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Assignment - Chapter 1

Due: 11:59pm on Thursday, January 14, 2010

Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy

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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 dimensions associated with them. Thus, your height is not 74, but rather 74 inches, often expressed as6 feet 2 inches. Although feet and inches are different units they have the same dimension--length.

Part A

In classical mechanics there are three base dimensions. Length is one of them. What are the other two?

Hint A.1 MKS system

Hint not displayed

ANSWER: acceleration and mass

acceleration and time

acceleration and charge

mass and time

mass and charge

time and charge

Correct

There are three dimensions used in mechanics: length ( ), mass ( ), and time ( ). A combination of these three dimensions 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. One then derives a unit to measure the new physical quantity from thatequation, 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 a side of the square. Clearly . Plugging this into the equation gives .

Part B

Find the dimensions of volume.

Hint B.1 Equation for volume

Hint not displayed

Express your answer as powers of length ( ), mass ( ), and time ( ).

ANSWER: =

Correct

Part C

Find the dimensions of speed.

Hint C.1 Equation for speed

Hint not displayed

Hint C.2 Familiar units for speed

Hint not displayed


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.

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 perhour, as long as you convert both quantities to the same set of units before you actually compute the sum. You can use this rule to check your answers to any physics problem you work. If the answer involvesthe sum or difference of two quantities with different dimensions, then it must be incorrect.

This rule also ensures that the dimensions of any physical quantity will never involve sums or differences of the base dimensions. (As in the preceeding example, is not a valid dimension for a physical

quantitiy.) A valid dimension will only involve the product or ratio of powers of the base dimensions (e.g. ).

Part D

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentID...

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Find the dimensions of acceleration.

Hint D.1 Equation for acceleration

Hint not displayed


ANSWER:

=

Correct

Motion of Two Rockets

Learning Goal: To learn to use images of an object in motion to determine velocity and acceleration.

Two toy rockets are traveling in the same direction (taken to be the x axis). A diagram is shown of a time-exposure image where a stroboscope has illuminated the rockets at the uniform time intervals indicated.

Part A

At what time(s) do the rockets have the same velocity?

Hint A.1 How to determine the velocity

Hint not displayed

ANSWER: at time only

at time only

at times and

at some instant in time between and

at no time shown in the figure

Correct

Part B

At what time(s) do the rockets have the same x position?


at time only

at times and



Correct

Part C

At what time(s) do the two rockets have the same acceleration?

Hint C.1 How to determine the acceleration

Hint not displayed


at time only

at times and



Correct

Part D

The motion of the rocket labeled A is an example of motion with uniform (i.e., constant) __________.

ANSWER: and nonzero acceleration


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velocity

displacement

time

Correct

Part E

The motion of the rocket labeled B is an example of motion with uniform (i.e., constant) __________.

ANSWER: and nonzero acceleration

velocity

displacement

time

Correct

Part F

At what time(s) is rocket A ahead of rocket B?

Hint F.1 Use the diagram

Hint not displayed

ANSWER: before only

after only

before and after

between and

at no time(s) shown in the figure

Correct

Tactics Box 1.6 Using Significant Figures

Learning Goal: To practice Tactics Box 1.6 Using Significant Figures.

You can think of a significant figure as being a digit that is reliably known. For example, a length measurement of 6.2 has two significant figures, the 6 and the 2. The next decimal place—the

one-hundredths—is not reliably known and is thus not a significant figure.

Determining the proper number of significant figures is straightforward, but there a few definite rules to follow. These are summarized in this tactics box.

TACTICS BOX 1.6 Using significant figures.

It is acceptable to keep one or two extra digits during intermediate steps of a calculation. The goal here is to minimize round-off errors in the calculation. But the final answer must be reported with the proper

number of significant figures.

When multiplying or dividing several numbers, or when taking roots, the number of significant figures in the answer should match the number of significant figures of the least precisely known numberused in the calculation.

1.

When adding or subtracting several numbers, the number of decimal places in the answer should match the smallest number of decimal places of any number used in the calculation.2.It is acceptable to keep one or two extra digits during intermediate steps of a calculation, as long as the final answer is reported with the proper number of significant figures. The goal here is to

minimize round-off errors in the calculation. But keep only one or two extra digits, not the seven or eight shown in your calculator display.

