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University Physics with Modern Physics In SI Units

Hugh D Young; Roger A Freedman

Chapter 1

Units, Physical Quantities, and Vectors - all with Video Answers

Educators

+ 7 more educators

Chapter Questions

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Problem 1

How many years older will you be 1.00 gigasecond from now?

David Morris
David Morris
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Problem 2

You read in a brochure that Kensington Gardens, one of London's eight Royal Parks, covers an area of 265 acres. An acre is a British unit widely used in land measurement in the UK (1 acre $=43,560 \mathrm{ft}^{2}$ and $1 \mathrm{ft}=0.3048 \mathrm{~m}$, see Appendix $\mathrm{C}$ ). What is the area of Kensington Gardens in square meters? in square kilometers? in hectares (a hectare is a unit of area equal to $10,000 \mathrm{~m}^{2}$ used in land measurement)?

David Morris
David Morris
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Problem 3

How many nanoseconds does it take light to travel $0.3 \mathrm{~m}$ in vacuum?

Ankur S
Ankur S
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Problem 4

The density of silver is $10.5 \mathrm{~g} / \mathrm{cm}^{3}$. What is this value in kilograms per cubic meter?

David Morris
David Morris
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Problem 5

In the ancient Roman Empire, large distances were measured in miliarum. A miliarum was subdivided into 8 stadia, 1 stadium into 125 passus, 1 passus into 5 pes, 1 pes into 4 palmus, 1 palmus into 4 digitus. Knowing that 1 pes $=0.296 \mathrm{~m}$, find the number of (a) meters in 1.00 miliarum and (b) centimeters in 1.00 digitus.

David Morris
David Morris
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04:10

Problem 6

The speed limits in Ireland were changed from imperial (miles per hour, $\mathrm{mi} / \mathrm{h}$ ) to metric (kilometers per hour, $\mathrm{km} / \mathrm{h}$ ) on 20 January 2005. Use $1 \mathrm{mi}=5280 \mathrm{ft}$ and $1 \mathrm{ft}=30.48 \mathrm{~cm}$ to: (a) convert the for-
mer $30 \mathrm{mi} / \mathrm{h}$ speed limit for build-up areas to units of $\mathrm{km} / \mathrm{h}$ and compare this result with the new $50 \mathrm{~km} / \mathrm{h}$ speed limit; $(\mathrm{b})$ convert the new $120 \mathrm{~km} / \mathrm{h}$ speed limit for motorways to units of $\mathrm{mi} / \mathrm{h}$ and compare this result with the old $70 \mathrm{mi} / \mathrm{h}$ speed limit.

Vishal Gupta
Vishal Gupta
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Problem 7

Fuel consumption of a car is the amount of fuel used per unit distance. It is customary to give fuel consumption in liters per $100 \mathrm{~km}$ (L/100 km). For example, the 2019 Austin Mini Cooper has a fuel consumption of $5.5 \mathrm{~L} / 100 \mathrm{~km},$ that is, it uses 5.5 liters of petrol per every $100 \mathrm{~km}$ driven. (a) If this car's petrol tank holds $40 \mathrm{~L}$, how many tanks of petrol will you use to drive $1200 \mathrm{~km} ?$ (b) You read in an English car magazine that the classic 1964 Austin Mini Cooper has an average estimated "mileage of 30.5 miles per gallon" (where 1 mile $=1.609 \mathrm{~km}$ is a British unit of distance and 1 gallon $(\mathrm{UK})=4.546 \mathrm{~L}$ is a British unit of capacity, see Appendix $\mathrm{C}$ ). Find the fuel consumption of this car and compare it with the 2019 model.

David Morris
David Morris
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Problem 8

(a) The recommended daily allowance (RDA) of the trace metal magnesium is $410 \mathrm{mg} /$ day for males. Express this quantity in $\mu \mathrm{g} /$ day. (b) For adults, the RDA of the amino acid lysine is $12 \mathrm{mg}$ per $\mathrm{kg}$ of body weight. How many grams per day should a $77-\mathrm{kg}$ adult receive? (c) A typical multivitamin tablet can contain $2.0 \mathrm{mg}$ of vitamin $\mathrm{B}_{2}$ (riboflavin), and the RDA is $0.0030 \mathrm{~g} /$ day. How many such tablets should a person take each day to get the proper amount of this vitamin, if he gets none from other sources?
(d) The RDA for the trace element selenium is $0.000070 \mathrm{~g} /$ day. Express this dose in $\mathrm{mg} /$ day.

David Morris
David Morris
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03:05

Problem 9

Neptunium. In the fall of 2002 , scientists at Los Alamos National Laboratory determined that the critical mass of neptunium- 237 is about $60 \mathrm{~kg}$. The critical mass of a fissionable material is the minimum amount that must be brought together to start a nuclear chain reaction. Neptunium- 237 has a density of $19.5 \mathrm{~g} / \mathrm{cm}^{3}$. What would be the radius of a sphere of this material that has a critical mass?

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
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Problem 10

Bacteria. Bacteria vary in size, but a diameter of $2.0 \mu \mathrm{m}$ is not unusual. What are the volume (in cubic centimeters) and surface area (in square millimeters) of a spherical bacterium of that size? (Consult Appendix D for relevant formulas.)

David Morris
David Morris
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06:20

Problem 11

With a wooden ruler, you measure the length of a rectangular piece of sheet metal to be $12 \mathrm{~mm}$. With micrometer calipers, you measure the width of the rectangle to be $5.98 \mathrm{~mm}$. Use the correct number of significant figures: What are (a) the area of the rectangle; (b) the ratio of the rectangle's width to its length; (c) the perimeter of the rectangle;
(d) the difference between the length and the width; and (e) the ratio of the length to the width?

EO
Everardo Olide
Numerade Educator
02:28

Problem 12

The volume of a solid cylinder is given by $V=\pi r^{2} h,$ where $r$ is the radius and $h$ is the height. You measure the radius and height of a thin cylindrical wire and obtain the results $r=0.036 \mathrm{~cm}$ and $h=12.1 \mathrm{~cm} .$ What do your measurements give for the volume of the wire in $\mathrm{mm}^{3}$ ? Use the correct number of significant figures in your answer.

Nishant Kumar
Nishant Kumar
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02:57

Problem 13

A useful and easy-to-remember approximate value for the number of seconds in a year is $\pi \times 10^{7}$. Determine the percent error in this approximate value. (There are 365.24 days in one year.)

Zachary Warner
Zachary Warner
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05:11

Problem 14

Express each approximation of $\pi$ to six significant figures: (a) $22 / 7$
and (b) $355 / 113$. (c) Are these approximations accurate to that precision?

