Ch 01 HW
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Ch 01 HW
Due: 10:59pm on Tuesday, March 7, 2023
You will receive no credit for items you complete after the assignment is due. Grading Policy
Significant Figures Conceptual Question
In the parts that follow select whether the number presented in statement A is greater than, less than, or equal to the number
presented in statement B. Be sure to follow all of the rules concerning significant figures.
Part A
Statement A: 2.567 , to two significant figures.
Statement B: 2.567 , to three significant figures.
Determine the correct relationship between the statements.
Hint 1. Rounding and significant figures
Rounding to a different number of significant figures changes a number. For example, consider the number 3.4536.
This number has five significant figures. The following table illustrates the result of rounding this number to different
numbers of significant figures:
Four significant figures
3.454
Three significant figures
3.45
Two significant figures
3.5
One significant figure
3
Notice that, when rounding 3.4536 to one significant figure, since 0.4536 is less than 0.5, the result is 3, even though
if you first rounded to two significant figures (3.5), the result would be 4.
ANSWER:
Correct
Part B
Statement A: (2.567 + 3.146 ), to two significant figures.
Statement B: (2.567 , to two significant figures) + (3.146 , to two significant figures).
Determine the correct relationship between the statements.
Statement A is greater than Statement B.
Statement A is less than Statement B.
Statement A is equal to Statement B.
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ANSWER:
Correct
Evaluate statement A as follows: (2.567 + 3.146 ) = 5.713 to two significant figures is 5.7 .
Statement B evaluates as 2.6 + 3.1 = 5.7 . Therefore, the two statements are equal.
Part C
Statement A: Area of a rectangle with measured length = 2.536 and width = 1.4 .
Statement B: Area of a rectangle with measured length = 2.536 and width = 1.41 .
Since you are not told specific numbers of significant figures to round to, you must use the rules for multiplying numbers
while respecting significant figures. If you need a reminder, consult the hint.
Determine the correct relationship between the statements.
Hint 1. Significant figures and multiplication
When you multiply two numbers, the result should be rounded to the number of significant figures in the less
accurate of the two numbers. For instance, if you multiply 2.413 (four significant figures) times 3.81 (three significant
figures), the result should have three significant figures: . Similarly, ,
when significant figures are respected (i.e., 15.328646 rounded to one significant figure).
ANSWER:
Correct
Evaluate statement A as follows: (2.536 ) (1.4 ) = 3.5504 to two significant figures is 3.6 . Statement
B evaluates as (2.536 ) (1.41 ) = 3.57576 to three significant figures is 3.58 . Therefore, statement A
is greater than statement B.
Vector Components--Review
Learning Goal:
To introduce you to vectors and the use of sine and cosine for a triangle when resolving components.
Statement A is greater than Statement B.
Statement A is less than Statement B.
Statement A is equal to Statement B.
Statement A is greater than Statement B.
Statement A is less than Statement B.
Statement A is equal to Statement B.
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Vectors are an important part of the language of science, mathematics, and engineering. They are used to discuss multivariable
calculus, electrical circuits with oscillating currents, stress and strain in structures and materials, and flows of atmospheres and
fluids, and they have many other applications. Resolving a vector into components is a precursor to computing things with or
about a vector quantity. Because position, velocity, acceleration, force, momentum, and angular momentum are all vector
quantities, resolving vectors into components is the most important skill
required in a mechanics course.
shows the components of , and , along the x
and y
axes of
the coordinate system, respectively. The components of a vector
depend on the coordinate system's orientation, the key being the
angle between the vector and the coordinate axes, often designated
.
Part A
shows the standard way of measuring the angle. is measured to
the vector from
the x
axis, and counterclockwise is positive.
Express and in terms of the length of the vector and the angle , with the components separated by a
comma.
ANSWER:
, =
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Correct
In principle, you can determine the components of any
vector with these expressions. If lies in one of the other
quadrants of the plane, will be an angle larger than 90 degrees (or in radians) and and will
have the appropriate signs and values.
Unfortunately this way of representing , though mathematically correct, leads to equations that must be
simplified using trig identities such as
and
.
