Physics For Scientists And Engineers 6E - part 19

Problems

through a quarter of a circle of radius 3.70 cm that lies in
a north-south vertical plane. Find (a) the magnitude of
the total displacement of the object, and (b) the angle the
total displacement makes with the vertical.

Figure P3.35

75.0˚

60.0˚

80.0 N

120 N

Consider the two vectors A " 3ˆi # 2ˆj and B " # ˆi # 4ˆj.
Calculate (a) A % B, (b) A # B, (c)

!A % B!, (d) !A # B!,

and (e) the directions of A % B and A # B.

32. Consider the three displacement vectors A " (3ˆi # 3ˆj) m,

B " (ˆi # 4ˆj) m, and C " (# 2

i % 5ˆj) m. Use the compo-

nent method to determine (a) the magnitude and direc-
tion of the vector D " A % B % C, (b) the magnitude and
direction of E " # A # B % C.

A particle undergoes the following consecutive displace-
ments: 3.50 m south, 8.20 m northeast, and 15.0 m west.
What is the resultant displacement?

34. In a game of American football, a quarterback takes the

ball from the line of scrimmage, runs backward a distance
of  10.0  yards,  and  then  sideways  parallel  to  the  line  of
scrimmage  for  15.0  yards.  At  this  point,  he  throws  a  for-
ward  pass  50.0  yards  straight  downfield  perpendicular  to
the line of scrimmage. What is the magnitude of the foot-
ball’s resultant displacement?

35. The helicopter view in Fig. P3.35 shows two people pulling

on a stubborn mule. Find (a) the single force that is equiv-
alent to the two forces shown, and (b) the force that a
third person would have to exert on the mule to make the
resultant force equal to zero. The forces are measured in
units of newtons (abbreviated N).

33.

31.

36. A novice golfer on the green takes three strokes to sink the

ball. The successive displacements are 4.00 m to the north,
2.00 m northeast, and 1.00 m at 30.0° west of south. Start-
ing at the same initial point, an expert golfer could make
the hole in what single displacement?

37. Use the component method to add the vectors A and B

shown in Figure P3.15. Express the resultant A % B in
unit–vector notation.

38.

In an assembly operation illustrated in Figure P3.38, a ro-

bot moves an object first straight upward and then also to
the east, around an arc forming one quarter of a circle of
radius 4.80 cm that lies in an east-west vertical plane. The
robot then moves the object upward and to the north,

Figure P3.38

39.

Vector B has x, y, and z components of 4.00, 6.00, and 3.00

units, respectively. Calculate the magnitude of B and the
angles that B makes with the coordinate axes.

40.

You are standing on the ground at the origin of a coordi-

nate system. An airplane flies over you with constant velocity
parallel to the x axis and at a fixed height of 7.60 ) 10

At time t " 0 the airplane is directly above you, so that the
vector leading from you to it is P

(7.60 ) 10

m)ˆj. At

t " 30.0 s the position vector leading from you to the
airplane is P

(8.04 ) 10

m)ˆi % (7.60 ) 10

m)ˆj. De-

termine the magnitude and orientation of the airplane’s po-
sition vector at t " 45.0 s.

The vector A has x, y, and z components of 8.00, 12.0, and
#

4.00 units, respectively. (a) Write a vector expression for

A in unit–vector notation. (b) Obtain a unit–vector expres-
sion for a vector B one fourth the length of A pointing in
the same direction as A. (c) Obtain a unit–vector expres-
sion for a vector C three times the length of A pointing in
the direction opposite the direction of A.

42.

Instructions for finding a buried treasure include the fol-

lowing: Go 75.0 paces at 240°, turn to 135° and walk 125
paces, then travel 100 paces at 160°. The angles are mea-
sured counterclockwise from an axis pointing to the east,
the % x direction. Determine the resultant displacement
from the starting point.

43. Given the displacement vectors A " (3ˆi # 4ˆj % 4ˆk) m and

B " (2ˆi % 3ˆj # 7 ˆk) m, find the magnitudes of the vectors
(a) C " A % B and (b) D " 2A # B, also expressing each
in terms of its rectangular components.

