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