These must be used to reduce all trig functions present in your equations to either or . Unless you
perform this followup step flawlessly, you will fail to recoginze that
,
and your equations will not simplify so that you can progress further toward a solution. Therefore, it is best to
express all components in terms of either or , with between 0 and 90 degrees (or 0 and in
radians), and determine the signs of the trig functions by knowing in which quadrant the vector lies.
Part B
When you resolve a vector into components, the components must have the form
or . The signs
depend on which quadrant the vector lies in, and there will be one component with and the other with .
In real problems the optimal coordinate system is often rotated so that the x
axis is not horizontal. Furthermore, most
vectors will not lie in the first quadrant. To assign the sine and cosine correctly for vectors at arbitrary angles, you must
figure out which angle is and then properly reorient the definitional triangle.
As an example, consider the vector shown in labeled "tilted
axes," where you know the angle between and the y
axis.
Which of the various ways of orienting the definitional triangle
must be used to resolve into components in the tilted
coordinate system shown? (In the figures, the hypotenuse is blue
(long dashes), the side adjacent to is red (short dashes), and
the side opposite is yellow (solid).)
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Indicate the number of the figure with the correct orientation.
Hint 1. Recommended procedure for resolving a vector into components
First figure out the sines and cosines of , then figure out the signs from the quadrant the vector is in and write in the
signs.
Hint 2. Finding the trigonometric functions
Sine and cosine are defined according to the following convention, with the key lengths shown in green: The
hypotenuse (blue long dashes) has unit length, the side adjacent (red short dashes) to has length , and the
side opposite (yellow solid) has length .
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ANSWER:
Correct
Part C
Choose the correct procedure for determining the components of a vector in a given coordinate system from this list:
ANSWER:
Correct
Part D
1
2
3
4
Align the adjacent side of a right triangle with the vector and the hypotenuse along a coordinate direction with as the included angle.
Align the hypotenuse of a right triangle with the vector and an adjacent side along a coordinate direction with as the included angle.
Align the opposite side of a right triangle with the vector and the hypotenuse along a coordinate direction with as the included angle.
Align the hypotenuse of a right triangle with the vector and the opposite side along a coordinate direction with as the included angle.
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The space around a coordinate system is conventionally divided into four numbered quadrants
depending on the signs of
the x
and y
coordinates . Consider the following conditions:
A. , B. , C. , D. , Which of these lettered conditions are true in which the numbered
quadrants shown in ?
Write
the
answer in the following way: If A were true in the third quadrant, B in the second, C in the first, and D in the fourth,
enter "
3, 2, 1, 4
" as your response.
ANSWER:
Correct
Part E
Now find the components and of in the tilted coordinate system of Part B
.
Express your answer in terms of the length of the vector and the angle , with the components separated by a
comma.
ANSWER:
Correct
Exercise 1.22 - Enhanced - with Feedback
1,4,2,3
, =
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Part A
For the vectors and in the figure , use a scale drawing to
find the magnitude of the vector sum .
Express your answer in meters.
ANSWER:
Correct
Part B
Find the direction of the vector sum .
Express your answer in degrees.
ANSWER:
Correct
Part C
Find the magnitude of the vector difference .
Express your answer in meters.
ANSWER:
Correct
Part D
=
9.0
angle =
34
counterclockwise from -axis
=
22
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Find the direction of the vector difference .
Express your answer in degrees.
ANSWER:
Correct
Part E
Use your answers to find the magnitude of .
Express your answer in meters.
ANSWER:
Correct
Part F
Find the direction of .
Express your answer in degrees.
ANSWER:
Correct
Part G
Find the magnitude of .
Express your answer in meters.
ANSWER:
Correct
Part H
angle =
250
counterclockwise from -axis
| =
9.0
angle =
214
counterclockwise from -axis
=
22
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Find the direction of .
Express your answer in degrees.
ANSWER:
Correct
± Vector Addition and Subtraction
In general it is best to conceptualize vectors as arrows in space, and then to make calculations with them using their
components. (You must first specify a coordinate system in order to find the components of each arrow.) This problem gives you
some practice with the components.
