Assignment 1.pdf

2/16/2015 Assignment 1

https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=3473134 1/17

Assignment 1Due: 12:32pm on Tuesday, February 17, 2015

You will receive no credit for items you complete after the assignment is due. Grading Policy

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 dimensionsassociated with them. Thus, your height is not 74, but rather 74 inches, often expressed as 6 feet 2 inches. Althoughfeet and inches are different units they have the same dimensionlength.

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 SystèmeInternational). In the past this system was called the mks system for its base units: meter, kilogram, andsecond. What are the dimensions of these quantities?

ANSWER:

Correct

There are three dimensions used in mechanics: length ( ), mass ( ), and time ( ). A combination of these threedimensions 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. Onethen 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 aside of the square. Clearly . Plugging this into the equation gives .

acceleration and mass

acceleration and time

acceleration and charge

mass and time

mass and charge

time and charge

l m t

A [A] = l2 x [x]A = s2 s

[s] = l [A] = [s =]2l2



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 theseequations will give you the dimensions for volume. You can find the dimensions most easily from thevolume of a cube , where is the length of the edge of the cube.

ANSWER:

Correct

Part C

Find the dimensions of speed.


Hint 1. Equation for speed

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). Thinkabout the dimensions for miles and hours. If you divide the dimensions for miles by the dimensions forhours, 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.

[V ]

l m t

V = e3 e

= [V ] l3

[v]

l m t

v d t

v = dt

[v] = [d]/[t]

= [v] lt

Δv



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 convertboth quantities to the same set of units before you actually compute the sum. You can use this rule to check youranswers to any physics problem you work. If the answer involves the sum or difference of two quantities with differentdimensions, then it must be incorrect.

This rule also ensures that the dimensions of any physical quantity will never involve sums or differences of the basedimensions. (As in the preceeding example, is not a valid dimension for a physical quantitiy.) A valid dimensionwill only involve the product or ratio of powers of the base dimensions (e.g. ).

Part D

Find the dimensions of acceleration.


Hint 1. Equation for acceleration

In physics, acceleration is defined as the change in velocity in a certain time. This is shown by theequation . The is a symbol that means "the change in."

ANSWER:

Correct

Geometric vs Componentwise Vector Addition

Learning Goal:

To understand that adding vectors by using geometry and by using components gives the same result, and thatmanipulating vectors with components is much easier.

Vectors may be manipulated either geometrically or using components. In this problem we consider the addition of twovectors using both of these two methods.

The vectors and have lengths and , respectively, and makes an angle from the direction of .

Vector addition using geometry

Vector addition using geometry is accomplished by putting the tail of one vector (in this case ) on the tip of the other () and using the laws of plane geometry to find the length , and angle , of the resultant (or sum) vector,

:

1. ,

2.

l + tm2/3 l2 t−2

[a]

l m t

aa = Δv/Δt Δ

= [a] l

t2

A B A B B θ A

B

A C ϕ

= +C A B

C = + − 2AB cos(c)A2 B2− −−−−−−−−−−−−−−−−√

ϕ = ( ).sin−1 B sin(c)C



Vector addition using components

Vector addition using components requires the choice of a coordinate system. In this problem, the x axis is chosenalong the direction of . Then the x and y components of are and respectively. This means that thex and y components of are given by

3. ,4. .

Part A

Which of the following sets of conditions, if true, would show that the expressions 1 and 2 above define the samevector as expressions 3 and 4?

Check all that apply.

ANSWER:

A B B cos(θ) B sin(θ)

C

= A + B cos(θ)Cx

= B sin(θ)Cy

C



Correct

To show that the two pairs of expressions (for and and for and ) define the same vector youcan show that any of the sets of conditions listed above are met except

They give the same length and x component for .They give the same length and y component for .

If you consider just the first set of conditions, showing that the two sets of expressions have the same lengthand x component will imply that the y component has the correct magnitude. However, there is no way toknow the sign (i.e., direction) of the y component. To show that the pairs of expressions given above definethe same vector , we would need to show that they give the same length and the same x and ycomponents. We will do this in the questions that follow.

