introduction and mathematical concepts

Introduction and Mathematical Concepts

Chapter 1

1.1 The Nature of Physics

Physics has developed out of the effortsof men and women to explain our physicalenvironment.

Physics encompasses a remarkable variety of phenomena:

planetary orbitsradio and TV wavesmagnetismlasers

many more!

1.1 The Nature of Physics

Physics predicts how nature will behavein one situation based on the results of experimental data obtained in anothersituation.

Newton’s Laws → Rocketry

Maxwell’s Equations → Telecommunications

1.1.1. Which of the following individuals did not make significant contributions in physics?

a) Galileo Galilei

b) Isaac Newton

c) James Clerk Maxwell

d) Neville Chamberlain

1.2 Units

Physics experiments involve the measurementof a variety of quantities.

These measurements should be accurate andreproducible.

The first step in ensuring accuracy andreproducibility is defining the units in whichthe measurements are made.

1.2 Units

SI unitsmeter (m): unit of length

kilogram (kg): unit of mass

second (s): unit of time

MEASUREMENTLet’s watch this brief clip on measurement

1.2 Units

1.2 Units

The units for length, mass, and time (aswell as a few others), are regarded asbase SI units.

These units are used in combination to define additional units for other importantphysical quantities such as force and energy.

1.1.2. Which of the following statements is not a reason that physics is a required course for students in a wide variety of disciplines?

a) There are usually not enough courses for students to take.

b) Students can learn to think like physicists.

c) Students can learn to apply physics principles to a wide range of problems.

d) Physics is both fascinating and fundamental.

e) Physics has important things to say about our environment.

1.2.1. The text uses SI units. What do the “S” and the “I” stand for?

a) Système International

b) Science Institute

c) Swiss Institute

d) Systematic Information

e) Strong Interaction

1.2.2. Which of the following units is not an SI base unit?

a) slug

b) meter

c) kilogram

d) second

1.2.3. Complete the following statement: The standard meter is defined in terms of the speed of light because

a) all scientists have access to sunlight.

b) no agreement could be reached on a standard meter stick.

c) the yard is defined in terms of the speed of sound in air.

d) the normal meter is defined with respect to the circumference of the earth.

e) it is a universal constant.

1.3 The Role of Units in Problem Solving

THE CONVERSION OF UNITS

1 ft = 0.3048 m

1 mi = 1.609 km

1 hp = 746 W

1 liter = 10-3 m3


Example 1 The World’s Highest Waterfall

The highest waterfall in the world is Angel Falls in Venezuela,with a total drop of 979.0 m. Express this drop in feet.

Since 3.281 feet = 1 meter, it follows that

(3.281 feet)/(1 meter) = 1

feet 3212meter 1

feet 281.3meters 0.979 Length


Reasoning Strategy: Converting Between Units

1. In all calculations, write down the units explicitly.

2. Treat all units as algebraic quantities. When identical units are divided, they are eliminated algebraically.

3. Use the conversion factors located on the pagefacing the inside cover. Be guided by the fact that multiplying or dividing an equation by a factor of 1does not alter the equation.


Example 2 Interstate Speed Limit

Express the speed limit of 65 miles/hour in terms of meters/second.

Use 5280 feet = 1 mile and 3600 seconds = 1 hour and 3.281 feet = 1 meter.

second

feet95s 3600

hour 1 mile

feet 5280hourmiles 6511

hourmiles 65 Speed

secondmeters29

feet 3.281meter 1

secondfeet951

secondfeet95 Speed


DIMENSIONAL ANALYSIS

[L] = length [M] = mass [T] = time

221 vtx

Is the following equation dimensionally correct?

TLTTLL 2


Is the following equation dimensionally correct?

vtx

LTTLL

1.3.1. Which one of the following statements concerning unit conversion is false?

a) Units can be treated as algebraic quantities.

b) Units have no numerical significance, so 1.00 kilogram = 1.00 slug.

c) Unit conversion factors are given inside the front cover of the text.

d) The fact that multiplying an equation by a factor of 1 does not change an equation is important in unit conversion.

e) Only quantities with the same units can be added or subtracted.