Let vectors , , and . Calculate the following, and express your answers as
ordered triplets of values separated by commas.
Part A
ANSWER:
Correct
Part B
ANSWER:
Correct
Part C
ANSWER:
Correct
angle =
70
counterclockwise from -axis
=
3,-5,-4
=
-5,4,0
=
-6,4,3
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Part D
ANSWER:
Correct
Part E
ANSWER:
Correct
Part F
ANSWER:
Correct
Tracking a Plane
A radar station, located at the origin of xy
plane, as shown in , detects an
airplane coming straight at the station from the east. At first observation
(point A), the position of the airplane relative to the origin is . The
position vector has a magnitude of 360 and is located at exactly
40
above the horizon. The airplane is tracked for another 123
in the
vertical east-west plane for 5.0 , until it has passed directly over the
station and reached point B. The position of point B relative to the origin is
(the magnitude of is 880 ). The contact points are shown in the
diagram, where the x
axis represents the ground and the positive y
direction is upward.
Part A
Define the displacement of the airplane while the radar was tracking it: . What are the components of
?
=
-3,-2,-11
=
-11,14,8
=
17,-12,-6
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Express in terms of the unit vectors and , in meters.
Hint 1. How to approach the problem
Keep in mind that . According to the rules of vector addition and subtraction, the x
component of
is .
Hint 2. Finding the components of What are the components of in the x
and y
directions?
Express your answer in terms of the unit vectors and , in meters.
ANSWER:
Hint 3. Finding the components of What are the components of in the x
and y
directions?
Express your answer in terms of the unit vectors and , in meters.
ANSWER:
ANSWER:
Correct
± Vector Dot Product
Let vectors , , and .
Calculate the following:
Part A
Hint 1. Remember the dot product equation
If and , then
=
=
=
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.
ANSWER:
Correct
Part B
What is the angle between and ?
Express your answer using one significant figure.
Hint 1. Remember the definition of dot products
, where is the angle between and .
ANSWER:
Correct
Part C
ANSWER:
Correct
Part D
ANSWER:
Correct
Part E
Which of the following can be computed?
=
-10
= 2
=
30
=
30
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Hint 1. Dot product operator
The dot product operates only on two vectors. The dot product of a vector and a scalar is not defined.
ANSWER:
Correct
and are different vectors with lengths and respectively. Find the following:
Part F
Express your answer in terms of Hint 1. What is the angle between a vector and itself?
The angle between a vector and itself is 0.
Hint 2. Remember the definition of dot products
, where is the angle between and .
ANSWER:
Correct
Part G
If and are perpendicular,
Hint 1. What is the angle between perpendicular vectors?
The angle between vectors that are perpendicular is equal to radians or 90 degrees.
=
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ANSWER:
Correct
Part H
If and are parallel,
Express your answer in terms of and .
Hint 1. What is the angle between parallel vectors?
The angle between vectors that are parallel is equal to 0.
ANSWER:
Correct
Dimensions of Physical Quantities
Learning Goal:
To introduce the idea of physical dimensions and to learn how to find them.
Physical quantities are generally not purely numerical: They have a particular dimension
or combination of dimensions
associated with them. Thus, your height is not 74, but rather 74 inches, often expressed as 6 feet 2 inches. Although feet and
inches are different units
they have the same dimension
--
length
.
Part A
In classical mechanics there are three base dimensions. Length is one of them. What are the other two?
Hint 1. MKS system
The current system of units is called the International System (abbreviated SI from the French Système
International). In the past this system was called the mks system for its base units: meter, kilogram, and second.
What are the dimensions of these quantities?
ANSWER:
=
=
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There are three dimensions used in mechanics: length (
), mass (
), and time (
). A combination of these three dimensions
suffices to express any physical quantity, because when a new physical quantity is needed (e.g., velocity), it always obeys an
equation that permits it to be expressed in terms of the units used for these three dimensions. One then derives a unit to
measure the new physical quantity from that equation, and often its unit is given a special name. Such new dimensions are
called derived dimensions and the units they are measured in are called derived units.
