jain

Numerade Educator

Problem 42

Two racing boats set out from the same dock and speed away at the same constant speed of $101 \mathrm{km} / \mathrm{h}$ for half an hour $(0.500 \mathrm{h}),$ the blue boat headed $25.0^{\circ}$ south of west, and the green boat headed $37.0^{\circ}$ south of west. During this half hour (a) how much farther west does the blue boat travel, compared to the green boat, and (b) how much farther south does the green boat travel, compared to the blue boat? Express your answers in km.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 43

The magnitude of the force vector $\overrightarrow{\mathbf{F}}$ is 82.3 newtons. The $x$ component of this vector is directed along the $+x$ axis and has a magnitude of 74.6 newtons. The $y$ component points along the $+y$ axis. (a) Find the direction of $\overrightarrow{\mathbf{F}}$ relative to the $+x$ axis. (b) Find the component of $\overrightarrow{\mathbf{F}}$ along the $+y$ axis.

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 44

Available on WileyPLUS.

Manish Jain

Numerade Educator

Problem 45

Consult Multiple-Concept Example 9 in preparation for this problem. A golfer, putting on a green, requires three strokes to "hole the ball." During the first putt, the ball rolls $5.0 \mathrm{m}$ due east. For the second putt, the ball travels $2.1 \mathrm{m}$ at an angle of $20.0^{\circ}$ north of east. The third putt is $0.50 \mathrm{m}$ due north. What displacement (magnitude and direction relative to due east) would have been needed to "hole the ball" on the very first putt?

Narayan Hari

Numerade Educator

Problem 46

The three displacement vectors in the drawing have magnitudes of $A=5.00 \mathrm{m}, B=5.00 \mathrm{m},$ and $C=4.00 \mathrm{m} .$ Find the resultant (magnitude and directional angle) of the three vectors by means of the component method. Express the directional angle as an angle above the positive or negative $x$ axis.

Prashant Bana

Numerade Educator

Problem 47

role in this problem. Two forces are applied to a tree stump to pull it out of the ground. Force $\overrightarrow{\mathbf{F}}_{\mathbf{A}}$ has a magnitude of 2240 newtons and points $34.0^{\circ}$ south of east, while force $\overrightarrow{\mathbf{F}}_{\mathbf{B}}$ has a magnitude of 3160 newtons and points due south. Using the component method, find the magnitude and direction of the resultant force $\overrightarrow{\mathbf{F}}_{\mathrm{A}}+\overrightarrow{\mathbf{F}}_{\mathrm{B}}$ that is applied to the stump. Specify the direction with respect to due east.

Vishal Gupta

Numerade Educator

Problem 48

A baby elephant is stuck in a mud hole. To help pull it out, game keepers use a rope to apply a force $\overrightarrow{\mathbf{F}}_{\mathbf{A}},$ as part $a$ of the drawing shows. By itself, however, force $\overrightarrow{\mathbf{F}}_{\mathbf{A}}$ is insufficient. Therefore, two additional forces $\overrightarrow{\mathbf{F}}_{\mathbf{B}}$ and $\overrightarrow{\mathbf{F}}_{\mathbf{C}}$ are applied, as in part $b$ of the drawing. Each of these additional forces has the same magnitude $F$. The magnitude of the resultant force acting on the elephant in part $b$ of the drawing is $k$ times larger than that in part $a$. Find the ratio $F / F_{\mathrm{A}}$ when $k=2.00.$

Vishal Gupta

Numerade Educator

Problem 49

Displacement vector $\overrightarrow{\mathbf{A}}$ points due east and has a magnitude of $2.00 \mathrm{km} .$ Displacement vector $\overrightarrow{\mathbf{B}}$ points due north and has a magnitude of 3.75 km. Displacement vector $\overrightarrow{\mathbf{C}}$ points due west and has a magnitude of $2.50 \mathrm{km} .$ Displacement vector $\overrightarrow{\mathbf{D}}$ points due south and has a magnitude of $3.00 \mathrm{km} .$ Find the magnitude and direction (relative to due west) of the resultant vector $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}+\overrightarrow{\mathbf{D}}$.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 50

Multiple-Concept Example 9 provides background pertinent to this problem. The magnitudes of the four displacement vectors shown in the drawing are $A=16.0 \mathrm{m}, B=11.0 \mathrm{m}, C=12.0 \mathrm{m},$ and $D=26.0 \mathrm{m} .$ Determine the magnitude and directional angle for the resultant that occurs when these vectors are added together.

