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 km, to two significant figures. Statement B: 2.567 km, 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: Statement A is
greater than less than equal to Statement B.
Correct Part B
Statement A: (2.567 km + 3.146 km), to two significant figures. Statement B: (2.567 km, to two significant figures) + (3.146 km, to two significant figures).
Determine the correct relationship between the statements. ANSWER: Statement A is
greater than
less than equal to Statement B.
Correct Evaluate statement A as follows: (2.567 km + 3.146 km ) = 5.713 km to two significant figures is 5.7 km . Statement B evaluates as 2.6 km + 3.1 km = 5.7 km . Therefore, the two statements are equal.
Part C Statement A: Area of a rectangle with measured length = 2.536 m and width = 1.4 m. Statement B: Area of a rectangle with measured length = 2.536 m and width = 1.41 m. 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: 2.413×3.81=9.19. Similarly, 2×7.664323=20, when significant figures are respected (i.e., 15.328646 rounded to one significant figure). ANSWER: Statement A is
greater than less than equal to Statement B.
Correct Evaluate statement A as follows: (2.536 m ) (1.4 m ) = 3.5504 m2 to two significant figures is 3.6 m2 . Statement B evaluates as (2.536 m ) (1.41 m ) = 3.57576 m2 to three significant figures is 3.58 m2 . 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. 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.
The figure shows the components of F , Fx and Fy, 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 θ.
Part A
The figure shows the standard way of measuring the angle. θ is measured to the vector from the x axis, and counterclockwise is positive. Express Fx and Fy in terms of the length of the vector F and the angle θ, with the components separated by a comma. ANSWER:
Fx, Fy = Fcos(θ),Fsin(θ) Correct
In principle, you can determine the components of any vector with these expressions. If F lies in one of the other quadrants of the plane, θ will be an angle larger than 90 degrees (or π/2 in radians) and cos(θ) and sin(θ) will have the appropriate signs and values. Unfortunately this way of representing F , though mathematically correct, leads to equations that must be simplified using trig identities such as
sin(180∘+ϕ)=−sin(ϕ) and
cos(90∘+ϕ)=−sin(ϕ). These must be used to reduce all trig functions present in your equations to either sin(ϕ) or cos(ϕ). Unless you perform this followup step flawlessly, you will fail to recoginze that sin(180∘+ϕ)+cos(270∘−ϕ)=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 either sin(ϕ) or cos(ϕ), with ϕ between 0 and 90 degrees (or 0 and π/2 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 F into components, the components must have the form |F |cos(θ) or |F |sin(θ). The signs depend on which quadrant the vector lies in, and there will be one component with sin(θ) and the other with cos(θ). 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 θ and then properly reorient the definitional triangle. As an example, consider the vector N shown in the
diagram labeled "tilted axes," where you know the angle θbetween N and the y axis. Which of the various ways of orienting the definitional triangle must be used to resolve N into components in the tilted coordinate system shown? (In the figures, the hypotenuse is orange, the side adjacent to θ is 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 components First figure out the sines and cosines of θ, 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 has unit length, the side adjacent to θ has length cos(θ), and the side opposite has length sin(θ). 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 Correct Part C Choose 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 with θ as the included angle. Align the hypotenuse of a right triangle with the vector and an adjacent side along a coordinate direction with θ as the included angle. Align the opposite side of a right triangle with the vector and the hypotenuse along a coordinate direction with θ as the included angle. Align the hypotenuse of a right triangle with the vector and the opposite side along a coordinate direction with θ as the included angle. Correct Part D
The space around a coordinate system is conventionally divided into four numbered quadrants depending
on the signs of the x and y coordinates the following conditions:
. Consider
A. x>0, y>0 B. x>0, y<0 C. x<0, y>0 D. x<0, y<0 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: 1,4,2,3
Correct Part E
Now find the components Nx and Ny of N
in the tilted coordinate system of Part B.
Express your answer in terms of the length of the vector separated by a comma.
N and the angle θ, with the components
ANSWER:
Nx, Ny = −{|N|sin(θ)},|N|cos(θ) Correct
Exercise 1.12 Part A The recommended daily allowance (RDA) of the trace metal magnesium is 410 Express this quantity in μg/day.
mg/day for males.
Express your answer using two significant figures. ANSWER:
4.1×105 μg/day Correct Part B For adults, the RDA of the amino acid lysine is 12 should a 80kg adult receive?
mg per kg of body weight. How many grams per day
Express your answer using two significant figures. ANSWER:
0.96 g/day Correct Part C A typical multivitamin tablet can contain 2.0 mg of vitamin B2 (riboflavin), and the RDA is 0.0030 g/day. 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. ANSWER:
2 tablets Correct Part D The RDA for the trace element selenium is 0.000070
g/day. Express this dose in mg/day.
Express your answer using two significant figures. ANSWER:
7.0×10−2 mg/day Correct
Exercise 1.7 Part A How many years older will you be 1.10 billion seconds from now? (Assume a 365-day year.) ANSWER:
34.9 years Correct
Exercise 1.24 You are using water to dilute small amounts of chemicals in the laboratory, drop by drop.
Part A How many drops of water are in a 1.0 L bottle? (Hint: Start by estimating the diameter of a drop of water.) ANSWER:
∼10 drops of water ∼102 drops of water ∼105 drops of water ∼108 drops of water 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 A =(1,0,−3), B =(−2,5,1), and C =(3,1,1). Calculate the following, and express your answers as ordered triplets of values separated by commas.