For example, area has derived dimensions . (Note that "dimensions of variable " is symbolized as .) You can
find these dimensions by looking at the formula for the area of a square , where is the length of a side of the square.
Clearly . Plugging this into the equation gives .
Part B
Find the dimensions of volume.
Express your answer as powers of length (
), mass (
), and time (
).
Hint 1. Equation for volume
You have likely learned many formulas for the volume of various shapes in geometry. Any of these equations will give
you the dimensions for volume. You can find the dimensions most easily from the volume of a cube , where is the length of the edge of the cube.
ANSWER:
Correct
Part C
Find the dimensions of speed.
Express your answer as powers of length (
), mass (
), and time (
).
Hint 1. Equation for speed
acceleration and mass
acceleration and time
acceleration and charge
mass and time
mass and charge
time and charge
=
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Speed is defined in terms of distance and time as
.
Therefore, .
Hint 2. Familiar units for speed
You are probably accustomed to hearing speeds in miles per hour (or possibly kilometers per hour). Think about the
dimensions for miles and hours. If you divide the dimensions for miles by the dimensions for hours, you will have the
dimensions for speed.
ANSWER:
Correct
The dimensions of a quantity are not changed by addition or subtraction of another quantity with the same dimensions. This
means that , which comes from subtracting two speeds, has the same dimensions as speed.
It does not make physical sense to add or subtract two quanitites that have different dimensions, like length plus time. You can
add quantities that have different units, like miles per hour and kilometers per hour, as long as you convert both quantities to the
same set of units before you actually compute the sum. You can use this rule to check your answers to any physics problem you
work. If the answer involves the sum or difference of two quantities with different dimensions, then it must be incorrect.
This rule also ensures that the dimensions of any physical quantity will never
involve sums or differences of the base
dimensions. (As in the preceeding example, is not a valid dimension for a physical quantitiy.) A valid dimension will only
involve the product or ratio of powers of the base dimensions (e.g. ).
Part D
Find the dimensions of acceleration.
Express your answer as powers of length (
), mass (
), and time (
).
Hint 1. Equation for acceleration
In physics, acceleration is defined as the change in velocity in a certain time. This is shown by the equation
. The is a symbol that means "the change in."
ANSWER:
Correct
Consistency of Units
=
=
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In physics, every physical quantity is measured with respect to a unit
. Time is measured in seconds, length is measured in
meters, and mass is measured in kilograms. Knowing the units of physical quantities will help you solve problems in physics.
Part A
Gravity causes objects to be attracted to one another. This attraction keeps our feet firmly planted on the ground and causes
the moon to orbit the earth. The force of gravitational attraction is represented by the equation
,
where is the magnitude of the gravitational attraction on either body, and are the masses of the bodies, is the
distance between them, and is the gravitational constant. In SI units, the units of force are , the units of mass
are , and the units of distance are . For this equation to have consistent units, the units of must be which of the
following?
Hint 1. How to approach the problem
To solve this problem, we start with the equation
.
For each symbol whose units we know, we replace the symbol with those units. For example, we replace with
. We now solve this equation for .
ANSWER:
Correct
Part B
One consequence of Einstein's theory of special relativity is that mass is a form of energy. This mass-energy relationship is
perhaps the most famous of all physics equations:
,
where is mass, is the speed of the light, and is the energy. In SI units, the units of speed are . For the preceding
equation to have consistent units (the same units on both sides of the equation), the units of must be which of the
following?
Hint 1. How to approach the problem
To solve this problem, we start with the equation
.
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For each symbol whose units we know, we replace the symbol with those units. For example, we replace with .
We now solve this equation for .
ANSWER:
Correct
To solve the types of problems typified by these examples, we start with the given equation. For each symbol
whose units we know, we replace the symbol with those units. For example, we replace with . We now solve
this equation for the units of the unknown variable.
Measurements in SI Units
Familiarity with SI units will aid your study of physics and all other sciences.
Part A
What is the approximate height of the average adult in centimeters?
Hint 1. Converting between feet and centimeters
The distance from your elbow to your fingertips is typically about 50 .