1.3.2. Which one of the following pairs of units may not be added together, even after the appropriate unit conversions have been made?

a) feet and centimeters

b) seconds and slugs

c) meters and miles

d) grams and kilograms

e) hours and years

1.3.3. Which one of the following terms is used to refer to the physical nature of a quantity and the type of unit used to specify it?

a) scalar

b) conversion

c) dimension

d) vector

e) symmetry

1.3.4. In dimensional analysis, the dimensions for speed are

a)

b)

c)

d)

e)

TL 2

2T

L

2

2

T

L

TL

TL

1.4 Trigonometry

1.4 Trigonometry

hhosin

hhacos

a

o

hh

tan

1.4 Trigonometry

m2.6750tan oh

m0.80m2.6750tan oh

a

o

hh

tan

1.4 Trigonometry

hho1sin

hha1cos

a

o

hh1tan

1.4 Trigonometry

a

o

hh1tan 13.9

m0.14m25.2tan 1

1.4 Trigonometry

222ao hhh Pythagorean theorem:

1.4.1. Which one of the following terms is not a trigonometric function?

a) cosine

b) tangent

c) sine

d) hypotenuse

e) arc tangent

1.4.2. For a given angle , which one of the following is equal to the ratio of sin /cos ?

a) one

b) zero

c) sin1

d) arc cos

e) tan

1.4.3. Referring to the triangle with sides labeled A, B, and C as shown, which of the following ratios is equal to the sine of the angle ?

a)

b)

c)

d)

e)

BA

CA

CB

AB

BC

1.4.4. Referring to the triangle with sides labeled A, B, and C as shown, which of the following ratios is equal to the tangent of the angle ?

a)

b)

c)

d)

e)

BA

CA

CB

AB

BC

1.4.5. Which law, postulate, or theorem states the following: “The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.”

a) Snell’s law

b) Pythagorean theorem

c) Square postulate

d) Newton’s first law

e) Triangle theorem

1.5 Scalars and Vectors

A scalar quantity is one that can be describedby a single number:

temperature, speed, mass

A vector quantity deals inherently with both magnitude and direction:

velocity, force, displacement


By convention, the length of a vectorarrow is proportional to the magnitudeof the vector.

8 lb4 lb

Arrows are used to represent vectors. Thedirection of the arrow gives the direction ofthe vector.

1.5.1. Which one of the following statements is true concerning scalar quantities?

a) Scalar quantities have both magnitude and direction.

b) Scalar quantities must be represented by base units.

c) Scalar quantities can be added to vector quantities using rules of trigonometry.

d) Scalar quantities can be added to other scalar quantities using rules of ordinary addition.

e) Scalar quantities can be added to other scalar quantities using rules of trigonometry.

1.5.2. Which one of the following quantities is a vector quantity?

a) the age of the pyramids in Egypt

b) the mass of a watermelon

c) the sun's pull on the earth

d) the number of people on board an airplane

e) the temperature of molten lava

1.5.3. A vector is represented by an arrow. What is the significance of the length of the arrow?

a) Long arrows represent velocities and short arrows represent forces.

b) The length of the arrow is proportional to the magnitude of the vector.

c) Short arrows represent accelerations and long arrows represent velocities.

d) The length of the arrow indicates its direction.

e) There is no significance to the length of the arrow.

1.5.4. Which one of the following situations involves a vector quantity?

a) The mass of the Martian soil probe was 250 kg.

b) The overnight low temperature in Toronto was 4.0 C.

c) The volume of the soft drink can is 0.360 liters.

d) The velocity of the rocket was 325 m/s, due east.

e) The light took approximately 500 s to travel from the sun to the earth.