Part B

We begin by investigating whether the lengths are the same.Find the length of the vector starting from the components given in Equations 3 and 4.

Express in terms of , , and .

Hint 1. Apply the Pythagorean theorem

You are given the x and y components in Equations 3 and 4. Simply square these, add them, and take thesquare root (i.e., apply the Pythagorean theorem) to find the length of vector .

ANSWER:

Correct

Part C

The following reasons might explain why the equation for the length just obtained using components is the same asthe answer obtained using geometry (Equation 1 above):

The two pairs of expressions give the same length and direction for .

The two pairs of expressions give the same length and x component for .

The two pairs of expressions give the same direction and x component for .

The two pairs of expressions give the same length and y component for .

The two pairs of expressions give the same direction and y component for .

The two pairs of expressions give the same x and y components for .

C

C

C

C

C

C

C ϕ Cx Cy C

C

C

C

C

C A B θ

C

= C + + 2ABcosθA2 B2− −−−−−−−−−−−−−−−√



1. and are supplementary angles, that is, .2. The cosine function satisfies .3. Cosine is an even function of its argument, so the extra negative sign in one expression does not

matter.Which of these reasons is/are necessary to show that

?

ANSWER:

Correct

Having shown that the length of obtained using either geometrical addition OR componentwise addition are the same,all that remains is to show that any one of the other conditions from part A is satisfied. In the last two parts of theproblem, we'll show that the y component of determined geometrically is equal to that given above.

Part D

We begin by finding the y component of from its length and the angle it makes with the x axis, that is, fromgeometry.

Express the y component of in terms of and .

ANSWER:

Correct

Part E

Now express the y component of just found by using the geometrical approach in terms of rather than .

Express in terms of , and .

Hint 1. Law of sines

c θ θ + c = πcos(α) = − cos(π − α)

=+ − 2AB cos(c)A2 B2− −−−−−−−−−−−−−−−−√ + + 2AB cos(θ)A2 B2

− −−−−−−−−−−−−−−−−√

1 only

2 only

3 only

1 and 2

2 and 3

1 and 3

1 and 2 and 3

C

C

C

C C ϕ

= Cy Csinϕ

C θ ϕ

Cy A B θ

sin (ϕ) sin (c)



The law of sines can be used to relate and .

Express in terms of , , and .

ANSWER:

Hint 2. Supplementary angles

Angles and are supplementary, which means that . What is the relationship of and ?

ANSWER:

Correct

This is the same as Equation 4 above, obtained using components. Thinking in terms of components, theresult is fairly obvious: The y component of is equal to the the y component of since has no ycomponent in the chosen coordinate system.

At this point you have actually shown that the two vectors are equal by showing that the overall lengths areequal, and also that the y components are equal. You would only have to argue that the x component ispositive to complete the proof of equality.

Nevertheless, we are asking you to complete the algebra to show that the x components are equal as well.

Part F

Now find the x component of .

Express your answer in terms of and only.

ANSWER:

Correct

± Vector Addition

Consider the following three vectors:

,

sin (ϕ) sin (c)

sin (ϕ) sin (c) C B

= sin (ϕ)

B

sin(c)C

θ c θ + c = π sin(x)sin(π − x)

= Cy Bsinθ

C B A

C

C ϕ

= Cx Ccosϕ

= (2, −1, 1)A

= (3, 0, 5)



,and

.Calculate the following combinations. Express your answers as ordered triplets [e.g., ].

Part A

Hint 1. How to add vectors

Add vectors by adding the x components, y components, and z components individually.

ANSWER:

Correct

Part B

ANSWER:

Correct

Part C

ANSWER:

Correct

Part D

Hint 1. Remember the order of precedence

means multiply the vector by the constant 3, which you can do by multiplying each component by 3separately. Follow normal rules of mathematical precedence; that is, multiply before adding vectors.

ANSWER:

= (3, 0, 5)B

= (1, 4, −2)C (9, 4, −2)

= 5,1,6+A B

= 4,4,3+B C

= 6,3,4+ +A B C

3A A



Correct

Part E

ANSWER:

Correct

Part F

ANSWER:

Correct

Vector Cross Product

Let vectors , , and .Calculate the following, expressing your answers as ordered triples (three commaseparated numbers).