Sheh Lit Chang

University of Washington

Problem 51

Available on WileyPLUS.

Manish Jain

Numerade Educator

Problem 52

Two geological field teams are working in a remote area. A global positioning system (GPS) tracker at their base camp shows the location of the first team as $38 \mathrm{km}$ away, $19^{\circ}$ north of west, and the second team as 29 km away, $35^{\circ}$ east of north. When the first team uses its GPS to check the position of the second team, what does the GPS give for the second team's (a) distance from them and (b) direction, measured from due east?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 53

A sailboat race course consists of four legs, defined by the displacement vectors $\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}}, \overrightarrow{\mathbf{C}},$ and $\overrightarrow{\mathbf{D}},$ as the drawing indicates. The magnitudes of the first three vectors are $A=$ $3.20 \mathrm{km}, B=5.10 \mathrm{km},$ and $C=4.80 \mathrm{km}$ The finish line of the course coincides with the starting line. Using the data in the drawing, find the distance of the fourth leg and the angle $\theta$.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 54

Multiple-Concept Example 9 deals with the concepts that are important in this problem. A grasshopper makes four jumps. The displacement vectors are (1) $27.0 \mathrm{cm},$ due west; $(2) 23.0 \mathrm{cm}, 35.0^{\circ}$ south of west; (3) $28.0 \mathrm{cm}$ $55.0^{\circ}$ south of east; and $(4) 35.0 \mathrm{cm}, 63.0^{\circ}$ north of east. Find the magnitude and direction of the resultant displacement. Express the direction with respect to due west.

Vishal Gupta

Numerade Educator

Problem 55

Available on WilevPLUS.

Manish Jain

Numerade Educator

Problem 56

The route followed by a hiker consists of three displacement vectors $\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}},$ and $\overrightarrow{\mathbf{C}}$. Vector $\overrightarrow{\mathbf{A}}$ is along a measured trail and is $1550 \mathrm{m}$ in a direction $25.0^{\circ}$ north of east. Vector $\overrightarrow{\mathbf{B}}$ is not along a measured trail, but the hiker uses a compass and knows that the direction is $41.0^{\circ}$ east of south. Similarly, the direction of vector $\overrightarrow{\mathbf{C}}$ is $35.0^{\circ}$ north of west. The hiker ends up back where she started. Therefore, it follows that the resultant displacement is zero, or $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}=\mathbf{0} .$ Find the magnitudes of (a) vector $\overrightarrow{\mathbf{B}}$ and (b) vector $\overrightarrow{\mathbf{C}}$.

Anand Jangid

Numerade Educator

Problem 57

Available on WileyPLUS.

Manish Jain

Numerade Educator

Problem 58

A monkey is chained to a stake in the ground. The stake is $3.00 \mathrm{m}$ from a vertical pole, and the chain is $3.40 \mathrm{m}$ long. How high can the monkey climh mp the male?

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 59

Available on WileyPLUS.

Manish Jain

Numerade Educator

Problem 60

The volume of liquid flowing per second is called the volume flow rate $Q$ and has the dimensions of $[\mathrm{L}]^{3} /[\mathrm{T}] .$ The flow rate of a liquid through a hypodermic needle during an injection can be estimated with the following equation:
$$Q=\frac{\pi R^{n}\left(P_{2}-P_{1}\right)}{8 \eta L}$$
The length and radius of the needle are $L$ and $R,$ respectively, both of which have the dimension [L]. The pressures at opposite ends of the needle are $P_{2}$ and $P_{1},$ both of which have the dimensions of $[\mathrm{M}] /\left\{[\mathrm{L}][\mathrm{T}]^{2}\right\} .$ The symbol $\eta$ represents the viscosity of the liquid and has the dimensions of $[\mathrm{M}] /\{[\mathrm{L}][\mathrm{T}]\} .$ The symbol $\pi$ stands for pi and, like the number 8 and the exponent $n,$ has no dimensions. Using dimensional analysis, determine the value of $n$ in the expression for $Q$.