Part A ANSWER:
A −B = 3,-5,-4 Correct Part B ANSWER:
B −C = -5,4,0
Correct Part C ANSWER:
−A +B −C = -6,4,3 Correct Part D ANSWER:
3A −2C = -3,-2,-11 Correct Part E ANSWER:
−2A +3B −C = -11,14,8 Correct Part F ANSWER:
2A −3(B −C ) = 17,-12,-6 Correct
Tracking a Plane A radar station, located at the origin of xz plane, as shown in the
figure , 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 R A. The position vector R A has a magnitude of 360 m and is located at exactly 40 degrees above the horizon. The airplane is tracked for another 123 degrees in the vertical east-west plane for 5.0 s, until it has passed directly over the station and reached point B. The position of point B relative to the origin
is R B (the magnitude of R B is 880 m). The contact points are shown in the diagram, where the x axis represents the ground and the positive z direction is upward.
Part A Define the displacement of the airplane while the radar was tracking it: components of R BA?
R
BA=R B−R A. What are the
Express R BA in meters as an ordered pair, separating the x and z components with a comma, to two significant figures.
Hint 1. How to approach the problem Keep in mind that R BA=R B−R A.According to the rules of vector addition and subtraction, the x component of R BA is (R BA)x=(R B)x−(R A)x.
Hint 2. Finding the components of R A What are the components of R A in the i^ and k^ directions? Express your answer in meters as an ordered pair, separating the x and z values with commas, to three significant figures. ANSWER:
RAx, RAz = 276,231
m
Hint 3. Finding the components of R B What are the components of R B in the i^ and k^ directions? Express your answer in meters as an ordered pair, separating the x and z components with a comma, to three significant figures. ANSWER:
RBx, RBz = -842,257
m
ANSWER:
R BA = -1100,26 Correct
m
± Vector Dot Product Let vectors A =(2,1,−4), B Calculate the following:
=(−3,0,1), and C =(−1,−1,2).
Part A Hint 1. Remember the dot product equation If M
=(Mx,My,Mz) and N =(Nx,Ny,Nz), then M ⋅N =MxNx+MyNy+MzNz.
ANSWER:
A ⋅B = -10 Correct Part B What is the angle θAB between A
and B ?
Express your answer using one significant figure.
Hint 1. Remember the definition of dot products
A ⋅B =|A ||B |cos(θ), where θ is the angle between A and B . ANSWER:
θAB = 2 radians Correct Part C ANSWER:
2B ⋅3C = 30 Correct Part D ANSWER:
2(B ⋅3C ) = 30 Correct Part E Which of the following can be computed?
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:
A ⋅B ⋅C A ⋅(B ⋅C ) A ⋅(B +C ) 3⋅A Correct
V
1 and
V
2 are different vectors with lengths
V1 and V2 respectively. Find the following:
Part F Express your answer in terms of V1
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
A ⋅B =|A ||B |cos(θ), where θ is the angle between A and B . ANSWER:
V 1⋅V 1 = V12 Correct Part G If V 1 and V 2 are perpendicular,
Hint 1. What is the angle between perpendicular vectors? The angle between vectors that are perpendicular is equal to π/2 radians or 90 degrees. ANSWER:
V 1⋅V 2 = 0 Correct Part H If V 1 and V 2 are parallel, Express your answer in terms of V1 and V2.
Hint 1. What is the angle between parallel vectors? The angle between vectors that are parallel is equal to 0. ANSWER:
V 1⋅V 2 = V1V2 Correct
Finding the Cross Product
The figure shows two vectors
You are given that T
T and U separated by an angle θTU.
=(3,1,0), U =(2,4,0), and T ×U =V .
Part A Express V
as an ordered triplet of values, separated by commas.
ANSWER:
V = 0,0,10 Correct Part B Find the magnitude of V . ANSWER:
|V | = 10 Correct Part C Find the sine of the angle between T
and U .
ANSWER:
sin(θTU) = 0.707 Correct
Exercise 1.40 In each case, find the x- and y-components of vector A .
Part A
A = 3.40 i^− 5.90 j^
Enter your answers numerically separated by a comma. Express your answers using three significant figures. ANSWER:
Ax, Ay = 3.40,-5.90 Correct Part B
A = 18.2 j^− 7.91 i^ Enter your answers numerically separated by a comma. Express your answers using three significant figures. ANSWER:
Ax, Ay = -7.91,18.2 Correct Part C
A =− 14.0 i^+ 22.4 j^ Enter your answers numerically separated by a comma. Express your answers using three significant figures. ANSWER:
Ax, Ay = -14.0,22.4 Correct Part D
A =5.0B^, where B^= 4 i^− 4 j^ Enter your answers numerically separated by a comma. Express your answers using one significant figure. ANSWER:
Ax, Ay = 20,-20 Correct
Problem 1.81 While following a treasure map, you start at an old oak tree. You first walk 825 m directly south, then turn and walk 1.25 km at 30.0 ∘ west of north, and finally walk 1.00 km at 40.0 ∘ north of east, where you find the treasure: a biography of Isaac Newton!
Part A To return to the old oak tree, in what direction should you head ? Use components to solve this problem. ANSWER:
θ = 8.90 Correct
∘
west of south
Part B To return to the old oak tree, how far will you walk? Use components to solve this problem. ANSWER:
D = 911 m Correct Score Summary: Your score on this assignment is 101%. You received 11.06 out of a possible total of 11 points.