ANSWER:
100 200 300
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Correct
If you're not familiar with metric units of length, you can use your body to develop intuition for them. The average
height of an adult is 5 6.4 . The distance from elbow to fingertips on the average adult is about 50 .
Ten (1 ) is about the width of this adult's little finger and 10 is about the width of the average hand.
Part B
Approximately what is the mass of the average adult in kilograms?
Hint 1. Converting between pounds and kilograms
Something that weighs 1 has a mass of about .
ANSWER:
Correct
Something that weighs 1 has a mass of about . This is a useful conversion to keep in mind!
Exercise 1.18 - Enhanced - with Feedback
Four astronauts are in a spherical space station.
Part A
If, as is typical, each of them breathes about 500 of air with each breath, approximately what volume of air (in cubic
meters) do these astronauts breathe in a year? Assume the typical respiratory rate for a healthy adult at rest is 15 breaths
per minute.
ANSWER:
80 500 1200
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Correct
Part B
What would the diameter (in meters) of the space station have to be to contain all this air?
ANSWER:
Correct
Components of Vectors
Shown is a 10 by 10 grid, with coordinate axes and .
The grid runs from -5 to 5 on both axes. Drawn on this grid are four
vectors, labeled through . This problem will ask you various
questions about these vectors. All answers should be in decimal
notation, unless otherwise specified.
Part A
What is the component of ?
Express your answer to the nearest integer.
Hint 1. How to derive the component
A component of a vector is its length (but with appropriate sign) along a particular coordinate axis, the axes being
specfied in advance. You are asked for the component of that lies along the axis, which is horizontal in this
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problem. Imagine two lines perpendicular to the axis running from the head (end with the arrow) and tail of down
to the axis. The length of the axis between the points where these lines intersect is the component of . In
this problem, the component is the coordinate at which the perpendicular from the head of the vector hits the
axis (because the tail of the vector is at the origin).
ANSWER:
Correct
Part B
What is the component of ?
Express your answer to the nearest integer.
ANSWER:
Correct
Part C
What is the component of ?
Express your answer to the nearest integer.
Hint 1. Consider the direction
Don't forget the sign.
ANSWER:
Correct
Part D
What is the component of ?
Express your answer to the nearest integer.
Hint 1. How to find the start and end points of the vector components
= 3
= 3
= -3
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A vector is defined only by its magnitude and direction. The starting point of the vector is of no consequence to its
definition. Therefore, you need to somehow eliminate the starting point from your answer. You can run two
perpendiculars to the axis, one from the head (end with the arrow) of , and another to the tail, with the component being the difference between coordinates of head and tail (negative if the tail is to the right of the
head). Another way is to imagine bringing the tail of to the origin, and then using the same procedure you used
before to find the components of and . This is equivalent to the previous method, but it might be easier to
visualize.
ANSWER:
Correct
The following questions will ask you to give both components of vectors using the ordered pairs method. In this method, the component is written first, followed by a comma, and then the component. For example, the components of would be written
3,3 in ordered pair notation.
The answers below are all integers, so estimate the components to the nearest whole number.
Part E
In ordered pair notation, write down the components of vector .
Express your answers to the nearest integer.
ANSWER:
Correct
Part F
In ordered pair notation, write down the components of vector .
Express your answers to the nearest integer.
ANSWER:
Correct
Part G
What is true about and ?
ANSWER:
= -2
, = 2,-3
, = 2,-3
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Exercise 1.27 - Enhanced - with Feedback
Vector has -component = +18.0 . makes an angle of 42.0 counterclockwise from the -axis.
Part A
What is the -component of ?
Express your answer with the appropriate units.
ANSWER:
Correct
Part B
What is the magnitude of ?
Express your answer with the appropriate units.
ANSWER:
Correct
Exercise 1.38 - Enhanced - with Feedback
You are given two vectors and . Let counterclockwise angles be positive.
They have different components and are not the same vectors.
They have the same components but are not the same vectors.
They are the same vectors.
= -16.2 = 24.2
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Part A
What angle , where , does make with the -axis?
Express your answer in degrees.
ANSWER:
Correct
Part B
What angle , where , does make with the -axis?
Express your answer in degrees.
