Significant Figures Conceptual QuestionIn 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: 2.567km, to two significant figures. Statement B:
2.567km, to three significant figures.Determine the correct
relationship between the statements.Hint 1.Rounding and significant
figuresRounding 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 figures3.454
Three significant figures3.45
Two significant figures3.5
One significant figure3
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:Statement A isgreater than
less than
equal to
Statement B.CorrectPart B Statement A: (2.567km+ 3.146km), to
two significant figures. Statement B: (2.567km, to two significant
figures) + (3.146km, to two significant figures).Determine the
correct relationship between the statements.ANSWER:Statement A
isgreater than
less than
equal to
Statement B.CorrectEvaluate statement A as follows:
(2.567km+3.146km) =5.713kmto two significant figures is5.7km.
Statement B evaluates as2.6km+3.1km=5.7km. Therefore, the two
statements are equal.Part C Statement A: Area of a rectangle with
measured length = 2.536mand width = 1.4m. Statement B: Area of a
rectangle with measured length = 2.536mand width = 1.41m.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 multiplicationWhen 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:2.4133.81=9.19. Similarly,27.664323=20, when significant
figures are respected (i.e., 15.328646 rounded to one significant
figure).ANSWER:Statement A isgreater than
less than
equal to
Statement B.CorrectEvaluate statement A as follows: (2.536m)
(1.4m) =3.5504m2to two significant figures is3.6m2. Statement B
evaluates as (2.536m) (1.41m) =3.57576m2to three significant
figures is3.58m2. Therefore, statement A is greater than statement
B.Vector Components--ReviewLearning Goal:To introduce you to
vectors and the use of sine and cosine for a triangle when
resolving components.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 isthe most important
skillrequired in a mechanics course.The figureshows the components
ofF,FxandFy, along thexandyaxes 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 AThe figureshows the
standard way of measuring the angle.is measuredtothe
vectorfromthexaxis, and counterclockwise is
positive.ExpressFxandFyin terms of the length of the vectorFand the
angle, with the components separated by a
comma.ANSWER:Fx,Fy=Fcos(),Fsin()
CorrectIn principle, you can determine the components
ofanyvector with these expressions. IfFlies in one of the other
quadrants of the plane,will be an angle larger than 90 degrees
(or/2in radians) andcos()andsin()will have the appropriate signs
and values.Unfortunately this way of representingF, though
mathematically correct, leads to equations that must be simplified
using trig identities such assin(180+)=sin()andcos(90+)=sin().These
must be used to reduce all trig functions present in your equations
to eithersin()orcos().Unless you perform this followup step
flawlessly, you will fail to recoginze thatsin(180+)+cos(270)=0,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 eithersin()orcos(), withbetween 0 and 90 degrees (or 0
and/2in radians), and determine the signs of the trig functions by
knowing in which quadrant the vector lies.Part BWhen you resolve a
vectorFinto components, the componentsmust have the
form|F|cos()or|F|sin(). The signs depend on which quadrant the
vector lies in, and there will be one component withsin()and the
other withcos().In real problems the optimal coordinate system is
often rotated so that thexaxis 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 isand then properly reorient the definitional
triangle.As an example, consider the vectorNshown in the
diagramlabeled "tilted axes," where you know the anglebetweenNand
theyaxis.Which of the various ways of orienting the definitional
triangle must be used to resolveNinto components in the tilted
coordinate system shown? (In the figures, the hypotenuse is orange,
the side adjacent tois red, and the side opposite is
yellow.)Indicate the number of the figure with the correct
orientation.Hint 1.Recommended procedure for resolving a vector
into componentsFirst 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 functionsSine and
cosine are defined according to the following convention, with the
key lengths shown in green: The hypotenuse has unit length, the
side adjacent tohas lengthcos(), and the side opposite has
lengthsin(). The colors are chosen to remind you that the vector
sum of the two orthogonal sides is the vector whose magnitude is
the hypotenuse;red + yellow = orange.
ANSWER:1
2
3
4
CorrectPart CChoose the correct procedure for determining the
components of a vector in a given coordinate system from this
list:ANSWER:Align the adjacent side of a right triangle with the
vector and the hypotenuse along a coordinate direction withas the
included angle.
Align the hypotenuse of a right triangle with the vector and an
adjacent side along a coordinate direction withas the included
angle.
Align the opposite side of a right triangle with the vector and
the hypotenuse along a coordinate direction withas the included
angle.
Align the hypotenuse of a right triangle with the vector and the
opposite side along a coordinate direction withas the included
angle.
CorrectPart DThe space around a coordinate system is
conventionally divided into four numberedquadrantsdepending on the
signs of thexandycoordinates. Consider the following
conditions:
A. x>0,y>0B. x>0,y