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 61

An ocean liner leaves New York City and travels $18.0^{\circ}$ north of east for $155 \mathrm{km} .$ How far east and how far north has it gone? In other words, what are the magnitudes of the components of the ship's displacement vector in the directions (a) due east and (b) due north?

Vishal Gupta

Numerade Educator

Problem 62

A pilot flies her route in two straight-line segments. The displacement vector $\overrightarrow{\mathbf{A}}$ for the first segment has a magnitude of $244 \mathrm{km}$ and a direction $30.0^{\circ}$ north of east. The displacement vector $\overrightarrow{\mathbf{B}}$ for the second segment has a magnitude of $175 \mathrm{km}$ and a direction due west. The resultant displacement vector is $\overrightarrow{\mathbf{R}}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ and makes an angle $\theta$ with the direction due east. Using the component method, find the magnitude of $\overrightarrow{\mathbf{R}}$ and the directional angle $\theta$.

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 63

Available on WileyPLUS.

Manish Jain

Numerade Educator

Problem 64

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Manish Jain

Numerade Educator

Problem 65

Vector $\overrightarrow{\mathbf{A}}$ has a magnitude of 6.00 units and points due east. Vector $\overrightarrow{\mathbf{B}}$ points due north. (a) What is the magnitude of $\overrightarrow{\mathbf{B}},$ if the vector $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ points $60.0^{\circ}$ north of east? (b) Find the magnitude of $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$.

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 66

Three forces act on an object, as indicated in the drawing. Force $\overrightarrow{\mathbf{F}}_{1}$ has a magnitude of 21.0 newtons $(21.0 \mathrm{N})$ and is directed $30.0^{\circ}$ to the left of the $+y$ axis. Force $\overrightarrow{\mathbf{F}}_{2}$ has a magnitude of $15.0 \mathrm{N}$ and points along the $+x$ axis. What must be the magnitude and direction (specified by the angle $\theta$ in the drawing ) of the third force $\overrightarrow{\mathbf{F}}_{3}$ such that the vector sum of the three forces is $0 \mathrm{N}$ ?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 67

Available on WileyPLUS.

Manish Jain

Numerade Educator

Problem 68

friend lives in the other building. The two of you are having a discussion about the heights of the buildings, and your friend claims that the height of his building is more than 1.50 times the height of yours. To resolve the issue you climb to the roof of your building and estimate that your line of sight to the top edge of the other building makes an angle of $21^{\circ}$ above the horizontal, whereas your line of sight to the base of the other building makes an angle of $52^{\circ}$ below the horizontal. Determine the ratio of the height of the taller building to the height of the shorter building. State whether your friend is right or wrong.

Alexandra Brown

Alexandra Brown

Numerade Educator

Problem 69

Available on WileyPLUS.

Manish Jain

Numerade Educator

Problem 70

The figure shows two displacement vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$. Vector $\overrightarrow{\mathbf{A}}$ points at an angle of $22^{\circ}$ above the $x$ axis and has an unknown magnitude. Vector $\overrightarrow{\mathbf{B}}$ has an $x$ component $B_{x}=35.0 \mathrm{m}$ and has an unknown $y$ component $B_{y} .$ These two vectors are equal. Concepts: (i) What does the condition that vector $\overrightarrow{\mathbf{A}}$ equals $\overrightarrow{\mathbf{B}}$ imply about the magnitudes and directions of the vectors? (ii) What does the condition that vector $\overrightarrow{\mathbf{A}}$ equals $\overrightarrow{\mathbf{B}}$ imply about the $x$ and $y$ components of the vectors? Calculations: Find the magnitude of $\overrightarrow{\mathbf{A}}$ and the value of $B_{\gamma}$.