44.

A radar station locates a sinking ship at range 17.3 km and

bearing 136° clockwise from north. From the same station
a rescue plane is at horizontal range 19.6 km, 153° clock-
wise from north, with elevation 2.20 km. (a) Write the po-
sition vector for the ship relative to the plane, letting ˆi
represent east, ˆj north, and ˆk up. (b) How far apart are the
plane and ship?

41.

45.

As it passes over Grand Bahama Island, the eye of a hurri-

cane is moving in a direction 60.0° north of west with a
speed of 41.0 km/h. Three hours later, the course of the
hurricane suddenly shifts due north, and its speed slows to
25.0 km/h. How far from Grand Bahama is the eye 4.50 h
after it passes over the island?

46.

(a) Vector E has magnitude 17.0 cm and is directed 27.0°

counterclockwise from the % x axis. Express it in unit–
vector notation. (b) Vector F has magnitude 17.0 cm and
is directed 27.0° counterclockwise from the % y axis.
Express it in unit–vector notation. (c) Vector G has magni-
tude 17.0 cm and is directed 27.0° clockwise from the # y
axis. Express it in unit–vector notation.

Vector A has a negative x component 3.00 units in length
and a positive y component 2.00 units in length. (a) Deter-
mine  an  expression  for  A in  unit–vector  notation.
(b)  Determine  the  magnitude  and  direction  of  A.
(c)  What  vector  B when  added  to  A gives  a  resultant
vector with no x component and a negative y component
4.00 units in length?

48. An airplane starting from airport A flies 300 km east, then

350 km at 30.0° west of north, and then 150 km north to
arrive finally at airport B. (a) The next day, another plane
flies directly from A to B in a straight line. In what direc-
tion should the pilot travel in this direct flight? (b) How
far will the pilot travel in this direct flight? Assume there is
no wind during these flights.

Three displacement vectors of a croquet ball are

49.

47.

52.

Two vectors A and B have precisely equal magnitudes. In

order for the magnitude of A % B to be larger than the
magnitude of A # B by the factor n, what must be the an-
gle between them?

53.

A vector is given by R " 2ˆi % ˆj % 3 ˆk. Find (a) the mag-

nitudes of the x, y, and z components, (b) the magnitude
of R, and (c) the angles between R and the x, y, and z
axes.

54.

The biggest stuffed animal in the world is a snake 420 m

long,  constructed  by  Norwegian  children.  Suppose  the
snake is laid out in a park as shown in Figure P3.54, form-
ing two straight sides of a 105° angle, with one side 240 m
long. Olaf and Inge run a race they invent. Inge runs di-
rectly from the tail of the snake to its head and Olaf starts
from the same place at the same time but runs along the
snake.  If  both  children  run  steadily  at  12.0 km/h,  Inge
reaches  the  head  of  the  snake  how  much  earlier  than
Olaf?

CHAPTE R 3 • Vectors

Figure P3.54

Figure P3.49

45.0

50. If A " (6.00ˆi # 8.00ˆj) units, B " (# 8.00ˆi % 3.00ˆj) units,

and C " (26.0ˆi % 19.0ˆj) units, determine a and b such
that aA % bB % C " 0.

Additional Problems

51.

Two vectors A and B have precisely equal magnitudes. In

order for the magnitude of A % B to be one hundred
times larger than the magnitude of A # B, what must be
the angle between them?

55.

An air-traffic controller observes two aircraft on his radar

screen.  The  first  is  at  altitude  800 m,  horizontal  distance
19.2 km, and 25.0° south of west. The second aircraft is at
altitude  1 100 m,  horizontal  distance  17.6 km,  and  20.0°
south  of  west.  What  is  the  distance  between  the  two  air-
craft? (Place the x axis west, the y axis south, and the z axis
vertical.)

56.

A ferry boat transports tourists among three islands. It sails

from the first island to the second island, 4.76 km away, in
a direction 37.0° north of east. It then sails from the sec-
ond island to the third island in a direction 69.0° west of
north. Finally it returns to the first island, sailing in a di-
rection 28.0° east of south. Calculate the distance between
(a) the second and third islands (b) the first and third
islands.