ANSWER:
Correct
Part C
Vector is the sum of and , so . What angle , where , does make with the
-axis?
Express your answer in degrees.
ANSWER:
Correct
Exercise 1.40 - Enhanced - with Feedback
Part A
Given two vectors and , find the scalar product of the two vectors and .
ANSWER:
= 121
= 21.8
= 74.1
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Part B
Find the angle between these two vectors.
Express your answer in degrees.
ANSWER:
Correct
Vector Cross Product
Let vectors , , and .
Calculate the following, expressing your answers as ordered triples (three comma-separated numbers).
Part A
Hint 1. The cross product
If and , then
.
ANSWER:
Correct
Part B
ANSWER:
=
7.52
= 80.7
=
4,5,-17
=
-4,-5,17
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Correct
Part C
ANSWER:
Correct
Part D
ANSWER:
Correct
Part E
ANSWER:
Correct
and are different vectors with lengths and respectively. Find the following, expressing your answers in terms of
given quantities.
Part F
If and are perpendicular,
Hint 1. What is the angle between perpendicular vectors?
The angle between vectors that are perpendicular is equal to radians or 90 degrees.
Hint 2. Magnitude of the cross product
, where is the angle between and .
ANSWER:
=
24,30,-102
=
15,5,5
=
55
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Part G
If and are parallel,
Hint 1. What is the angle between two parallel vectors?
The angle between vectors that are parallel is equal to 0.
ANSWER:
Correct
Problem 1.50 - Enhanced - with Feedback
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 in 100,000 years. (The large number of significant figures given for the frequency simply
illustrates the remarkable accuracy to which it has been measured.)
Part A
What is the time for one cycle of the radio wave?
Express your answer to three significant figures and include the appropriate units.
ANSWER:
Correct
Part B
How many cycles occur in 1.1 ?
ANSWER:
=
=
0
=
7.04×10
−10
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Part C
How many cycles would have occurred during the age of the earth, which is estimated to be years?
ANSWER:
Correct
Part D
By how many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth?
Express your answer in seconds.
ANSWER:
Correct
Problem 1.61 - Enhanced - with Feedback
A patient with a dislocated shoulder is put into a traction apparatus as shown in . The pulls and have equal magnitudes and
must combine to produce an outward traction force of 11.7 on the
patient's arm.
=
5.6×10
12
cycles
=
2.1×10
26
cycles
= 4.6×10
4
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Part A
How large should these pulls be?
Express your answer with the appropriate units.
ANSWER:
Correct
Problem 1.81 - Enhanced - with Feedback
Vector has magnitude 13.0 and vector has magnitude 13.0 . The scalar product is 117 .
Part A
What is the magnitude of the vector product between these two vectors?
Express your answer with the appropriate units.
ANSWER:
Correct
Key Example Variation Problem 1.10
Be sure to review Examples 1.9 and 1.10 (Section 1.10) before attempting these problems.
VP 1.10.1
Part A
Vector has magnitude 5.00 and is at an angle of 36.9
south of east. Vector has magnitude 6.40 and is at an angle of
20.0
west of north. Choose the positive -direction to the east and the positive -direction to the north. Find the
components of .
Enter the and components of the vector separated by a comma.
ANSWER:
= 6.90 = 122
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Part B
Find the components of .
Enter the and components of the vector separated by a comma.
ANSWER:
Correct
Part C
Calculate the scalar product .
ANSWER:
Correct
VP 1.10.2
Part D
Vector has magnitude 6.50 and is at an angle of 55.0
measured counterclockwise from the -axis toward the
-axis. Vector has components and . Calculate the scalar product .
ANSWER:
Correct
Part E
Find the angle between the vectors and .
Express your answer in degrees to three significant figures.
ANSWER:
, = 4.00,-3.00
, = -2.19,6.01
=
-26.8
=
-26.8
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VP 1.10.3
Part F
Vector has components , , and . Vector has components , ,
and . Find the angle between the two vectors.
Express your answer in degrees to two significant figures.