1.6 Vector Addition and Subtraction

Often it is necessary to add one vector to another.


5 m 3 m

8 m


2.00 m

6.00 m


2.00 m

6.00 m

222 m 00.6m 00.2 R

R

m32.6m 00.6m 00.2 22 R


2.00 m

6.00 m

6.32 m

00.600.2tan

4.1800.600.2tan 1


When a vector is multiplied by -1, the magnitude of the vector remains the same, but the direction of the vector is reversed.


A

B

BA

A

B

BA

1.6.1. A and B are vectors. Vector A is directed due west and vector B is directed due north. Which of the following choices correctly indicates the directions of vectors A and B?

a) A is directed due west and B is directed due north

b) A is directed due west and B is directed due south

c) A is directed due east and B is directed due south

d) A is directed due east and B is directed due north

e) A is directed due north and B is directed due west

1.6.2. Which one of the following statements concerning vectors and scalars is false?

a) In calculations, the vector components of a vector may be used in place of the vector itself.

b) It is possible to use vector components that are not perpendicular.

c) A scalar component may be either positive or negative.

d) A vector that is zero may have components other than zero.

e) Two vectors are equal only if they have the same magnitude and direction.

1.7 The Components of a Vector

. ofcomponent vector theandcomponent vector thecalled are and r

yx

yx


.AAA

AA

A

yx

that soy vectoriall together add and

axes, and the toparallel are that and vectors

larperpendicu twoare of components vector The

yxyx


It is often easier to work with the scalar components rather than the vector components.

. of

componentsscalar theare and

A

yx AA

1. magnitude with rsunit vecto are ˆ and ˆ yx

yxA ˆˆ yx AA


Example

A displacement vector has a magnitude of 175 m and points atan angle of 50.0 degrees relative to the x axis. Find the x and ycomponents of this vector.

rysin

m 1340.50sinm 175sin ry

rxcos m 1120.50cosm 175cos rx

yxr ˆm 134ˆm 112

1.7.1. A, B, and, C are three vectors. Vectors B and C when added together equal the vector A. In mathematical form, A = B + C. Which one of the following statements concerning the components of vectors B and C must be true if Ay = 0?

a) The y components of vectors B and C are both equal to zero.

b) The y components of vectors B and C when added together equal zero.

c) By Cy = 0 or Cy By = 0

d) Either answer a or answer b is correct, but never both.

e) Either answer a or answer b is correct. It is also possible that both are correct.

1.7.2. Vector r has a magnitude of 88 km/h and is directed at 25 relative to the x axis. Which of the following choices indicates the horizontal and vertical components of vector r?

rx ry

a) +22 km/h +66 km/h

b) +39 km/h +79 km/h

c) +79 km/h +39 km/h

d) +66 km/h +22 km/h

e) +72 km/h +48 km/h

1.7.3. A, B, and, C are three vectors. Vectors B and C when added together equal the vector A. Vector A has a magnitude of 88 units and it is directed at an angle of 44 relative to the x axis as shown. Find the scalar components of vectors B and C.

Bx By Cx Cy

a) 63 0 0 61

b) 0 61 63 0

c) 63 0 61 0

d) 0 63 0 61

e) 61 0 63 0

1.8 Addition of Vectors by Means of Components

BAC

yxA ˆˆ yx AA

yxB ˆˆ yx BB

1.8 Addition of Vectors by Means of Components

yx

yxyxCˆˆ

ˆˆˆˆ

yyxx

yxyx

BABA

BBAA

xxx BAC yyy BAC

1.8.1. Vector A has scalar components Ax = 35 m/s and Ay = 15 m/s. Vector B has scalar components Bx = 22 m/s and By = 18 m/s. Determine the scalar components of vector C = A B.

Cx Cy

a) 13 m/s 3 m/s

b) 57 m/s 33 m/s

c) 13 m/s 33 m/s

d) 57 m/s 3 m/s

e) 57 m/s 3 m/s

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