3.

Part A

Your bedroom has a rectangular shape, and you want to measure its area. You use a tape that is precise to 0.001 and find that the shortest wall in the room is 3.547 long. The tape, however, is too short

to measure the length of the second wall, so you use a second tape, which is longer but only precise to 0.01 . You measure the second wall to be 4.79 long. Which of the following numbers is the most

precise estimate that you can obtain from your measurements for the area of your bedroom?

Hint A.1 Find how many significant figures should be in your answer

Hint not displayed

ANSWER: 17.0

16.990

16.99

16.9

16.8

Correct

Part B

Using the measurements described in Part A, which of the following numbers is the most precise estimate for the perimeter of your bedroom?

Hint B.1 Find how many decimal places should be in your answer

Hint not displayed


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ANSWER: 16.674

16.67

16.68

16.7

17

Correct

Part C

If your bedroom has a circular shape, and its diameter measured 6.32 , which of the following numbers would be the most precise value for its area?

Hint C.1 The area of a circle

Hint not displayed

Hint C.2 Find how many significant figures should be in your answer

Hint not displayed

ANSWER: 30

31.0

31.4

31.37

31.371

Correct

To calculate the area of a circle given its diameter , you need to first divide the diameter by 2, then multiply your result by itself (or take the square of it), and finally multiply everything by :

.

Since here the number 2 and are exact numbers, they do not change the accuracy of the measured numbers involved in the calculation. Therefore, your answer should be expressed to the same number of

significant figures as that used in the given diameter.

Converting Units: The Magic of 1

Learning Goal: To learn how to change units of physical quantities.

Quantities with physical dimensions like length or time must be measured with respect to a unit, a standard for quantities with this dimension. For example, length can be measured in units of meters or feet, timein seconds or years, and velocity in meters per second.

When solving problems in physics, it is necessary to use a consistent system of units such as the International System (abbreviated SI, for the French Système International) or the more cumbersome Englishsystem. In the SI system, which is the preferred system in physics, mass is measured in kilograms, time in seconds, and length in meters. The necessity of using consistent units in a problem often forces you to

convert some units from the given system into the system that you want to use for the problem.

The key to unit conversion is to multiply (or divide) by a ratio of different units that equals one. This works because multiplying any quantity by one doesn't change it. To illustrate with length, if you know that

, you can write

.

To convert inches to centimeters, you can multiply the number of inches times this fraction (since it equals one), cancel the inch unit in the denominator with the inch unit in the given length, and come up with a

value for the length in centimeters. To convert centimeters to inches, you can divide by this ratio and cancel the centimeters.

For all parts, notice that the units are already written after the answer box; don't try to write them in your answer also.

Part A

How many centimeters are there in a length 32.40 ?

Express your answer in centimeters to three significant figures.

ANSWER: 82.3

Correct

Sometimes you will need to change units twice to get the final unit that you want. Suppose that you know how to convert from centimeters to inches and from inches to feet. By doing both, in order, you can

convert from centimeters to feet.

Part B

Suppose that a particular artillery piece has a range = 9520 . Find its range in miles. Use the facts that and .

Hint B.1 Convert yards to feet

Hint not displayed

Express your answer in miles to three significant figures.

ANSWER: 9520 = 5.41

Correct


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Often speed is given in miles per hour ( ), but in physics you will almost always work in SI units. Therefore, you must convert to meters per second ( ).

Part C

What is the speed of a car going in SI units? Notice that you will need to change from miles to meters and from hours to seconds. You can do each conversion separately. Use the facts that

and .

Hint C.1 Convert miles to meters

Hint not displayed

Hint C.2 Convert hours to seconds

Hint not displayed

Express your answer in meters per second to three significant figures.

ANSWER: = 0.447

Correct

Notice that by equating the two values for , you get . It might be valuable to remember this, as you may frequently need to convert from miles per hour into more useful SI units. By

remembering this relationship in the future, you can reduce this task to a single conversion.

Consistency of Units

In physics, every physical quantity is measured with respect to a unit. Time is measured in seconds, length is measured in meters, and mass is measured in kilograms. Knowing the units of physical quantities

will help you solve problems in physics.