ANSWER:
Correct
VP 1.10.4
Part G
If a force acts on an object as that object moves through a displacement , the work
done by that force equals the scalar
product of and : . A certain object moves through displacement . As it moves
it is acted on by force , which has -component = -12.0 (1 = 1 newton is the SI unit of force). The work done by
this force is 26.0 = 26.0 (1 = 1 joule = 1 newton-meter is the SI unit of work). Find the -component of .
Express your answer with the appropriate units.
ANSWER:
Correct
Part H
Find the angle between and .
Express your answer in degrees.
ANSWER:
Correct
= 115
= 91
= 14.8 = 77.7
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Key Example Variation Problem 1.7
Be sure to review Example 1.7 (Section 1.8) before attempting these problems.
Choose the -axis as east and the -axis as north. Consider the following vectors , , and :
: 72.4 , 32.0
east of north
: 57.3 , 36.0
south of west
: 17.8 due south
VP 1.7.1
Part A
If a fourth vector is added to , the result is zero: . Find the magnitude of .
Express your answer with the appropriate units.
ANSWER:
Correct
Part B
State the direction of in terms of an angle measured counterclockwise from the positive -axis.
Express your answer in degrees.
ANSWER:
Correct
Part C
State in which quadrant the angle from part B lies.
ANSWER:
= 12.7 = -51.1
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VP 1.7.2
Part D
Calculate the magnitude of the vector . (
Hint
: The components of are just the negatives of the
components of .)
Express your answer with the appropriate units.
ANSWER:
Correct
Part E
State the direction of in terms of an angle measured counterclockwise from the positive -axis.
Express your answer in degrees.
ANSWER:
Correct
Part F
State in which quadrant the angle from part E lies.
ANSWER:
is in the second quadrant.
is in the first quadrant.
is in the third quadrant.
is in the fourth quadrant.
= 115 = 42.4
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VP 1.7.3
Part G
Find the components of the vector .
Express the and components of the vector in meters separated by a comma.
ANSWER:
Correct
Part H
Find the magnitude of .
Express your answer with the appropriate units.
ANSWER:
Correct
Part I
State the direction of in terms of an angle measured counterclockwise from the positive -axis.
Express your answer in degrees.
ANSWER:
Correct
is in the second quadrant.
is in the third quadrant.
is in the fourth quadrant.
is in the first quadrant.
, = -8.0,-7.9
= 11.2 = 225
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Part J
State in which quadrant the angle from part I lies.
ANSWER:
Correct
VP 1.7.4
Part K
A hiker undergoes the displacement shown in the inroduction. The hiker then undergoes a second displacement such
that she ends up 38.0 from her starting point, in a direction from her starting point that is 37.0
west of north. Find the
magnitude of this second displacement .
Express your answer with the appropriate units.
ANSWER:
Correct
Part L
State the direction of in terms of an angle measured counterclockwise from the positive -axis.
Express your answer in degrees.
ANSWER:
Correct
Part M
State in which quadrant the angle from part L lies.
ANSWER:
is in the first quadrant.
is in the second quadrant.
is in the fourth quadrant.
is in the third quadrant.
= 68.7 = 206.9
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Exercise 1.8
Part A
The recommended daily allowance (RDA) of the trace metal magnesium is 410 for males. Express this quantity in
.
Express your answer in micrograms per day.
ANSWER:
Correct
Part B
For adults, the RDA of the amino acid lysine is 12 per of body weight. How many grams per day should a 70 adult receive?
Express your answer in grams per day.
ANSWER:
Correct
Part C
A typical multivitamin tablet can contain 2.0 of vitamin B
2
(riboflavin), and the RDA is 0.0030 . How many such
tablets should a person take each day to get the proper amount of this vitamin, assuming that he gets none from any other
sources?
Express your answer as an integer.
is in the first quadrant.
is in the third quadrant.
is in the second quadrant.
is in the fourth quadrant.
4.1×10
5
0.84
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ANSWER:
Correct
Part D
The RDA for the trace element selenium is 0.000070 . Express this dose in .
Express your answer in milligrams per day.
ANSWER:
Correct
Bridging Problem: Vectors on the Roof
An air-conditioning unit is fastened to a roof that slopes at an angle of = 37 above the horizontal . Its weight is a force on
the air conditioner that is directed vertically downward. In order that
the unit not crush the roof tiles, the component of the unit’s weight
perpendicular to the roof cannot exceed 485 . (One newton, or 1 ,
is the SI unit of force. It is equal to 0.2248 .) (a) What is the
maximum allowed weight of the unit? (b) If the fasteners fail, the unit
slides = 1.90 along the roof before it comes to a halt against a
ledge. How much work does the weight force do on the unit during its
slide if the unit has the weight calculated in part (a)? The work done
by a force on an object that undergoes a displacement is
.
IDENTIFY
and SET UP
Part A
This problem involves vectors and components. Assuming the direction of the axes sketched in the diagram, which quantity
is known?
Hint 1. Safe condition
2 tablets
7.0×10
−2
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Although the magnitude of the critical weight is not given directly, there is a safe condition that gives us an idea of
one of its component. Also, note that the direction of the axes were chosen for conveniece; as long as they are
mutually perpendicular, they don't have to always be horizontal and vertical.
ANSWER:
Correct
You have correctly identified that we know the -component of vector .
Part B
Which aspect of the weight vector (magnitude, direction, or particular components) represent the target variable for question
(a)?
Hint 1. Phrasing of the problem
The problem asks for the "maximum allowed weight of the unit".
ANSWER:
Correct
You have correctly identified that we want to solve for the magnitude of vector .
Part C
Which aspect(s) of the weight vector must you know directly to answer question (b)?
Check all that apply.
Hint 1. Definition of the scalar product
The magnitude of vector The -component of vector The -component of vector The -component of vector The magnitude of vector The direction of vector The -component of vector
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The scalar product of vectors and is .
ANSWER:
Correct
You have correctly identified what you need in order to obtain the scalar product.
Part D
Choose the two equations that represent the and -component of the weight force.
Hint 1. Choosing the correct angle
Check your sketch and the figure on the left to see which value of the angle you should use.
ANSWER:
Correct
You have correctly identified the relationships between the magnitude, direction, and components of the weight
vector.
EXECUTE
Part E
The magnitude of vector The direction of vector The -component of vector The -component of vector
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Use the relationship between the magnitude and direction of a vector and its components to solve for the target variable in
question (a).
Express your answer with the appropriate units.
Hint 1. Magnitude of the vector
You will have to find the magnitude of the vector. Irrespective of the vector direction, its magnitude is non-negative,
by definition.
Hint 2. Significant figures
The number of significant figures expected from your answer depends on the number of significant figures in the
input data.
ANSWER:
Correct
You now know the value of the magnitude of vector .
Part F
Use the definition of the scalar product to solve for the target variable in question (b).
Express your answer with the appropriate units.
Hint 1. Formulas for the scalar product
The scalar product of vectors and , in the two-dimensional case, can be calculated either as ,
where is the angle between the two vectors; or as .
Hint 2. Direction of the vector Vector will be parallel to the -axis. This means one of the components of vector will be zero.
ANSWER:
Correct
You successfully calculated the value of the scalar product.
EVALUATE
Part G
Did your answer to question (a) include a vector component whose absolute value is greater than the magnitude of the
vector?
= 610 = 690
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ANSWER:
Correct
The answer to question (a) is really greater than the value of 435 given in the statement. But the point is that
the given value is a component and the found value is the magnitide of the vector, not vice versa.
Part H
Is it possible for the absolute value of a vector's component to be greater than the magnitude of the vector?
ANSWER:
Correct
Part I
There are two ways to find the scalar product of two vectors, one of which you used to answer question (b). Check your
answer by repeating the calculation, using the other way. Do you get the same answer?
ANSWER:
Correct
For the steps and strategies involved in solving this problem, you may view the Video Tutor Solution
.
Score Summary:
Your score on this assignment is 95.7%.
You received 21.06 out of a possible total of 22 points.
no
yes
The absolute value of the vector component cannot be greater than the magnitude of the vector.
The absolute value of the vector component can be greater than the magnitude of the vector.
The answers are different
The answer is the same
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