EO
Everardo Olide
Numerade Educator
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Problem 15

Which of the values given below are plausible for an average adult male giraffe: (a) height of $2 \mathrm{~m}$ or $5 \mathrm{~m}$ ? (b) mass of $1500 \mathrm{~kg}$ or 5000 $\mathrm{kg}$ ? (c) maximal speed of $15 \mathrm{~m} / \mathrm{s}$ or $45 \mathrm{~m} / \mathrm{s} ?$

David Morris
David Morris
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Problem 16

How many liters of petrol are used in Italy in one day? Assume that there are two cars for every three people, that each car is driven an average of 10,000 kilometers per year, and that the petrol consumption of an average car is 8 liters per 100 kilometers. Assume that the population of Italy is approximately 60 million.

David Morris
David Morris
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02:56

Problem 17

In Wagner's opera Das Rheingold, the goddess Freia is ransomed for a pile of gold just tall enough and wide enough to hide her from sight. Estimate the monetary value of this pile. The density of gold is $19.3 \mathrm{~g} / \mathrm{cm}^{3}$, and take its value to be about $$\$ 40$$ per gram.

Sachin Rao
Sachin Rao
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09:47

Problem 18

Four astronauts are in a spherical space station. (a) If, as is typical, each of them breathes about $500 \mathrm{~cm}^{3}$ of air with each breath, approximately what volume of air (in cubic meters) do these astronauts breathe in a year? (b) What would the diameter (in meters) of the space station have to be to contain all this air?

EO
Everardo Olide
Numerade Educator
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Problem 19

You are using water to dilute small amounts of chemicals in the laboratory, drop by drop. How many drops of water are in a $1.0 \mathrm{~L}$ bottle?

David Morris
David Morris
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Problem 20

How many times does a human heart beat during a person's lifetime? How many liters of blood does it pump? (Estimate that the heart pumps $50 \mathrm{~cm}^{3}$ of blood with each beat and assume a 365 -day year.)

David Morris
David Morris
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Problem 21

A postal employee drives a delivery truck along the route shown in Fig. E1.21. Determine the magnitude and direction of the resultant displacement by drawing a scale diagram.

David Morris
David Morris
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05:39

Problem 22

For the vectors $\vec{A}$ and $\vec{B}$ in Fig. E1.22, use a scale drawing to find the magnitude and direction of (a) the vector $\operatorname{sum} \overrightarrow{\boldsymbol{A}}+\overrightarrow{\boldsymbol{B}}$ and $(\mathrm{b})$
the vector difference $\vec{A}-\vec{B}$. Use your answers to find the magnitude and direction of $(\mathrm{c})-\vec{A}-\vec{B}$ and
(d) $\overrightarrow{\boldsymbol{B}}-\overrightarrow{\boldsymbol{A}}$. (See also Exercise 1.29 for a different approach.)

Supratim Pal
Supratim Pal
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Problem 23

A spelunker is surveying a cave. She follows a passage 180 $\mathrm{m}$ straight west, then $210 \mathrm{~m}$ in a direction $45^{\circ}$ east of south, and then $280 \mathrm{~m}$ at $30^{\circ}$ east of north. After a fourth displacement, she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the fourth displacement.

David Morris
David Morris
Numerade Educator
02:43

Problem 24

Let $\theta$ be the angle that the vector $\vec{A}$ makes with the $+x$ axis, measured counterclockwise from that axis. Find angle $\theta$ for a vector that has these components:
(a) $A_{x}=2.00 \mathrm{~m}, A_{y}=-1.00 \mathrm{~m} ;$
(b) $A_{x}=2.00 \mathrm{~m}, A_{y}=1.00 \mathrm{~m} ;$
(c) $A_{x}=-2.00 \mathrm{~m}, A_{y}=1.00 \mathrm{~m} ;$
(d) $A_{x}=-2.00 \mathrm{~m}, A_{y}=-1.00 \mathrm{~m}$

Derek Walkama
Derek Walkama
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02:21

Problem 25

Compute the $x$ - and $y$ -components of the vectors $\overrightarrow{\boldsymbol{A}}, \overrightarrow{\boldsymbol{B}}, \overrightarrow{\boldsymbol{C}},$ and $\overrightarrow{\boldsymbol{D}}$ in Fig. $\mathrm{E} 1.22$.

Khaled Yasein
Khaled Yasein
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Problem 26

Vector $\overrightarrow{\boldsymbol{A}}$ is in the direction $40.0^{\circ}$ clockwise from the $-y$ -axis. The $x$ -component of $\vec{A}$ is $A_{x}=-20.0 \mathrm{~m}$. (a) What is the $y$ -component of $\vec{A}$ ? (b) What is the magnitude of $\vec{A}$ ?

David Morris
David Morris
Numerade Educator
02:17

Problem 27

Vector $\overrightarrow{\boldsymbol{A}}$ has $y$ -component $A_{y}=+13.0 \mathrm{~m} . \overrightarrow{\boldsymbol{A}}$ makes an angle of $32.0^{\circ}$ counterclockwise from the $+y$ -axis. (a) What is the $x$ -component of $\vec{A} ?$ (b) What is the magnitude of $\vec{A}$ ?

Nishant Kumar
Nishant Kumar
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Problem 28

A postal employee drives a delivery truck over the route shown in Fig. E1.21. Use the method of components to determine the magnitude and direction of her resultant displacement. In a vector addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:41

Problem 29

For the vectors $\overrightarrow{\boldsymbol{A}}$ and $\overrightarrow{\boldsymbol{B}}$ in Fig. E1.22, use the method of components to find the magnitude and direction of (a) the vector sum $\vec{A}+\vec{B} ;(b)$ the vector $\operatorname{sum} \overrightarrow{\boldsymbol{B}}+\overrightarrow{\boldsymbol{A}} ;(\mathrm{c})$ the vector difference $\overrightarrow{\boldsymbol{A}}-\overrightarrow{\boldsymbol{B}}$
(d) the vector difference $\overrightarrow{\boldsymbol{B}}-\overrightarrow{\boldsymbol{A}}$.