57.

The rectangle shown in Figure P3.57 has sides parallel to

the x and y axes. The position vectors of two corners are
A " 10.0 m at 50.0° and B " 12.0 m at 30.0°. (a) Find the

shown in Figure P3.49, where

!A! " 20.0 units, !B! " 40.0

units, and

!C! " 30.0 units. Find (a) the resultant in

unit–vector notation and (b) the magnitude and direction
of the resultant displacement.

(30.0 m, # 20.0 m), (60.0 m, 80.0 m), (# 10.0 m, # 10.0 m),
(40.0 m,  # 30.0 m),  and  (# 70.0 m,  60.0 m),  all  measured
relative  to  some  origin,  as  in  Figure  P3.62.  His  ship’s  log
instructs you to start at tree A and move toward tree B, but
to cover only one half the distance between A and B. Then
move  toward  tree  C,  covering  one  third  the  distance  be-
tween  your  current  location  and  C.  Next  move  toward  D,
covering  one  fourth  the  distance  between  where  you  are
and  D.  Finally  move  towards  E,  covering  one  fifth  the  dis-
tance  between  you  and  E,  stop,  and  dig.  (a)  Assume  that
you  have  correctly  determined  the  order  in  which  the  pi-
rate labeled the trees as A, B, C, D, and E, as shown in the
figure.  What  are  the  coordinates  of  the  point  where  his
treasure  is  buried?  (b) What  if you  do  not  really  know  the
way the pirate labeled the trees? Rearrange the order of the
trees  [for  instance,  B(30 m,  # 20 m),  A(60 m,  80  m),
E(# 10 m,  # 10 m),  C(40 m,  # 30 m),  and  D(# 70 m,
60 m)]  and  repeat  the  calculation  to  show  that  the  answer
does  not  depend  on  the  order  in  which  the  trees  are
labeled.

perimeter of the rectangle. (b) Find the magnitude and
direction of the vector from the origin to the upper right
corner of the rectangle.

Problems

Figure P3.57

Figure P3.59

End

200 m

60.0

30.0

150 m

300 m

100 m

Start

63.

Consider a game in which N children position themselves

at equal distances around the circumference of a circle. At
the center of the circle is a rubber tire. Each child holds a
rope attached to the tire and, at a signal, pulls on his rope.
All children exert forces of the same magnitude F. In the
case N " 2, it is easy to see that the net force on the tire
will be zero, because the two oppositely directed force vec-
tors add to zero. Similarly, if N " 4, 6, or any even integer,
the  resultant  force  on  the  tire  must  be  zero,  because  the
forces  exerted  by  each  pair  of  oppositely  positioned  chil-
dren  will  cancel.  When  an  odd  number  of  children  are
around  the  circle,  it  is  not  so  obvious  whether  the  total
force on the central tire will be zero. (a) Calculate the net
force on the tire in the case N " 3, by adding the compo-
nents  of  the  three  force  vectors.  Choose  the  x axis  to  lie
along  one  of  the  ropes.  (b)  What  If?  Determine  the  net
force for the general case where N is any integer, odd or
even,  greater  than  one.  Proceed  as  follows:  Assume  that
the total force is not zero. Then it must point in some par-
ticular direction. Let every child move one position clock-
wise. Give a reason that the total force must then have a di-
rection  turned  clockwise  by  360°/N.  Argue  that  the  total
force  must  nevertheless  be  the  same  as  before.  Explain
that  the  contradiction  proves  that  the  magnitude  of  the
force is zero. This problem illustrates a widely useful tech-
nique  of  proving  a  result  “by  symmetry ”—by  using  a  bit
of the mathematics of group theory. The particular situation

58.

Find the sum of these four vector forces: 12.0 N to the

right at 35.0° above the horizontal, 31.0 N to the left at
55.0° above the horizontal, 8.40 N to the left at 35.0° be-
low the horizontal, and 24.0 N to the right at 55.0° below
the horizontal. Follow these steps: Make a drawing of this
situation and select the best axes for x and y so you have
the least number of components. Then add the vectors by
the component method.

A person going for a walk follows the path shown in Fig.
P3.59. The total trip consists of four straight-line paths. At
the end of the walk, what is the person’s resultant displace-
ment measured from the starting point?

59.

60.

The instantaneous position of an object is specified by its

position vector r leading from a fixed origin to the loca-
tion of the point object. Suppose that for a certain object
the position vector is a function of time, given by
r " 4ˆi % 3ˆj # 2t ˆk, where r is in meters and t is in seconds.
Evaluate d r/dt. What does it represent about the object?

61. A jet airliner, moving initially at 300 mi/h to the east, sud-

denly enters a region where the wind is blowing at 100 mi/h
toward the direction 30.0° north of east. What are the new
speed and direction of the aircraft relative to the ground?

62.

Long John Silver, a pirate, has buried his treasure on an is-

land with five trees, located at the following points:

Figure P3.62

CHAPTE R 3 • Vectors

is actually encountered in physics and chemistry when an
array of electric charges (ions) exerts electric forces on an
atom at a central position in a molecule or in a crystal.

64.

A rectangular parallelepiped has dimensions a, b, and c, as

in Figure P3.64. (a) Obtain a vector expression for the
face diagonal vector R

. What is the magnitude of this vec-

tor? (b) Obtain a vector expression for the body diagonal
vector R

. Note that R

, c

k, and R

make a right triangle

and prove that the magnitude of R

√

67.

A point P is described by the coordinates (x, y) with re-

spect to the normal Cartesian coordinate system shown in
Fig. P3.67. Show that (x(, y(), the coordinates of this point
in the rotated coordinate system, are related to (x, y) and
the rotation angle * by the expressions

x( " x cos * % y sin *

y( " # x sin * % y cos *

Figure P3.64

Figure P3.66

Figure P3.67

′

65.

Vectors A and B have equal magnitudes of 5.00. If the

sum of A and B is the vector 6.00ˆj, determine the angle be-
tween A and B.

66.

In Figure P3.66 a spider is resting after starting to spin its

web. The gravitational force on the spider is 0.150 newton
down. The spider is supported by different tension forces
in  the  two  strands  above  it,  so  that  the  resultant  vector
force on the spider is zero. The two strands are perpendic-
ular  to  each  other,  so  we  have  chosen  the  x and  y direc-
tions  to  be  along  them.  The  tension  T

is 0.127 newton.

Find (a) the tension T

, (b) the angle the x axis makes

with the horizontal, and (c) the angle the y axis makes
with the horizontal.

Answers to Quick Quizzes

3.1 Scalars: (a), (d), (e). None of these quantities has a direc-

tion. Vectors: (b), (c). For these quantities, the direction is
necessary to specify the quantity completely.

3.2 (c). The resultant has its maximum magnitude A % B "

12 % 8 " 20 units when vector A is oriented in the same
direction as vector B. The resultant vector has its mini-
mum magnitude A # B " 12 # 8 " 4 units when vector A
is oriented in the direction opposite vector B.

3.3 (a). The magnitudes will add numerically only if the vec-

tors are in the same direction.

3.4 (b) and (c). In order to add to zero, the vectors must point

in opposite directions and have the same magnitude.

3.5 (b). From the Pythagorean theorem, the magnitude of a

vector is always larger than the absolute value of each com-
ponent, unless there is only one nonzero component, in
which case the magnitude of the vector is equal to the ab-
solute value of that component.

3.6 (b). From the Pythagorean theorem, we see that the mag-

nitude of a vector is nonzero if at least one component is
nonzero.

3.7 (d). Each set of components, for example, the two x com-

ponents A

and B

, must add to zero, so the components

must be of opposite sign.

3.8 (c). The magnitude of C is 5 units, the same as the z com-

ponent. Answer (b) is not correct because the magnitude
of any vector is always a positive number while the y com-
ponent of B is negative.

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