Part A

Gravity causes objects to be attracted to one another. This attraction keeps our feet firmly planted on the ground and causes the moon to orbit the earth. The force of gravitational attraction is represented by theequation

,

where is the magnitude of the gravitational attraction on either body, and are the masses of the bodies, is the distance between them, and is the gravitational constant. In SI units, the units or

force are , the units of mass are , and the units of distance are . For this equation to have consistent units, the units of must be which of the following?

Hint A.1 How to approach the problem

Hint not displayed

ANSWER:

Correct

Part B

One consequence of Einstein's theory of special relativity is that mass is a form of energy. This mass-energy relationship is perhaps the most famous of all physics equations:

,

where is mass, is the speed of the light, and is the energy. In SI units, the units of speed are . For the preceding equation to have consistent units (the same units on both sides of the equation), the

units of must be which of the following?

Hint B.1 How to approach the problem

Hint not displayed

ANSWER:

Correct

To solve the types of problems typified by these examples, we start with the given equation. For each symbol whose units we know, we replace the symbol with those units. For example, we replace

with . We now solve this equation for the units of the unknown variable.


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Conceptual Question 1.1

How many significant figures does each of the following numbers have?

Part A

6.21

Express your answer using one significant figure.

ANSWER: = 3

Correct

Part B

62.1


ANSWER: = 3

Correct

Part C

0.620


ANSWER: = 3

Correct

Part D

0.062


ANSWER: = 2

Correct


Part A

Determine the sign (positive or negative) of the position for the particle in the figure in the textbook.

ANSWER: Positive

Negative

Correct

Part B

Determine the sign (positive or negative) of the velocity for the particle in the figure.

ANSWER: Negative

Positive

Correct

Part C

Determine the sign (positive or negative) of the acceleration for the particle in the figure.

ANSWER: Negative

Positive

Correct



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Part A

Determine the sign (positive or negative) of the position for the particle in the figure in the textbook.

ANSWER: Positive

Negative

Correct

Part B

Determine the sign (positive or negative) of the velocity for the particle in the figure.

ANSWER: Positive

Negative

Correct

Part C

Determine the sign (positive or negative) of the acceleration for the particle in the figure.

ANSWER: Positive

Negative

Correct

Tactics Box 1.1 Vector Addition

Learning Goal: To practice Tactics Box 1.1 Vector Addition.

Vector addition obeys rules that are different from those for the addition of two scalar quantities. When you add two vectors, their directions, as well as their magnitudes, must be taken into account. This TacticsBox explains how to add vectors graphically.

TACTICS BOX 1.1 Vector addition

To add to , perform these steps:

Draw . 1.

Place the tail of at the tip of .2.

Draw an arrow from the tail of to the tip of . This is vector .3.

Part A

Create the vector by following the steps in the Tactics Box above. When moving vector , keep in mind that its direction should remain unchanged.

The location, orientation, and length of your vectors will be graded.

ANSWER:


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View Correct

Part B

Create the vector by following the steps in the Tactics Box above. When moving vector , keep in mind that its direction should remain unchanged.


ANSWER:

View Correct

Tactics Box 1.2 Vector Subtraction

Learning Goal: To practice Tactics Box 1.2 Vector Subtraction.

Vector subtraction has some similarities to the subtraction of two scalar quantities. With numbers, subtraction is the same as the addition of a negative number. For example, is the same as .

Similarly, . We can use the rules for vector addition and the fact that is a vector opposite in direction to to form rules for vector subtraction, as explained in this tactics box.

TACTICS BOX 1.2 Vector subtraction

To subtract from , perform these steps:

Draw .1.

Place the tail of at the tip of .2.

Draw an arrow from the tail of to the tip of . This is vector . 3.

Part A

Find vector by following the steps in the tactics box above. When drawing , keep in mind that it has the same magnitude as but opposite direction.


ANSWER:

View Correct

Part B

Find vector by following the steps in the tactics box above. When drawing , keep in mind that it has the same magnitude as but opposite direction.



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ANSWER:

View Correct


Part A

Does the object represented in the figure have positive or negative value of ? Assume that -axis points to the right.

ANSWER: Negative

Positive

Correct

Part B

Explain.