Supratim Pal
Supratim Pal
Numerade Educator
09:16

Problem 30

Find the magnitude and direction of the vector represented by the following pairs of components:
(a) $A_{x}=-8.60 \mathrm{~cm}, A_{y}=5.20 \mathrm{~cm} ;$
(b) $A_{x}=-9.70 \mathrm{~m}, A_{y}=-2.45 \mathrm{~m} ;$ (c) $A_{x}=7.75 \mathrm{~km}, A_{y}=-2.70 \mathrm{~km} .$

Zachary Warner
Zachary Warner
Numerade Educator
03:34

Problem 31

A disoriented physics professor drives $3.25 \mathrm{~km}$ north, then $2.20 \mathrm{~km}$ west, and then $1.50 \mathrm{~km}$ south. Find the magnitude and direction of the resultant displacement, using the method of components. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
03:57

Problem 32

Vector $\overrightarrow{\boldsymbol{A}}$ has magnitude $8.00 \mathrm{~m}$ and is in the $x y$ -plane at an angle of $127^{\circ}$ counterclockwise from the $+x$ -axis $\left(37^{\circ}\right.$ past the $+y$ -axis). What are the magnitude and direction of vector $\overrightarrow{\boldsymbol{B}}$ if the $\operatorname{sum} \overrightarrow{\boldsymbol{A}}+\overrightarrow{\boldsymbol{B}}$ is in the $-y$ -direction and has magnitude $12.0 \mathrm{~m}$ ?

Supratim Pal
Supratim Pal
Numerade Educator
03:13

Problem 33

Vector $\vec{A}$ is $2.80 \mathrm{~cm}$ long and is $60.0^{\circ}$ above the $x$ -axis in the first quadrant. Vector $\overrightarrow{\boldsymbol{B}}$ is $1.90 \mathrm{~cm}$ long and is $60.0^{\circ}$ below the $x$ -axis in the fourth quadrant (Fig. E1.33). Use components to find the magnitude and direction of
(a) $\vec{A}+\vec{B} ;$ (b) $\vec{A}-\vec{B} ;(c) \vec{B}-\vec{A} .$ In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your sketch.

Supratim Pal
Supratim Pal
Numerade Educator
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Problem 34

In each case, find the $x$ - and $y$ -components of vector $\vec{A}$ :
(a) $\overrightarrow{\boldsymbol{A}}=3.20 \hat{\imath}-6.50 \hat{\jmath}$;
(b) $\overrightarrow{\boldsymbol{A}}=18.2 \hat{\boldsymbol{j}}-7.91 \hat{\imath}$
(c) $\vec{A}=-12.0 \hat{\imath}+$
$21.2 \hat{\jmath} ;$ (d) $\overrightarrow{\boldsymbol{A}}=5.0 \overrightarrow{\boldsymbol{B}},$ where $\overrightarrow{\boldsymbol{B}}=8 \hat{\imath}-4 \hat{\jmath}$

David Morris
David Morris
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05:25

Problem 35

Write each vector in Fig. E1.22 in terms of the unit vectors $\hat{\imath}$ and $\hat{\jmath}$.

Vishal Gupta
Vishal Gupta
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01:14

Problem 36

Given two vectors $\overrightarrow{\boldsymbol{A}}=4.00 \hat{\imath}+7.00 \hat{\jmath}$ and $\overrightarrow{\boldsymbol{B}}=5.00 \hat{\imath}-$
$2.00 \hat{\jmath},$ (a) find the magnitude of each vector; (b) use unit vectors to write an expression for the vector difference $\overrightarrow{\boldsymbol{A}}-\overrightarrow{\boldsymbol{B}} ;$ and $(\mathrm{c})$ find the magnitude and direction of the vector difference $\overrightarrow{\boldsymbol{A}}-\overrightarrow{\boldsymbol{B}}$. (d) In a vector diagram show $\overrightarrow{\boldsymbol{A}}, \overrightarrow{\boldsymbol{B}},$ and $\overrightarrow{\boldsymbol{A}}-\overrightarrow{\boldsymbol{B}},$ and show that your diagram agrees qualitatively with your answer to part (c).

Dominador Tan
Dominador Tan
Numerade Educator
05:17

Problem 37

(a) Write each vector in Fig. $\mathrm{E} 1.37$ in terms of the unit vectors $\hat{\imath}$ and $\hat{j}$. (b) Use unit vectors to express vector $\overrightarrow{\boldsymbol{C}},$ where $\overrightarrow{\boldsymbol{C}}=3.00 \overrightarrow{\mathrm{A}}-4.00 \overrightarrow{\boldsymbol{B}} .$ (c) Find the
magnitude and direction of $\overrightarrow{\boldsymbol{C}}$.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:05

Problem 38

You are given two vectors $\vec{A}=-3.00 \hat{\imath}+6.00 \hat{\jmath} \quad$ and
$\vec{B}=7.00 \hat{\imath}+2.00 \hat{\jmath} . \quad$ Let $\quad$ coun-
terclockwise angles be positive.
(a) What angle does $A$ make with the $+x$ -axis? (b) What angle does $\overrightarrow{\boldsymbol{B}}$ make with the $+x$ -axis?
(c) Vector $\overrightarrow{\boldsymbol{C}}$ is the sum of $\vec{A}$ and $\vec{B}$, so $\overrightarrow{\boldsymbol{C}}=\overrightarrow{\boldsymbol{A}}+\overrightarrow{\boldsymbol{B}}$. What angle does $\boldsymbol{C}$ make with the $+x$ -axis?

Sachin Rao
Sachin Rao
Numerade Educator
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Problem 39

Given two vectors $\overrightarrow{\boldsymbol{A}}=-2.00 \hat{\imath}+3.00 \hat{\jmath}+4.00 \hat{k} \quad$ and
$\overrightarrow{\boldsymbol{B}}=3.00 \hat{\imath}+1.00 \hat{\jmath}-3.00 \hat{k},$ (a) find the magnitude of each vector;
(b) use unit vectors to write an expression for the vector difference $\vec{A}-\vec{B} ;$ and $(c)$ find the magnitude of the vector difference $\vec{A}-\vec{B} .$ Is this the same as the magnitude of $\overrightarrow{\boldsymbol{B}}-\overrightarrow{\boldsymbol{A}}$ ? Explain.

David Morris
David Morris
Numerade Educator
01:44

Problem 40

(a) Find the scalar product of the vectors $\vec{A}$ and $\vec{B}$ given in Exercise $1.36 .(\mathrm{b})$ Find the angle between these two vectors.

Derek Walkama
Derek Walkama
Numerade Educator
06:20

Problem 41

For the vectors $\vec{A}, \boldsymbol{B},$ and $\boldsymbol{C}$ in Fig. $\mathrm{E} 1.22,$ find the scalar products (a) $\vec{A} \cdot \vec{B} ;(b) \vec{B} \cdot \vec{C} ;(c) \vec{A} \cdot \vec{C}$.

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
01:10

Problem 42

Find the vector product $\vec{A} \times \vec{B}$ (expressed in unit vectors) of the two vectors given in Exercise $1.36 .$ What is the magnitude of the vector product?

Derek Walkama
Derek Walkama
Numerade Educator
03:25

Problem 43

Find the angle between each of these pairs of vectors:
(a) $\overrightarrow{\boldsymbol{A}}=-2.00 \hat{\imath}+6.00 \hat{\jmath} \quad$ and $\quad \overrightarrow{\boldsymbol{B}}=2.00 \hat{\imath}-3.00 \hat{\jmath}$
(b) $\overrightarrow{\boldsymbol{A}}=3.00 \hat{\imath}+5.00 \hat{\jmath} \quad$ and $\quad \overrightarrow{\boldsymbol{B}}=10.00 \hat{\imath}+6.00 \hat{\jmath}$
(c) $\overrightarrow{\boldsymbol{A}}=-4.00 \hat{\imath}+2.00 \hat{\jmath}$ and $\overrightarrow{\boldsymbol{B}}=7.00 \hat{\imath}+14.00 \hat{\jmath}$

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
03:03

Problem 44

For the two vectors in Fig. E1.33, find the magnitude and direction of (a) the vector product $\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}} ;$ (b) the vector product $\overrightarrow{\boldsymbol{B}} \times \overrightarrow{\boldsymbol{A}}$.

Derek Walkama
Derek Walkama
Numerade Educator
03:01

Problem 45

For the two vectors $\overrightarrow{\boldsymbol{A}}$ and $\overrightarrow{\boldsymbol{D}}$ in Fig. E1.22, find the magnitude and direction of (a) the vector product $\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{D}} ;$ (b) the vector product $\vec{D} \times \vec{A}$.

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
03:26

Problem 46

For the two vectors $\overrightarrow{\boldsymbol{A}}$ and $\overrightarrow{\boldsymbol{B}}$ in Fig. $\mathrm{E} 1.37$, find (a) the scalar product $\overrightarrow{\boldsymbol{A}} \cdot \overrightarrow{\boldsymbol{B}} ;$ (b) the magnitude and direction of the vector product $\vec{A} \times \vec{B}$.

Zachary Warner
Zachary Warner
Numerade Educator
03:04

Problem 47

The vector product of vectors $\vec{A}$ and $\overrightarrow{\boldsymbol{B}}$ has magnitude $16.0 \mathrm{~m}^{2}$ and is in the $+z$ -direction. If vector $\vec{A}$ has magnitude $8.0 \mathrm{~m}$ and is in the $-x$ -direction, what are the magnitude and direction of vector $\overrightarrow{\boldsymbol{B}}$ if it has no $x$ -component?

Sachin Rao
Sachin Rao
Numerade Educator
02:59

Problem 48

The angle between two vectors is $\theta$. (a) If $\theta=30.0^{\circ}$, which has the greater magnitude: the scalar product or the vector product of the two vectors? (b) For what value (or values) of $\theta$ are the magnitudes of the scalar product and the vector product equal?

Sachin Rao
Sachin Rao
Numerade Educator
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Problem 49

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

David Morris
David Morris
Numerade Educator
02:42

Problem 50

A maser is a laser-type device that produces electromagnetic waves with frequencies in the microwave and radio-wave bands of the electromagnetic spectrum. You can use the radio waves generated by a hydrogen maser as a standard of frequency. The frequency of these waves is 1,420,405,751.786 hertz. (A hertz is another name for one cycle per second.) A clock controlled by a hydrogen maser is off by only $1 \mathrm{~s}$ in 100,000 years. For the following questions, use only three significant figures. (The large number of significant figures given for the frequency simply illustrates the remarkable accuracy to which it has been measured.) (a) What is the time for one cycle of the radio wave? (b) How many cycles occur in $1 \mathrm{~h} ?$ (c) How many cycles would have occurred during the age of the earth, which is estimated to be $4.6 \times 10^{9}$ years? (d) By how many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth?

Derek Walkama
Derek Walkama
Numerade Educator
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Problem 51

An Earthlike Planet. In January 2006 astronomers reported the discovery of a planet, comparable in size to the earth, orbiting another star and having a mass about 5.5 times the earth's mass. It is believed to consist of a mixture of rock and ice, similar to Neptune. If this planet has the same density as Neptune $\left(1.76 \mathrm{~g} / \mathrm{cm}^{3}\right),$ what is its radius expressed (a) in kilometers and (b) as a multiple of earth's radius? Consult the back of the book for astronomical data.

David Morris
David Morris
Numerade Educator
01:29

Problem 52

A rectangular piece of aluminum is $7.60 \pm 0.01 \mathrm{~cm}$ long and $1.50 \pm 0.01 \mathrm{~cm}$ wide. (a) Find the area of the rectangle and the uncertainty in the area.
(b) Verify that the fractional uncertainty in the area is equal to the sum of the fractional uncertainties in the length and in the width. (This is a general result.)

Dominador Tan
Dominador Tan
Numerade Educator
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Problem 53

Estimate the number of atoms in your body. (Hint: Based on what you know about biology and chemistry, what are the most common types of atom in your body? What is the mass of each type of atom? Appendix $\mathrm{F}$ gives the atomic masses of different elements, measured in atomic mass units; you can find the value of an atomic mass unit, or $1 \mathrm{u}$, in Appendix $\mathrm{B}$.)

Lainey Roebuck
Lainey Roebuck
Numerade Educator
04:35

Problem 54

Biological tissues are typically made up of $98 \%$ water. Given that the density of water is $1.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$, estimate the mass of (a) the
heart of an adult human; (b) a cell with a diameter of $0.5 \mu \mathrm{m} ;(\mathrm{c})$ a honeybee.

Bruce Edelman
Bruce Edelman
Numerade Educator
03:00

Problem 55

Vector $\overrightarrow{\boldsymbol{A}}=3.0 \hat{\imath}-4.0 \hat{\boldsymbol{k}} .$ (a) Construct a unit vector that is parallel to $\vec{A}$. (b) Construct a unit vector that is antiparallel to $\vec{A}$.
(c) Construct two unit vectors that are perpendicular to $\vec{A}$ and that have no $y$ -component.

Sachin Rao
Sachin Rao
Numerade Educator
05:43

Problem 56

Three horizontal ropes pull on a large stone stuck in the ground, producing the vector forces $\overrightarrow{\boldsymbol{A}}, \overrightarrow{\boldsymbol{B}},$ and $\overrightarrow{\boldsymbol{C}}$ shown in Fig. $\mathbf{P} 1 . \mathbf{5} 6$. Find the magnitude and direction of a fourth force on the stone that will make the vector sum of the four forces zero.

Sachin Rao
Sachin Rao
Numerade Educator
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Problem 57

As noted in Exercise 1.23 , a spelunker is surveying a cave. She follows a passage $177 \mathrm{~m}$ straight west, then $215 \mathrm{~m}$ in a direction $45^{\circ}$ east of south, and then $271 \mathrm{~m}$ at $30^{\circ}$ east of north. After a fourth displacement, she finds herself back where she started. Use the method of components to determine the magnitude and direction of the fourth displacement. Draw the vector-addition diagram and show that it is in qualitative agreement with your numerical solution.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
04:53

Problem 58

A plane leaves the airport in Galisteo and flies $145 \mathrm{~km}$ at $68.0^{\circ}$ east of north; then it changes direction to fly $250 \mathrm{~km}$ at $48.0^{\circ}$ south of east, after which it makes an immediate emergency landing in a pasture. When the airport sends out a rescue crew, in which direction and how far should this crew fly to go directly to this plane?

Aishwarya Krishnakumar
Aishwarya Krishnakumar
Numerade Educator
05:30

Problem 59

A charged object with electric charge $q$ produces an electric field. The SI unit for electric field is $\mathrm{N} / \mathrm{C},$ where $\mathrm{N}$ is the SI unit for force and $\mathrm{C}$ is the SI unit for charge. If at point $P$ there are electric fields from two or more charged objects, then the resultant field is the vector sum of the fields from each object. At point $P$ the electric field $\overrightarrow{\boldsymbol{E}}_{1}$ from charge $q_{1}$ is $450 \mathrm{~N} / \mathrm{C}$ in the $+y$ -direction, and the electric field $\overrightarrow{\boldsymbol{E}}_{2}$ from charge $q_{2}$ is $600 \mathrm{~N} / \mathrm{C}$ in the direction $36.9^{\circ}$ from the $-y$ -axis toward the $-x$ -axis. What are the magnitude and direction of the resultant field $\overrightarrow{\boldsymbol{E}}=\overrightarrow{\boldsymbol{E}}_{1}+\overrightarrow{\boldsymbol{E}}_{2}$ at point $P$ due to these two charges?

Sachin Rao
Sachin Rao
Numerade Educator
01:57

Problem 60

A sailor in a small sailboat encounters shifting winds. She sails $2.00 \mathrm{~km}$ east, next $3.50 \mathrm{~km}$ southeast, and then an additional distance in an unknown direction. Her final position is $5.80 \mathrm{~km}$ directly east of the starting point (Fig. P1.60). Find the magnitude and direction of the third leg of the journey. Draw the vector-addition diagram and show that it is in qualitative agreement with your numerical solution.

Derek Walkama
Derek Walkama
Numerade Educator
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Problem 61

Dislocated Shoulder. A patient with a dislocated shoulder is put into a traction apparatus as shown in Fig. P1.61. The pulls $\vec{A}$ and $\vec{B}$ have equal magnitudes and must combine to produce an outward traction force of $5.52 \mathrm{~N}$ on the patient's arm. How large should these pulls be?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
05:40

Problem 62

On a training flight in Switzerland, a student pilot flies from Bern to Zurich, next to Lugano, and then to Sion (Fig. $\mathbf{P} 1 . \mathbf{6 2}$ ). The directions are shown relative to north: $0^{\circ}$ is north, $90^{\circ}$ is east, $180^{\circ}$ is south, and $270^{\circ}$ is west. Use the method of components to find (a) the distance she has to fly from Sion to get back to Bern, and (b) the direction (relative to north) she must fly to get there. Illustrate your solutions with a vector diagram.

Supratim Pal
Supratim Pal
Numerade Educator
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Problem 63

You leave the airport in Wagga Wagga and fly $23.0 \mathrm{~km}$ in a direction $34.0^{\circ}$ south of east. You then fly $46.0 \mathrm{~km}$ due north. How far and in what direction must you then fly to reach a private landing strip that is $32.0 \mathrm{~km}$ due west of the Wagga Wagga airport?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 64

Getting Back. An explorer in Antarctica leaves his shelter during a whiteout. He takes 43 steps northeast, next 80 steps at $60^{\circ}$ north of west, and then 52 steps due south. Assume all of his steps are equal in length. (a) Sketch, roughly to scale, the three vectors and their resultant. (b) Save the explorer from becoming hopelessly lost by giving him the displacement, calculated by using the method of components, that will return him to his shelter.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:32

Problem 65

As a test of orienteering skills, your physics class holds a contest in a large, open field. Each contestant is told to travel $20.8 \mathrm{~m}$ due north from the starting point, then $38.0 \mathrm{~m}$ due east, and finally $18.0 \mathrm{~m}$ in the direction $33.0^{\circ}$ west of south. After the specified displacements, a contestant will find a silver coin hidden under a rock. The winner is the person who takes the shortest time to reach the location of the silver coin. Remembering what you learned in class, you run on a straight line from the starting point to the hidden coin. How far and in what direction do you run?

Anand Jangid
Anand Jangid
Numerade Educator
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Problem 66

You are standing on a street corner with your friend. You then travel $14.0 \mathrm{~m}$ due west across the street and into your apartment building. You travel in the lift $22.0 \mathrm{~m}$ upward to your floor, walk $12.0 \mathrm{~m}$ north to the door of your apartment, and then walk $6.0 \mathrm{~m}$ due east to your balcony that overlooks the street. Your friend is standing where you left her. Now how far are you from your friend?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 67

You are lost at night in a large, open field. Your GPS tells you that you are $122.0 \mathrm{~m}$ from your car, in a direction $58.0^{\circ}$ east of south. You walk $72.0 \mathrm{~m}$ due west along a ditch. How much farther, and in what direction, must you walk to reach your car?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
04:13

Problem 68

You live in a town where the streets are straight but are in a variety of directions. On Saturday you go from your apartment to the grocery store by driving $0.60 \mathrm{~km}$ due north and then $1.40 \mathrm{~km}$ in the direction $60.0^{\circ}$ west of north. On Sunday you again travel from your apartment to the same store but this time by driving $0.80 \mathrm{~km}$ in the direction $50.0^{\circ}$ north of west and then in a straight line to the store.
(a) How far is the store from your apartment?
(b) On which day do you travel the greater distance, and how much farther do you travel? Or, do you travel the same distance on each route to the store?

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
04:01

Problem 69

While following a treasure map, you start at an old oak tree. You first walk $825 \mathrm{~m}$ directly south, then turn and walk $1.25 \mathrm{~km}$ at $30.0^{\circ}$ west of north, and finally walk $1.00 \mathrm{~km}$ at $32.0^{\circ}$ north of east, where you find the treasure: a biography of Isaac Newton! (a) To return to the old oak tree, in what direction should you head and how far will you walk? Use components to solve this problem. (b) To see whether your calculation in part (a) is reasonable, compare it with a graphical solution drawn roughly to scale.

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
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Problem 70

A fence post is $59.0 \mathrm{~m}$ from where you are standing, in a direction $38.0^{\circ}$ north of east. A second fence post is due south from you. How far are you from the second post if the distance between the two posts is $70.0 \mathrm{~m} ?$

Lainey Roebuck
Lainey Roebuck
Numerade Educator
05:17

Problem 71

A dog in an open field runs $12.0 \mathrm{~m}$ east and then $28.0 \mathrm{~m}$ in a direction $50.0^{\circ}$ west of north. In what direction and how far must the dog then run to end up $10.0 \mathrm{~m}$ south of her original starting point?

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
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Problem 72

Ricardo and Jane are standing under a tree in the middle of a pasture. An argument ensues, and they walk away in different directions. Ricardo walks $30.0 \mathrm{~m}$ in a direction $60.0^{\circ}$ west of north. Jane walks $15.0 \mathrm{~m}$ in a direction $30.0^{\circ}$ south of west. They then stop and turn to face each other. (a) What is the distance between them? (b) In what direction should Ricardo walk to go directly toward Jane?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:12

Problem 73

You are camping with Joe and Karl. Since all three of you like your privacy, you don't pitch your tents close together. Joe's tent is $21.0 \mathrm{~m}$ from yours, in the direction $23.0^{\circ}$ south of east. Karl's tent is $32.0 \mathrm{~m}$ from yours, in the direction $37.0^{\circ}$ north of east. What is the distance between Karl's tent and Joe's tent?

Ronald Prasad
Ronald Prasad
Numerade Educator
01:55

Problem 74

In the methane molecule, $\mathrm{CH}_{4}$, each hydrogen atom is at a corner of a regular tetrahedron with the carbon atom at the center. In coordinates for which one of the $\mathrm{C}-\mathrm{H}$ bonds is in the direction of $\hat{\imath}+\hat{\jmath}+\hat{k},$ an adjacent $\mathrm{C}-\mathrm{H}$
bond is in the $\hat{\imath}-\hat{\jmath}-\hat{k}$ direction. Calculate the angle between these two bonds.

Derek Walkama
Derek Walkama
Numerade Educator
03:01

Problem 75

The work $W$ done by a constant force $\overrightarrow{\boldsymbol{F}}$ on an object that undergoes displacement $\vec{s}$ from point 1 to point 2 is $W=\vec{F} \cdot \vec{s}$. For $F$ in newtons $(\mathrm{N})$ and $s$ in meters $(\mathrm{m}), W$ is in joules $(\mathrm{J}) .$ If, during a displacement of the object, $\overrightarrow{\boldsymbol{F}}$ has constant direction $60.0^{\circ}$ above the $-x$ -axis and constant magnitude $5.00 \mathrm{~N}$ and if the displacement is $0.800 \mathrm{~m}$ in the $+x$ -direction, what is the work done by the force $\overrightarrow{\boldsymbol{F}} ?$

Sachin Rao
Sachin Rao
Numerade Educator
02:54

Problem 76

Magnetic fields are produced by moving charges and exert forces on moving charges. When a particle with charge $q$ is moving with velocity $\overrightarrow{\boldsymbol{v}}$ in a magnetic field $\overrightarrow{\boldsymbol{B}},$ the force $\overrightarrow{\boldsymbol{F}}$ that the field exerts on the particle is given by $\overrightarrow{\boldsymbol{F}}=q \overrightarrow{\boldsymbol{v}} \times \overrightarrow{\boldsymbol{B}}$. The SI units are as follows: For charge it is the coulomb (C), for magnetic field it is tesla (T), for force it is newton $(\mathrm{N}),$ and for velocity it is $\mathrm{m} / \mathrm{s}$. If $q=-8.00 \times 10^{-6} \mathrm{C}, \overrightarrow{\boldsymbol{v}}$
is $3.00 \times 10^{4} \mathrm{~m} / \mathrm{s}$ in the $+x$ -direction, and $\overrightarrow{\boldsymbol{B}}$ is $5.00 \mathrm{~T}$ in the $-y$ -direction, what are the magnitude and direction of the force that the magnetic field exerts on the charged particle?

Sachin Rao
Sachin Rao
Numerade Educator
01:57

Problem 77

Vectors $\vec{A}$ and $\vec{B}$ have scalar product -7.00 , and their vector product has magnitude $+9.00 .$ What is the angle between these two vectors?

Narayan Hari
Narayan Hari
Numerade Educator
02:54

Problem 78

Torque is a vector quantity that specifies the effectiveness of a force in causing the rotation of an object. The torque that a force $\vec{F}$ exerts on a rigid object depends on the point where the force acts and on the location of the axis of rotation. If $\vec{r}$ is the length vector from the axis to the point of application of the force, then the torque is $\overrightarrow{\boldsymbol{r}} \times \overrightarrow{\boldsymbol{F}}$. If $\overrightarrow{\boldsymbol{F}}$ is $22.0 \mathrm{~N}$ in the $-y$ -direction and if $\vec{r}$ is in the $x y$ -plane at an angle of $36^{\circ}$ from the $+y$ -axis toward the $-x$ -axis and has magnitude $4.0 \mathrm{~m}$, what are the magnitude and direction of the torque exerted by $\overrightarrow{\boldsymbol{F}} ?$

Sachin Rao
Sachin Rao
Numerade Educator
05:32

Problem 79

Vector $\vec{A}=a \hat{i}-b \hat{k}$ and vector $\vec{B}=-c \hat{\jmath}+d \hat{k}_{\rightarrow}$ (a) $\operatorname{In}$
terms of the positive scalar quantities $a, b, c,$ and $d,$ what are $\vec{A} \cdot \vec{B}$ and $\vec{A} \times \vec{B} ?(b)$ If $c=0,$ what is the magnitude of $\vec{A} \cdot \vec{B}$ and what are the magnitude and direction of $\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}$ ? Does your result for the direction for $\vec{A} \times \vec{B}$ agree with the result you get if you sketch $\vec{A}$ and $\vec{B}$ in the $x z$ -plane and apply the right-hand rule? The scalar product can be described as the magnitude of $\overrightarrow{\boldsymbol{B}}$ times the component of $\overrightarrow{\boldsymbol{A}}$ that is parallel to $\overrightarrow{\boldsymbol{B}}$. Does this agree with your result? The magnitude of the vector product can be described as the magnitude of $\overrightarrow{\boldsymbol{B}}$ times the component of $\vec{A}$ that is perpendicular to $\overrightarrow{\boldsymbol{B}}$. Does this agree with your result?

David González Cornejo
David González Cornejo
Numerade Educator
04:42

Problem 80

Vectors $\vec{A}$ and $\vec{B}$ are in the $x y$ -plane. Vector $\vec{A}$ is in the $+x$ direction, and the direction of vector $\overrightarrow{\boldsymbol{B}}$ is at an angle $\theta$ from the $+x$ -axis measured toward the $+y$ -axis. (a) If $\theta$ is in the range $0^{\circ} \leq \theta \leq 180^{\circ}$, for what two values of $\theta$ does the scalar product $\overrightarrow{\boldsymbol{A}} \cdot \overrightarrow{\boldsymbol{B}}$ have its maximum magnitude? For each of these values of $\theta,$ what is the magnitude of the vector product $\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}} ?$ (b) If $\theta$ is in the range $0^{\circ} \leq \theta \leq 180^{\circ}$, for what value of $\theta$ does the vector product $\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}$ have its maximum value? For this value of $\theta,$ what is the magnitude of the scalar product $\vec{A} \cdot \vec{B} ?(c)$ What is the angle $\theta$ in the range $0^{\circ} \leq \theta \leq 180^{\circ}$ for which $\vec{A} \cdot \vec{B}$ is twice $|\vec{A} \times \vec{B}| ?$

David González Cornejo
David González Cornejo
Numerade Educator
01:44

Problem 81

Vector $\overrightarrow{\boldsymbol{A}}$ has magnitude $12.0 \mathrm{~m},$ and vector $\overrightarrow{\boldsymbol{B}}$ has magnitude $12.0 \mathrm{~m}$. The scalar product $\overrightarrow{\boldsymbol{A}} \cdot \overrightarrow{\boldsymbol{B}}$ is $76.0 \mathrm{~m}^{2}$. What is the magnitude of the vector product between these two vectors?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
07:43

Problem 82

Vector $\vec{A}$ has magnitude $5.00 \mathrm{~m}$ and lies in the $x y$ -plane in a direction $53.0^{\circ}$ from the $+x$ -axis axis measured toward the $+y$ -axis. Vector $\vec{B}$ has magnitude $8.00 \mathrm{~m}$ and a direction you can adjust. (a) You want the vector product $\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}$ to have a positive $z$ -component of the largest possible magnitude. What direction should you select for vector $\overrightarrow{\boldsymbol{B}} ?$ (b) What is the direction of $\overrightarrow{\boldsymbol{B}}$ for which $\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}$ has the most negative $z$ -component?
(c) What are the two directions of $\overrightarrow{\boldsymbol{B}}$ for which $\vec{A} \times \vec{B}$ is zero?

David González Cornejo
David González Cornejo
Numerade Educator
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Problem 83

The scalar product of vectors $\overrightarrow{\boldsymbol{A}}$ and $\overrightarrow{\boldsymbol{B}}$ is $+59.0 \mathrm{~m}^{2}$. Vector $\overrightarrow{\boldsymbol{A}}$ has magnitude $9.00 \mathrm{~m}$ and direction $28.0^{\circ}$ west of south. If vector $\overrightarrow{\boldsymbol{B}}$ has direction $39.0^{\circ}$ south of east, what is the magnitude of $\overrightarrow{\boldsymbol{B}}$ ?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:57

Problem 85

You are given vectors $\overrightarrow{\boldsymbol{A}}=5.0 \hat{\imath}-6.5 \hat{\jmath}$ and $\overrightarrow{\boldsymbol{B}}=3.5 \hat{\imath}-7.0 \hat{\jmath}$.
A third vector, $\overrightarrow{\boldsymbol{C}},$ lies in the $x y$ -plane. Vector $\overrightarrow{\boldsymbol{C}}$ is perpendicular to vector $\vec{A},$ and the scalar product of $\overrightarrow{\boldsymbol{C}}$ with $\overrightarrow{\boldsymbol{B}}$ is $15.0 .$ From this information, find the components of vector $\overrightarrow{\boldsymbol{C}}$.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:49

Problem 86

Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes $A=3.00$ and $B=3.00$. Their vector product is $\vec{A} \times \vec{B}=-5.00 \hat{k}+2.00 \hat{\imath}$. What is the angle between $\overrightarrow{\boldsymbol{A}}$ and $\overrightarrow{\boldsymbol{B}}$ ?

Zachary Warner
Zachary Warner
Numerade Educator
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Problem 87

You are a team leader at a pharmaceutical company. Several technicians are preparing samples, and you want to compare the densities of the samples (density = mass/volume) by using the mass and volume values they have reported. Unfortunately, you did not specify what units to use. The technicians used a variety of units in reporting their values, as shown in the following table.
$$\begin{array}{lll}\hline \text { Sample ID } & {\text { Mass }} & {\text {Volume }} \\
\hline \text { A } & 8.00 \mathrm{~g} & 1.67 \times 10^{-6} \mathrm{~m}^{3} \\
\text { B } & 6.00 \mu \mathrm{g} & 9.38 \times 10^{6} \mu \mathrm{m}^{3} \\
\text { C } & 8.00 \mathrm{mg} & 2.50 \times 10^{-3} \mathrm{~cm}^{3} \\
\mathrm{D} & 9.00 \times 10^{-4} \mathrm{~kg} & 2.81 \times 10^{3} \mathrm{~mm}^{3} \\
\mathrm{E} & 9.00 \times 10^{4} \mathrm{ng} & 1.59 \times 10^{-2} \mathrm{~mm}^{3} \\
\mathrm{~F} & 6.00 \times 10^{-2} \mathrm{mg} & 1.25 \times 10^{-4} \mathrm{~cm}^{3}
\end{array}$$
List the sample IDs in order of increasing density of the sample.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 88

You are a mechanical engineer working for a manufacturing company. Two forces, $\overrightarrow{\boldsymbol{F}}_{1}$ and $\overrightarrow{\boldsymbol{F}}_{2}$, act on a component part of a piece of equipment. Your boss asked you to find the magnitude of the larger of these two forces. You can vary the angle between $\overrightarrow{\boldsymbol{F}}_{1}$ and $\overrightarrow{\boldsymbol{F}}_{2}$ from $0^{\circ}$ to $90^{\circ}$ while the magnitude of each force stays constant. And, you can measure the magnitude of the resultant force they produce (their vector sum), but you cannot directly measure the magnitude of each separate force. You measure the magnitude of the resultant force for four angles $\theta$ between the directions of the two forces as follows:
$$\begin{array}{lc}\hline \boldsymbol{\theta} & \text { Resultant force ( } \mathbf{N} \text { ) } \\\hline 0.0^{\circ} & 8.00 \\45.0^{\circ} & 7.43 \\60.0^{\circ} & 7.00 \\90.0^{\circ} & 5.83\end{array}$$
(a) What is the magnitude of the larger of the two forces? (b) When the equipment is used on the production line, the angle between the two forces is $30.0^{\circ}$. What is the magnitude of the resultant force in this case?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 89

Navigating in the Solar System. The Mars Polar Lander spacecraft was launched on January $3,1999 .$ On December 3 , 1999 , the day Mars Polar Lander impacted the Martian surface at high velocity and probably disintegrated, the positions of the earth and Mars were given by these coordinates:
$$\begin{array}{lccc}\hline & x & y & z \\
\hline \text { Earth } & 0.3182 \mathrm{AU} & 0.9329 \mathrm{AU} & -0.0000 \mathrm{AU} \\
\text { Mars } & 1.3087 \mathrm{AU} & -0.4423 \mathrm{AU} & -0.0414 \mathrm{AU}\end{array}$$
With these coordinates, the sun is at the origin and the earth's orbit is in the $x y$ -plane. The earth passes through the $+x$ -axis once a year on the autumnal equinox, the first day of autumn in the northern hemisphere (on or about September 22). One $\mathrm{AU},$ or astronomical unit, is equal to $1.496 \times 10^{8} \mathrm{~km},$ the average distance from the earth to the
(a) Draw the positions of the sun, the earth, and Mars on December 3 , sun.
1999. (b) Find these distances in AU on December 3,1999: from (i) the sun to the earth; (ii) the sun to Mars; (iii) the earth to Mars. (c) As seen from the earth, what was the angle between the direction to the sun and the direction to Mars on December $3,1999 ?$ (d) Explain whether Mars was visible from your current location at midnight on December 3 , $1999 .$ (When it is midnight, the sun is on the opposite side of the earth from you.)

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:18

Problem 90

You are testing parcel delivery by drone for a project. Your team uses vector displacements to record the route of the drone, with the origin taken to be the position of the control centre. During one test, the drone starts its flight at $+10 \hat{\imath}-50 \hat{\jmath},$ where the units are meters, $\hat{\imath}$ is to the east, and $\hat{\jmath}$ is to the north. Subsequent displacements of the drone are $+90 \hat{\imath},+110 \hat{\jmath},-60 \hat{\imath}+40 \hat{\jmath},$ and $+120 \hat{\imath}+180 \hat{\jmath}$. If the final destination of the drone is $-70 \hat{\jmath}$, how far and in which direction must the drone fly? (You are well advised to diagram the situation before solving this numerically.)

Khaled Yasein
Khaled Yasein
Numerade Educator
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Problem 91

All of the stars of the Big Dipper (part of the constellation Ursa Major) may appear to be the same distance from the earth, but in fact they are very far from each other. Figure $\mathbf{P} 1.91$ shows the distances from the earth to each of these stars. The distances are given in light-years (ly), the distance that light travels in one year. One light-year equals $9.461 \times 10^{15} \mathrm{~m}$.
(a) Alkaid and Merak are $25.6^{\circ}$ apart in the earth's sky. In a diagram, show the relative positions of Alkaid, Merak, and our sun. Find the distance in light-years from Alkaid to Merak.
(b) To an inhabitant of a planet orbiting Merak, how many degrees apart in the sky would Alkaid and our sun be?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 92

Calculating Lung Volume in Humans. In humans, oxygen and carbon dioxide are exchanged in the blood within many small sacs called alveoli in the lungs. Alveoli provide a large surface area for gas exchange. Recent careful measurements show that the total number of alveoli in a typical pair of lungs is about $480 \times 10^{6}$ and that the average volume of a single alveolus is $4.2 \times 10^{6} \mu \mathrm{m}^{3} .$ (The volume of a sphere is $V=\frac{4}{3} \pi r^{3},$ and the area of a sphere is $\left.A=4 \pi r^{2} .\right)$ .
What is total volume of the gas-exchanging region of the lungs?
(a) $2000 \mu \mathrm{m}^{3} ;$ (b) $2 \mathrm{~m}^{3} ;$ (c) $2.0 \mathrm{~L} ;$ (d) $120 \mathrm{~L}$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 93

Calculating Lung Volume in Humans. In humans, oxygen and carbon dioxide are exchanged in the blood within many small sacs called alveoli in the lungs. Alveoli provide a large surface area for gas exchange. Recent careful measurements show that the total number of alveoli in a typical pair of lungs is about $480 \times 10^{6}$ and that the average volume of a single alveolus is $4.2 \times 10^{6} \mu \mathrm{m}^{3} .$ (The volume of a sphere is $V=\frac{4}{3} \pi r^{3},$ and the area of a sphere is $\left.A=4 \pi r^{2} .\right)$
If we assume that alveoli are spherical, what is the diameter of a
typical alveolus? (a) $0.20 \mathrm{~mm} ;$ (b) $2 \mathrm{~mm} ;$ (c) $20 \mathrm{~mm} ;$ (d) $200 \mathrm{~mm}$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 94

Calculating Lung Volume in Humans. In humans, oxygen and carbon dioxide are exchanged in the blood within many small sacs called alveoli in the lungs. Alveoli provide a large surface area for gas exchange. Recent careful measurements show that the total number of alveoli in a typical pair of lungs is about $480 \times 10^{6}$ and that the average volume of a single alveolus is $4.2 \times 10^{6} \mu \mathrm{m}^{3} .$ (The volume of a sphere is $V=\frac{4}{3} \pi r^{3},$ and the area of a sphere is $\left.A=4 \pi r^{2} .\right)$
Individuals vary considerably in total lung volume. Figure $\mathbf{P} 1.94$ shows the results of measuring the total lung volume and average alveolar volume of six individuals. From these data, what can you infer about the relationship among alveolar size, total lung volume, and number of alveoli per individual? As the total volume of the lungs increases, (a) the number and volume of individual alveoli increase; (b) the number of alveoli increases and the volume of individual alveoli decreases;
(c) the volume of the individual alveoli remains constant and the number of alveoli increases; (d) both the number of alveoli and the volume of individual alveoli remain constant.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
05:27

Problem 184

Obtain a unit vector perpendicular to the two vectors given in Exercise 1.39 .

Zachary Warner
Zachary Warner
Numerade Educator