Ch 01 HW
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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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Ch 01 HW
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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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