ANSWER: My Answer:

Problem 1.18

The figure shows the motion diagram of a drag racer. The camera took one frame every 2 .

Part A

Make a position-versus-time graph for the drag racer.

ANSWER:

View

Correct

Conceptual Problem 1.6

Part A

An object has a velocity directed to the right, and an acceleration directed to the left. Which statement is true?


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ANSWER:

The position of the object is increasing to the right with time, and its speed is decreasing.

The position of the object is not changing in time.

The position of the object is increasing to the right with time, and its speed is increasing.

The position of the object is increasing to the left with time, and its speed is increasing.

Correct

True/False Problem 1.3

Part A

If a vector pointing to the right has a positive magnitude, then one pointing to the left has a negative magnitude.

ANSWER: True

False

Correct


Part A

If successive images of an object in a motion diagram get closer and closer together, then that object is accelerating.

ANSWER: True

False

Correct

PSS 1.1 Motion Diagrams

Learning Goal: To practice Problem-Solving Strategy 1.1 for motion diagram problems.

A car is traveling with constant velocity along a highway. The driver notices he is late for work, so he stomps down on the gas pedal and the car begins to speed up. The car has just achieved double its initialvelocity when the driver spots a police officer behind him and applies the brakes. The car then slows down, coming to rest at a stoplight ahead.

Draw a complete motion diagram for this situation.

PROBLEM-SOLVING STRATEGY 1.1 Motion diagrams

MODEL: Represent the moving object as a particle. Make simplifying assumptions when interpreting the problem statement.

VISUALIZE: A complete motion diagram consists of:

The position of the object in each frame of the film, shown as a dot. Use five or six dots to make the motion clear but without overcrowding the picture. More complex motions may need more dots.The average velocity vectors, found by connecting each dot in the motion diagram to the next with a vector arrow. There is one velocity vector linking each set of two position dots. Label the row ofvelocity vectors .

The average acceleration vectors, found using Tactics Box 1.3. There is one acceleration vector linking each set of two velocity vectors. Each acceleration vector is drawn at the dot between the two

velocity vectors it links. Use to indicate a point at which the acceleration is zero. Label the row of acceleration vectors .

Model

It is appropriate to use the particle model for the car. You should also make some simplifying assumptions.

Part A

The car's motion can be divided into three different stages: its motion before the driver realizes he's late, its motion after the driver hits the gas (but before he sees the police car), and its motion after the driversees the police car. Which of the following simplifying assumptions is it reasonable to make in this problem?

During each of the three different stages of its motion, the car is moving with constant acceleration.A.

During each of the three different stages of its motion, the car is moving with constant velocity.B.The highway is straight (i.e., there are no curves).C.The highway is level (i.e., there are no hills or valleys).D.

Enter all the correct answers in alphabetical order without commas. For example, if statements C and D are correct, enter CD.

ANSWER: ACD

Correct

In addition to the assumptions listed above, in the rest of this problem assume that the car is moving in a straight line to the right.


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Visualize

Part B

In the three diagrams shown to the left, the position of the car at five subsequent instants of time is represented by black dots, and the car's average velocity is represented by green arrows. Which of thesediagrams best describes the position and the velocity of the car before the driver notices he is late?

ANSWER: A

B

C

Correct

Part C

Which of the diagrams shown to the left best describes the position and the velocity of the car after the driver hits the gas, but before he notices the police officer?

ANSWER: A

B

C

Correct

Part D

Which of the diagrams shown to the left best describes the position and the velocity of the car after the driver notices the police officer?

ANSWER: A

B

C

Correct

Part E

Which of the diagrams shown below most accurately depicts the average acceleration vectors of the car during the events described in the problem introduction?

ANSWER: A

B

C

Correct

You can now draw a complete motion diagram for the situation described in this problem. Your diagram should look like this:


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Curved Motion Diagram

The motion diagram shown in the figure represents a pendulum released from rest at an angle of 45 from the vertical. The dots in the motion diagram

represent the positions of the pendulum bob at eleven moments separated by equal time intervals. The green arrows represent the average velocity

between adjacent dots. Also given is a "compass rose" in which directions are labeled with the letters of the alphabet.