Supratim Pal

Numerade Educator

Problem 71

The figure shows three displacement vectors $\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}},$ and $\overrightarrow{\mathbf{C}}$. These vectors are arranged in tail-to-head fashion, and they add together to give a resultant displacement $\overrightarrow{\mathbf{R}}$, which lies along the $x$ axis. Note that the vector $\overrightarrow{\mathbf{B}}$ is parallel to the $x$ axis. Concepts: (i) How is the magnitude of $\overrightarrow{\mathbf{A}}$ related to its scalar components $A_{x}$ and $A_{y} ?$ (ii) Do any of the vectors in the figure have a zero value for either their $x$ or $y$ components? (If so, which ones?) (iii) What does the fact that $\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}},$ and $\overrightarrow{\mathbf{C}}$ add together to give $\overrightarrow{\mathbf{R}}$ tell you about the components of these vectors? Calculations: What is the magnitude of the vector $\overrightarrow{\mathbf{A}}$ and its directional angle $\theta ?$

Ravindra Yadav

Numerade Educator

Problem 72

You and your team are exploring a river in South America when you come to the bottom of a tall waterfall. You estimate the cliff over which the water flows to be about 100 feet tall. You have to choose between climbing the cliff or backtracking and taking another route, but climbing the cliff would cut two hours off of your trip. There is only one experienced climber in the group: she would climb the cliff alone and drop a rope over the edge to lift supplies and allow the others to climb without packs. The climber estimates it will take her 45 minutes to get to the top. However, you are concerned that the rope might be too short to reach the bottom of the cliff (it is exactly $30.0 \mathrm{m}$ long $) .$ If it is too short, she'll have to climb back down (another 45 minutes) and you will be too far behind schedule to get to your destination before dark. As you contemplate how to determine whether the rope is long enough, you notice that the late afternoon shadow of the cliff grows as the sun descends over its edge. You suddenly remember your trigonometry. You measure the length of the shadow from the base of the cliff to the shadow's edge ($144 \mathrm{ft}$), and the angle subtended between the base and top of the cliff measured from the shadow's edge. The angle is $38.1^{\circ} .$ Do you send the climber, or start backtracking to take another route?

Manish Jain

Numerade Educator

Problem 73

Your South American expedition splits into two groups: one that stays at home base, and yours that goes off to set up a sensor that will monitor precipitation, temperature, and sunlight through the upcoming winter. The sensor must link up to a central communications system at base camp that simultaneously uploads the data from numerous sensors to a satellite. In order to set up and calibrate the sensor, you will have to communicate with base camp to give them specific location information. Unfortunately, the group's communication and navigation equipment has dwindled to walkie-talkies and a compass due to a river-raft mishap, which means your group must not exceed the range of the walkie-talkies ( 3.0 miles). However, you do have a laser rangefinder to help you measure distances as you navigate with the compass. After a few hours of hiking, you find the perfect plateau on which to mount the sensor. You have carefully mapped your path from base camp around lakes and other obstacles: $550 \mathrm{m}$ West $(\mathrm{W}), 275 \mathrm{m} \mathrm{S}, 750 \mathrm{m} \mathrm{W}$ $900 \mathrm{m} \mathrm{NE}, 800 \mathrm{m} \mathrm{W},$ and $400 \mathrm{m} 30.0^{\circ} \mathrm{W}$ of $\mathrm{S} .$ The final leg is due south, $2.20 \mathrm{km}$ up a constant slope and ending at a plateau that is $320 \mathrm{m}$ above the level of base camp. (a) How far are you from base camp? Will you be able to communicate with home base using the walkie-talkies? (b) What is the geographical direction from base camp to the sensor (expressed in the form $\theta^{\circ}$ south of west, etc.)? (c) What is the angle of inclination from base camp to the detector?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator