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Unformatted text preview: 01.9. The quantity 77 = 3.14159 . . . is a number with no dimen—
sions, since it is a ratio of two lengths. Describe two or three other
geometrical or physical quantities that are dimensionless. 01.19. What are the units of volume? Suppose another student tells
you that a cyiinder of radius r and height it has voiume given by
ﬁrah. Explain why this cannot be right. {21.11. Three archers each fire four arrows at a target. Joe’s four
arrows hit at points i0 cm above, 10 cm below, 10 cm to the left, and
:0 cm to the right of the center of the target. All four of Moe’s arrows
hit within 1 cm of a point 20 cm from the center, and Fio’s four
arrows all hit within 1 cm of the center. The contest judge says that
one of the archers is precise but not accurate, another archer is accu~
rate but not precise, and the third archer is both accurate and precise.
Which description goes with which archer? Explain your reasoning.
01.12. A circular racetrack has a radius of 500 in. What is the dis
placement of a bicyclist when she travels around the track from the
north side to the south side? When she makes one complete circle
around the track? Expiain your reasoning. (21.13. Can you ﬁnd two vectors with different lengths that have a
vector sum of zero? What length restrictions are required for three
vectors to have a vector sum of zero? Explain your reasoning.
{31.14. One sometimes speaks of the “direction of time,” evolving
from past to future. Does this mean that time is a vector quantity?
Explain your reasoning. (11.15. Air traffic controllers give instructions to airline pilots
teliing them in which direction they are to ﬂy. These instructions
are called “vectors.” If these are the only instructions given, is the
name “vector” used correctly? Why or why not? 01.16. Can you find a vector quantity that has a magnitude of zero
but components that are different from zero? Explain. Can the
magnitude of a vector be less than the magnitude of any of its com—
ponents? Explain. 01.17. (a) Does it make sense to say that a vector is negative?
Why? (b) Does it nitrite sense to say that one vector is the negative
of another"? Why? Does your answer here contradict what you said
in part (at? M w) A M _) {21.18. If C is the vector sum ofA and B, C = A + B, what must
be true ifC 2 A + B? What must be true ifC 2 0? q1.19.w)1i‘ 2i and E are nonzero vectors, is it possible for 3 J5? and
A x B both to be zero? Explain. 01.20. What doesmgi .174), the scalar product of a vector with itself,
give? What about A X the vector product of a vectorwith itseif?
01.21. Let X represent any nonzero vector. Why is Iii/A a unit vec
tor and what is its direction? 1ft? is the angle that 3 makes with the
+x—axis, explain why {Ail/i) ~i is called the direction cosine for
that axis. 02.22. Whigh of the fpllowing are legitimate mathematicai opera
tions; (20ft  (ME, —~ C); (1)} (2i —— E) X E; (c) 3’ (f? x 6‘);
(d) A x (B x (e) X x  In each case, give the reason
for your answer. QI.23._)CODS}C§BI the two repeated vector products A: x (I? x 6')
and (A x E) x E". Give an exampie that illustrates the generai
rule that these two vector products do not have the same magni—
tude or direction. Can you choose the vectors 3, B, and 6‘ such that
these two vector products are equal? If so, give an example. 01.24. Show that, no matter what X and E are, E  {If x Ii) 2 0.
(Hint: Do not look for an elaborate mathematical proof. Rather
look at the deiimnition of the direction of the cross product.) 01.25. (a) If A 45; ﬂ 0._ does it necessary foilow that A t—* 0 or
B m 0? Expiain. (ii) .If A X ii; = 0, does it necessary foliow that
A m 0 or B = 0? Explain. Exercises 29 01.26. If X m 0 for a vector in the xy piane, does it follow that
A, = way? What can you say about A“. and 14).? Exercises Section 1.3 Standards and Units Section 1.4 Unit Consistency and Conversions 1.3. Starting with the deﬁnition 'i in. m 2.54 cm, find the number
of (a) kilometers in i.00 mite and (b) feet in 1.00 ion. 1.2. According to the label on a bottle of salad dressing, the vol
ume of the contents is 0.473 liter (L). Using only the conversions
1 L = 1000 cm3 and lin. m 2.54 cm, express this volume in
cubic inches. 1.3. How many nanoseconds does it taice tight to travel 1.00 ft in
vacuum? (This result is a useful quantity to remember.) 1.13. The density of lead is 11.3 g/cm3. What is this value in idio—
grarns per cubic meter? 1.5. The most powerqu engine availabie for the ciassic 1963
Chevrolet Corvette Sting Ray developed 360 horsepower and had
a dispiacement of 327 cubic inches. Express this displacement in
liters (L) by using only the conversions l L = 1000 cm3 and
l in. = 2.54 cm. 1.6. A square ﬁeld measuring 100.0 m by l00.0 in has an area of
1.00 hectare. An acre has an area or 43,600 n2. If a country lot has
an area of 12.0 acres, what is the area in hectares? 1.7. How many years older wilt you be 1.00 biliion seconds from
now? (Assume a 365—day year.) 1.3. While driving in an exotic foreign land you see a speed limit
sign on a highway that reads i80,000 furlongs per fortnight. How
many miles per hour is this? (One furlong is i miie, and a fortnight
is 14 days. A furiong originally referred to the length of a plowed
furrow.) 1.9. A certain fuel—efﬁcient hybrid car gets gasoline mileage of
55.0 mpg (miles per gallon). (a) If you are driving this car in
Europe and want to compare its miieage with that of other Euro
pean cars, express this miieagc in km/L (L = liter). Use the con
version factors in Appendix E. (b) If this car’s gas tank holds 45 L,
how many tanks of gas will you use to drive 1500 km? 130. The following conversions occur frequently in physics and
are very useful. (a) Use 1 mi = 5280 ft and '1 h w 3600 s to con
vert 60 mph to units of ft/ s. (b) The acceleration of a freely falling
object is 32 ft/sz. Lisa 1 ft x 30.48 cm to expreSs this acceieration
in units of m/sz. (c) The density of water is 'i .0 g/crn3. Convert this
density to units of icg/m3. 1.11. Neptunium. In the felt of 2002, a group of scientists at Los
Alamos National Laboratory determined that the criticai mass of
neptuniumZB? is about 60 kg. The critical mass of a fissionable
materiai is the minimum amount that must be brought together to
start a chain reaction. This element has a density of 19.5 g/cma.
What would be the radius of a sphere of this material that has a
critical mass? Section 1.5 Uncertainty and Significant Figures 1.22. A useful and easyto‘reniernber approximate vaiue for the
number of seconds in a year is 17 X 107. Determine the percent
error in this approximate value. {There are 365.24 days in one year.)
1.13. Figure 1.7 shows the result of unacceptable error in the stop
ping position of a train. (a) If a train travels 890 ion from Beriin to
Paris and then overshoots the end of the track by 10 in, what is the
percent error in the total distance covered? (in) Is it correct to write
the total distance covered by the train as 890,010 In? Explain. 30 CHAPTER 1 Units, Physical Quantities, and Vectors 1.14. With a wooden ruier you measure the length of a rectanguiar
piece of sheet metal to be i2 mrn. You use micrometer calipers to
measure the width of the rectangle and obtain the value 5.98 mm.
Give your answers to the following questions to the correct number
of signiﬁcant ﬁgures. (a) What is the area of the rectangle? (is) What
is the ratio of the rectangle’s width to its length? (c) What is the
perimeter of the rectangle? (d) What is the difference between the
length and width? (e) What is the ratio of the length to the width?
1.15. Estimate the percent error in measuring (a) a distance of about
75 cm with a meter stick; (b) a mass of about 12 g with a chemical
balance; (c) a time interval of about 6 min with a stopwatch. £26. A rectangular piece of aiuminum is 5.10 :1: 0.01 cm long and
1.90 t 0.01 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
iength and in the width. (This is a general result; see Challenge
Problem 2.98.) 1.17. As you eat your way through a bag of chocolate chip cookies,
you ohserve that each cookie is a circuiar riisk with a diameter of
8.50 i 0.02 cm and a thickness of 0.050 m 0.005 cm. (a) Find the
average volume of a cookie and the uncertainty in the volume.
(in) Find the ratio of the diameter to the thickness and the uncer—
tainty in this ratio. Section 3.6 Estimates and Orders of Magnitude 138. How many gallons of gasoline are used in the United States
in one day? Assume two cars for every three peopie, that each car
is driven an average of i0,000 mi per year, and that the average car
gets 20 miles per galion. 1.19. A rather ordinary middle—aged man is in the hospitai for a
routine checkwup. The nurse writes the quantity 200 on his medical
chart but forgets to include the units. Which of the following quanw
tities could the 200 plausibly represent? (a) his mass in kilograms;
(b) his height in meters; (c) his height in centimeters; (d) his height
in millimeters; (e) his age in months. L20. How many kernels of corn does it take to ﬁll a 241. soft drink
bottle? 1.21. How many words are there in this hook? 1.22. Four astronauts are in a spherical space station. (a) if, as is
typical, each of them breathes about 500 cm3 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 ali this air? 1.23. How many times does a typical person blink her eyes in a
iifettme? $.24. How many times does a human heart beat during a lifetime?
How many gallons of blood does it pump? (Estimate that the heart
pumps 50 cm3 of blood with each beat.) 1.25. in Wagner’s opera Dos Rheingoid, the goddess Freia is ran—
somed for a pile of gold just tail enough and wide enough to hide
her from sight. Estimate the monetary value of this pile. The den~
sity of gold is 19.3 g/cma, and its vaiue is about $10 per gram
(although this varies). 1.26. You are using water to dilute small amounts of chemicals in
the laboratory, drop try drop. How many drops of water are in a
1.0 L bottie? (Hint: Start by estimating the diameter of a drop of
water.) 1.21. How many pizzas are consumed each academic year by stu«
dents at your school?  1.28. How many dollar biEls woutd you have to stack to reach the
moon? Would that he cheaper than building and iaunching a space craft? (Him: Start by folding a dollar bill to see how many thick~
nesses make 'i.0 mm.) 1.29. How much would it cost to paper the entire United States
{inciuding Aiaska and Hawaii) with dollar bills? What weald be
the cost to each person in the United States? Section 1.7 Vectors and Vector Addition 1.30. Hearing rattles from a snake, you make two rapid displace—
ments of magnitude 1.8 m and 2.4 m. In sketches (roughly to
scale), show how your two displacements might add up to give a
resuitant of magnitude (a) 4.2 rn; (b) 0.6 m; (c) 3.0 in. 1.31. A postal employee drives a delivery truck along the route
shown in Fig. 1.33. Determine the magnitude and direction of the Figure 1.33 Exercises 1.3i and 1.38. resultant displacement by drawing a scale diagram. (See also Exerw
cise 1.38 for a different approach to this same probiem.) 1.32. For the vectors 3 and 1.5
in Fig. 1.34, use a scale drawing
to ﬁnd the magnitude and direc
tion of (a) the vector sum K + Figure 1.34 Exercises 1.32,
1.35, 1.39, 1.47, 1.53, and
1.57, and Problem 1.72. and (min) the vector difference y
A ~B. Use your answers to ,§(15.0 m)
ﬁnd the magnitude and direction "" of (c) —Ai  a and (chit —A.
(See also Exercise i.39 for a dif—
ferent approach to this problem.)
1.33. A spelunker is surveying a
cave. She follows a passage
180 in straight west, then 2'10 m
in a direction 45" east of south, and then 280 m at 30° east of 502.0111)
north. After a fourth unmeasured
dispiacement, she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the
fourth dispiacement. (See aiso Probiem 1.73 for a different
approach to this problem.) .1' (3.00 on) Section 1.8 Components of Vectors 1.34. Use a scale drawing to ﬁnd the x and y—components of the
following vectors. For each vector the numbers given are the mag
nitude of the vector and the angle, measured in the sense from the
+x~axis toward the +yaxis, that it makes with the "exaxis:
(a) magnitude 9.30 in, angle 60.0“; (b) magnitude 22.0 km, angie
135°; (c) magnitude 6.35 cm, angie 307°. we 1.35. Compute the x— and ycomponents of the vectors 3, E, C,
and ES in Fig. 2.34. 1.36. Let the angle 6 be the angle that the vector 3 makes with the
+xaxis, measured counterciockwise from that axis. Find the angle
6 for a vector that has the following components: (a) AA. m 2.00 m,
Ay = ~t.00 m; (b) A,{ m 2.00 m, A). “m: 1.00 In; (c) A. 2 “200121,
Ay m i.00 m; (d) Ax ﬂ “0.00111, A). = ~l. .00 In. 1.37. A rocket ﬁres two engines simultaneously. One produces a
thrust of 725 N directly forward, While the other gives a 513N
thrust at 32.40 above the forward direction. Find the magnitude
and direction (relative to the forward direction} of the resultant
force that these engines exert on the rocket. 1.38. A postal emptoyee drives a delivery truck over the route
shown in Fig. 1.33. Use the method of components to determine
the magnitude and direction of her resuitant displacement. In a
vectoraaddition diagram (roughiy to scale), show that the result
ant displacement found from your diagram is in qualitative
agreement with the result you obtained using the method of
components. 1.39. For the vectors 2i and ii in Fig. :34, use the method of com
ponents to find the magnitude and direction of (a) the vector sum
.3 —i— E; (b) the vector sum ﬁ +3); (0) the vector difference
3 ~ (d) the vector difference ii  1.40. Find the magnitude and direction of the vector represented by the following pairs of components: (a) A,r 3 M860 cm,
A). m 5.20 cm; {0) A, x M070 m, Ay = —2.45 in; (c) A, = 7.75 km,
A). a “2.70 km. 1.41. A disoriented physics professor drives 3.25 km north, then
4.75 ion west, and then 1.50 km south. Find the magnitude and
direction of the resultant displacement, using the method of com
ponents. In a vector addition diagram (roughly to scale), show that
the resultant displacement found from your diagram is in qualita
h‘ve agreement with the i‘esuit you obtained using the method of
components. my 1.42;} VectorA has components Ax ﬂ 1.30 cm, A). m 2.25 cm; vec—
tor B has components 13r = cm“; 3,. = ~3.75 cm. Find (a) the
components gt" the vector sum A + B; (b) the magnitude anddireg
tion of A + B; (c) the components of the vector difference B — A;
(d) the magnitude and direction of 3 ~ 3. 1.43. Vector Si is 2.80 cm long
and is 600° above the xmaxis in
the ﬁrst quadrant. Vector B is
1.90 cm long and is 600°
below the xaxis in the fourth
quadrant (Fig. 1.35). Use com—
ponents to ﬁnd the magnitude
and direction of {a} X + 3;
(0)3 m a {0) it ~ 3. In each
case, sketch the vector addition
or subtraction and show that
your numerical answers are in
qualitative agreement with your
stretch. 1.44. A river ﬂows from south to
north at 5.0 km/h. On this river,
a boat is heading east to west
perpendicular to the current at
7.0 km/h. As viewed by an eagle hovering at rest over the shore,
how fast and in what direction is this boat traveling? MS. Use vector components to find the magnitude and direc—
tion of the vector needed to balance the two vectors shown in Figure 1.35 Exercises 1.43
and 1.59. ' .
4' ‘4 A (2.80 cm) 000° (1.90 cm) Exercises 3i Figure 1.36
Exercise L45. Figure 1.36. Let the 625N vector be along
the ~y—axis and let the +x~axis be perpen~
dicuiar to it toward the right. 1.46. Two ropes in a vertical plane exert
equal magnitude forces on a hanging
weight but pull with an angle of 860°
between them. What pull does each one
exert if their resultant pull is 372 N
directiy upward? Section 1.9 Unit Vectors
1.41. Write each vector in Fig. $.34 in terms of the unit vectors
2 and 1.48. In each case, find the x and ycomponents of vector K:
(202 e 5.02 ~ 6.3}; (b) A e 1L2} — 9.912; (c) it e —15.03 +
22.4}; (d) 21’ n 5.00, where r} z 42 m of. 1.49. (a) Write each vector in
Fig. 'i.37 in terms of the unit
vectors 2 and j. (0} Use unit
vectors to express the vector “c.
where E? ﬂ 3.00:4 — 4.003. {c)
Find the magnitude and direc
tion of 1.50. Given two vectors Kw
4.003 + 3.00,? and E x 5.003 —
2.00}. (a) ﬁnd the magnitude of
each vector; (b) write an expres
sion for the vector difference
3 w ii using unit vectors;
(c) ﬁnd the magnitude and direc
tion of the vector difference
K — (d) In a vector diagram
show and Ii — and also show that your diagram agrees
qualitatively with your answer in part (0}. £5}. (a) Is the vector (i +f + it) a unit vector? Justify your
answer. (b) Can a unit vector have any components with magni
tude greater than unity? Can it have any negative components? In
each case justify your answer. (c) If}; m 0(302 4: 413i), where a
is a constant, determine the value of a that makes A) a unit vector. Figure 1.37 Exercise 1.49
and Problem 1.86. Section 1.10 Products of Vectors 1.52. (a) Use vector components to prove that two vectors com—
mute for both addition and the scalar product. (b) Prove that two
vectors anticommute for the vector product; that is, prove that
X x is 2 ~13; x X. 1.53. For the vectors 2, E, and E in Fig. 1.30, ﬁnd the scaiar prod—
acts (a) Lid; an”)? .6; (mi of. 1.54. (a) Find the scalar product of the two vectors .3 and 1”}; given
in Exercise 1.50. (b) Find the angle between these two vectors.
1.55. Find the angle between each of the foilowing pairs of vectors: (21} X m “2.003 + 6.00j and ii m 2.003 — 300;“
(b) it m 3.002 + 5.00,? ii a 10.002 + 6.00;"
(o X = “4.002 + 2.00;“ ii = 7.002 a taooj 1.56. By making simple stretches of the appropriate vector prod
ucts, show that (a) X '3 can be interpreted as the product of the
magnitude of X times the component of :0 along or the magni—
tude of 13 times the componentof X along 3; (h) EX x ﬁl can be
interpreted as the product of the magnitude of X timesﬂthe compo
nent of ii perpendicuiar to X, or the magnitude of 5; times the
component of perpendicular to E. and
and VECTORS and SCALARS 13 (b) Here each vector is equal to but opposite in direction to the corresponding one in (a). The field there'
fore appears as in Fig.(b). In Fig.{a) the fieid has the appearance ot a £1de emerging from a point Source 0 and flowing in the
directions indicated. For this reason the field is called a Source field and 0 is a source. in Fig.(b)_ the field seems to be flowing toward 0, and the field is therefore called a sink field and 0
is a sink. In three dimensions the corresponding interpretation is that a fluid is emerging radialiy from (or pro
ceeding radially'toward) a line source (or line sink). The vector field is called two dimensional since it is independent of z. (c) Since the magnitude of each vector is V942 l y2 + 22. all points on the sphere 262+ 3/2 + 22 = a2, a > 0
have vectors of magnitude a associated with them. The field therefore takes on the appearance of that
of a fluid emerging from source 0 and proceeding in all directions in space. This is a three dimension
al course field. SUPPLEMENTARY PROBLEMS 31. Which of the following are scaiars and which are vectors? (:2) Kinetic energy, (6) electric field intensity.
(c) entropy, (d) work, (a) centrifngal force, (f) temperature, (g) gravitational potential, (it) charge. (i) shear
ing stress, (1‘) frequency. AM“ (a) scalar. (5) vector, (a) scalar. (d) scalar, (9) vector, (f) scalar, {g3 scalar, (h) scalar, (i) vector
(j) scalar 32. An airplane travels 200 miles due west and then 150 miles 60° north of west. Determine the resultant dis—
placement (a) graphically, (b) analytically.
Ans. magnitude 304.1 mi (501/3773, direction 2501'?" north of east (arc sin 31/111/74) 33. Find the resultant of the foliowing displacements: A, 20 miles 36°sonth of east; B, 50 miies due west;
(3, 40 miles northeast; D, 30miles 60° south of west.
Ans. magnitude 20.9 mi, direction 21°39’ south of West 34;. Show graphically that (A—B) a —A + B. 35. An object P is acted upon by three coplanar forces as shown in Fig.(a) below. Determine the force needed
to prevent P from moving. Arts. 323 lb directly opposite 150 1b force 36. Given vectors ALB, C and D (Fig.0) below). Construct (a) 3A— 213— (C MD) (b) é—C + gut—B + 21)). /\\ F1343) Fig.(b) ...
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