Part A

What is the direction of the acceleration of the object at moment 5?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Definition of acceleration

Hint not displayed

Hint A.3 Change of velocity: a graphical interpretation

Hint not displayed

Enter the letter of the arrow with this direction from the compass rose in the figure. Type Z if the acceleration vector has zero length.

ANSWER: A

Correct

Part B

What is the direction of the acceleration of the object at moments 0 and 10?

Hint B.1 Find the direction of the velocity

Hint not displayed

Hint B.2 Definition of acceleration

Hint not displayed

Hint B.3 Applying the definition of acceleration

Hint not displayed

Enter the letters corresponding to the arrows with these directions from the compass rose in the figure, separated by commas. Type Z if the acceleration vector has zero length.

ANSWER:directions at time step 0, time step 10 =

D,F

Correct

Reading Quiz 1.2

Part A

What quantities are shown on a complete motion diagram?

ANSWER: Only the position of the object in each frame of the film, shown as a dot

Only the average velocity vectors (found by connecting each dot in the motion diagram to the next with a vector arrow)

Only the average acceleration vectors (where each acceleration vector links every two velocity vectors)

All of the above


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Correct

Reading Quiz 1.3

Part A

Acceleration vectors

ANSWER: tell you how fast objects are going.

are constructed from two velocity vectors.

are the second derivative of the position.

are parallel or opposite to the direction of motion.

weren’t discussed in this chapter.

Correct

Reading Quiz 1.4

Part A

The pictorial representation of a physics problem consists of

ANSWER: just a sketch.

just a coordinate system.

only symbols.

only a list of known information.

all of the above.

Correct


Part A

Does the object represented in the figure have a positive or negative value of ? Assume that -axis points upward.

ANSWER: Negative

Positive

Correct

Part B

Explain.

ANSWER: My Answer:

Problem 1.11

Part A

Figure shows two dots of a motion diagram and vector . Draw the vector if the acceleration vector at dot 2 points up.

Draw the vector with its tail at the dot 2. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be

graded.

ANSWER:


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View Correct

Part B

Figure shows two dots of a motion diagram and vector . Draw the vector if the acceleration vector at dot 2 points down.

Draw the vector with its tail at the dot 2. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be

graded.

ANSWER:

View Correct

Problem 1.12

Part A

Figure shows two dots of a motion diagram and vector . Draw the vector if the acceleration vector at dot 3 points to the right.

Draw the vector with its tail at the dot 3. The orientation of your vector will be graded. The exact length of your vector will not be graded but the relative length of one to the other will be graded.

ANSWER:

View Correct

Part B

Figure shows two dots of a motion diagram and vector . Draw the vector if the acceleration vector at dot 3 points to the left.

Draw the vector with its tail at the dot 3. The orientation of your vector will be graded. The exact length of your vector will not be graded but the relative length of one to the other will be graded.

ANSWER:

View Correct

Problem 1.55

The figure shows a motion diagram of a car traveling down a street. The camera took one frame every 10 . A distance scale is provided.

Part A

Make a position-versus-time graph for the car.


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ANSWER:

View Correct

Problem 1.10

Part A

Find the average acceleration vector at point 1 of the threepoint motion diagram shown in the figure.

Draw the acceleration vector with its tail at the dot. The orientation of your vector will be graded. The exact length of your vector will not

be graded.

ANSWER:

View Correct

Part B

Is the object's average speed between points 1 and 2 greater than, less than, or equal to its average speed between points 0 and 1?

ANSWER: greater

less

equal

Correct

Part C

Explain how you can tell.

ANSWER: My Answer:


Part A

The change in the position vector of a moving object is equal to the distance it has moved.

ANSWER: True

False

Correct


Part A

Since vectors always have positive magnitudes, the sum of two vectors must have a magnitude greater than the magnitude of either one of them.

ANSWER: True


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False

Correct


Part A

If an object has a changing speed, its velocity must also be changing; but if it has a changing velocity, its speed is not necessarily changing.

ANSWER: True

False

Correct


Part A

All objects moving in a circle are accelerated.

ANSWER: True

False

Correct


Part A

If an object is accelerating toward a point, then it must be getting closer and closer to that point.

ANSWER: True

False

Correct

Score Summary:

Your score on this assignment is 96.3%.

You received 28.9 out of a possible total of 30 points.


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