Part A

Hint 1. The cross product

If and , then

.

ANSWER:

Correct

= 8,5,13 + 2A C

= 14,2,152 + 3 +A B C

= 16,10,112 + 3( + )A B C

= (1, 0, −3)A = (−2, 5, 1)B = (3, 1, 1)C

= ( , , )M Mx My Mz = ( , , )N Nx Ny Nz

× = ( − , − , − )M N My Nz Mz Ny Mz Nx Mx Nz Mx Ny My Nx

= 4,5,17×B C



Part B

ANSWER:

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 interms of given quantities.

Part F

If and are perpendicular,

= 4,5,17×C B

= 24,30,102(2 ) × (3 )B C

= 15,5,5× ( × )A B C

= 55⋅ ( × )A B C

V 1 V

2 V1 V2

V 1 V

2



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:

Correct

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

± Vector Dot Product

Let vectors , , and .Calculate the following:

Part A

Hint 1. Remember the dot product equation

If and , then

.

π/2

| × | = | | | | sin(θ)A B A B θ A B

= | × |V 1 V

2 V1 V2

V 1 V

2

= 0| × |V 1 V

2

= (2, 1, −4)A = (−3, 0, 1)B = (−1, −1, 2)C

= ( , , )M Mx My Mz = ( , , )N Nx Ny Nz

⋅ = + +M N Mx Nx My Ny Mz Nz



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


ANSWER:

Correct

Part C

ANSWER:

Correct

Part D

ANSWER:

Correct

Part E

Which of the following can be computed?

= 10⋅A B

θAB A B

⋅ = | | | | cos(θ)A B A B θ A B

= 2 θAB radians

= 302 ⋅ 3B C

= 302( ⋅ 3 )B C



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


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.

⋅ ⋅A B C

⋅ ( ⋅ )A B C

⋅ ( + )A B C

3 ⋅ A

V 1 V

2 V1 V2

V1

⋅ = | | | | cos(θ)A B A B θ A B

= ⋅V 1 V

1 V12

V 1 V

2

π/2



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

Exercise 1.12

A useful and easytoremember approximate value for the number of seconds in a year is .

Part A

Determine the percent error in this approximate value. (There are 365.24 days in one year.)

ANSWER:

Correct

Exercise 1.31

A postal employee drives a delivery truck along the route shown in the figure .

= ⋅V 1 V

2 0

V 1 V

2

V1 V2

= ⋅V 1 V

2 V1 V2

π × 107

0.446 %



Part A

Determine the magnitude of the resultant displacement by drawing a scale diagram.

Express your answer using two significant figures.

ANSWER:

Correct

Part B

Determine the direction of the resultant displacement.

Express your answer using two significant figures.

ANSWER:

Correct

Problem 1.63

Part A

Estimate the number of atoms in your body. (Hint: Based on what you know about biology and chemistry, what arethe most common types of atom in your body? What is the mass of each type of atom?)

7.8 km

38 North of East ∘



ANSWER:

Correct

Problem 1.83

While following a treasure map, you start at an old oak tree. You first walk 825 directly south, then turn and walk1.25 at 30.0 west of north, and finally walk 1.00 at 40.0 north of east, where you find the treasure: abiography 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:

Correct

Part B

To return to the old oak tree, how far will you walk? Use components to solve this problem.

ANSWER:

Correct

Multiple Choice Question 1.8

Part A

The components of vectors and are given as follows:

atoms

atoms

atoms

atoms

atoms

∼ 105

∼ 1010

∼ 1019

∼ 1028

∼ 1034

mkm ∘ km ∘

= 8.90 west of south θ ∘

= 911 D m

A→

B→



Ax = +5.7 Bx = 9.8Ay = 3.6 By = 6.5

The magnitude of the vector difference , is closest to:

ANSWER:

Correct

Score Summary:Your score on this assignment is 96.3%.You received 9.63 out of a possible total of 10 points.

−B→

A→

16

11

5.0

5.0

250

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Assignment 1.pdf