PHY121 Summer Session I, 2006

• Most of information is available at: http://nngroup.physics.sunysb.edu/~chiaki/PHY121-06. It will be frequently updated.• Homework assignments for each chapter due a week later (normally) and are delivered through WebAssign. Once the deadline has passed you cannot input answers on WebAssign. To gain access to WebAssign, you need to obtain access code and go to http://www.webassign.net. Your login username, institution name and password are: initial of your first name plus last name (such as cyanagisawa), sunysb, and the same as your username, respectively.

• In addition to homework assignments, there is a reading requirement of each chapter, which is very important.

• The lab session will start next Monday (June 5), for the first class go to A-117 at Physics Building. Your TAs will divide each group into two classes in alphabetic order.

Instructor : Chiaki Yanagisawa

Chapter 1: Introduction

Standards of Length, Mass and Time A physical quantity is measured in a unit which specifies the scale of the quantity.

• Fundamental unit of length : meter (m)

1 m = 100 cm = 1,000 mm, 1 km = 1,000 m,…1 inch = 2.54 cm = 0.0254 m, 1 foot = 30 cm = 0.30 m

• SI units (Systèm International), also known as MKS

A standard system of units for fundamental quantities ofscience an international committee agreed upon in 1960.

The meter was defined as the distance traveled by light invacuum during a time interval of 1/299,792,458 seconds in1980.

A physical quantity is measured in a unit which specifies the scale of the quantity (cont’d)

• Fundamental unit of mass : kilogram (kg)

1 kg = 1,000 g, 1 g = 1,000 mg, 1 ton = 1,000 kg1 pound = 0.454 kg = 454 g, 1 ounce = 28.3 g

The kilogram is defined as the mass of a specific platinumiridium alloy cylinder kept at the International Bureau ofWeights and Measures in France.

• Fundamental unit of time : second (s or sec)

1 sec = 1,000 msec = 1,000,000 sec,…1 hour = 60 min = 3,600 sec, 24 hours = 1 day

The second is defined as 9,192,631,700 times the periodof oscillation of radiation from cesium atom.

Standards of Length, Mass and Time

A physical quantity is measured in a unit which specifies the scale of the quantity (cont’d)

• Scale of some measured lengths in m

Distance from Earth to most remote normal galaxies 4 x 1025

Distance from Earth to nearest large galaxy (M31) 2 x 1022

Distance from Earth to closest star (Proxima Centauri) 4 x 1016

Distance for light to travel in one year (light year) 9 x 1015

Distance from Earth to Sun (mean) 2 x 1011

Mean radius of Earth 6 x 106

Length of football field 9 x 101

Size of smallest dust particle 2 x 10-4

Size of cells in most living organism 2 x 10-5

Diameter of hydrogen atom 1 x 10-10

Diameter of atomic nucleus 1 x 10-14

Diameter of proton 1 x 10-15

Standards of Length, Mass and Time

A physical quantity is measured in a unit which specifies the scale of the quantity (cont’d)

• Scale of some measured masses in kg

Observable Universe 1 x 1052

Milky Way Galaxy 7 x 1041

Sun 2 x 1030

Earth 6 x 1024

Human 7 x 101

Frog 1 x 10-1

Mosquito 1 x 10-5

Bacterium 1 x 10-15

Hydrogen atom 2 x 10-27

Electron 9 x 10-31

Standards of Length, Mass and Time

Other systems of units

• cgs : length in cm, mass in g, time in s area in cm2, volume in cm3, velocity in cm/s

• U.S. customary : length in ft , mass in lb, time in s area in ft2 , volume in ft3, velocity in ft/s

Standards of Length, Mass and Time

Prefix

10-12 10-9 10-6 10-3 10-2

pico- (p) nano- (n) micro- milli- (m) centi- (c)

1012 109 106 103 101

tera- (T) giga- (G) mega- (M) kilo- (k) deka- (da)

The Building Blocks of Matter History of model of atoms

electrons e-

nucleus (protons and neutrons)

Old view

Modern view

nucleusquarks

prot

onSemi-modern view

Dimensional Analysis In physics, the word dimension denotes the physical nature of a quantity

• The distance can be measured in feet, meters,… (different unit), which are different ways of expressing the dimension of length.

• The symbols that specify the dimensions of length, mass and time are L, M, and T.

dimension of velocity [v] = L/T (m/s) dimension of area [A] = L2 (m2)

Dimensional Analysis In physics, it is often necessary either to derive a mathematical expression or equation or to check its correctness. A useful procedure for this is called dimensional analysis.

• Dimensions can be treated as algebraic quantities:

dimension of distance [x] = L (m) dimension of velocity [v] = [x]/[t] = L/T (m/s) dimension of acceleration [] = [v]/[t] = (L/T)/T = L/T2 = [x]/[t]2 (m/s2)

Uncertainty in Measurement In physics, often laws in form of mathematics are tested by experiments. No physical quantity can be determined with complete accuracy.

• Accuracy of measurement depends on the sensitivity of the apparatus, the skill of the person conducting the measurement, and the number of times the measurement is repeated.

• For example, assume the accuracy of measuring length of a rectangular plate is +-0.1 cm. If a side is measured to be 16.3 cm, it is said that the length of the side is measured to be 16.3 cm +-0.1 cm. Therefore, the true value lies between 16.2 cm and 16.4 cm.

Significant figure : a reliably known digitIn the example above the digits 16.3 are reliably knowni.e. three significant digits with known uncertainty

Uncertainty in Measurement (cont’d)

• Area of a plate: length of sides 16.3+-0.1 cm, 4.5+-0.1 cm

The values of the area range between (16.3-0.1 cm)(4.5-0.1 cm)= (16.2 cm)(4.4cm)=71.28 cm2

=71 cm2 and (16.3+0.1 cm)(4.5+0.1 cm)=75.44 cm2= 75 cm2.

The mid-point between these two extreme values is 73 cm2 with uncertainty of +-2 cm2 .

Two significant figures! (Note that 0.1 has only one significant figure as 0 is simply a decimal point indicator.)

Uncertainty in Measurement (cont’d)

1) In multiplying (dividing) two or more quantities, the number of significant figures in the final product (quotient) is the same as

the number of significant figures in the least accurate of thefactors being combined, where least accurate means having thelowest number of significant figures.

Two rules of thumb to determine the significant figures

• Area of a plate: length of sides 16.3+-0.1 cm, 4.5+-0.1 cm16.3 x 4.5 = 73.35 = 73 (rounded to two significant figures).

three (two) significant figuresTo get the final number of significant digit, it is necessaryto do some rounding: If the last digit dropped is less than 5,simply drop the digit. If it is greater than or equal to 5, raisethe last retained digit by one

Uncertainty in Measurement (cont’d)

• A sum of two numbers 123 and 5.35: 123.xxx

+ 5.35x ------------

128.xxx123 + 5.35 = 128.35 = 128

zero decimal placestwo decimal places

zero decimal places

2) When numbers are added (subtracted), the number of decimalplaces in the result should equal the smallest number of decimalplaces of any term in the sum (difference).

Uncertainty in Measurement (cont’d)

• More complex example : 2.35 x 5.86/1.57

- 2.35 x 5.89 = 13.842 = 13.8 13.8 / 1.57 = 8.7898 = 8.79

- 5.89 / 1.57 = 3.7516 = 3.75 2.35 x 3.75 = 8.8125 = 8.81

- 2.35 / 1.57 = 1.4968 = 1.50 1.50 x 5.89 = 8.835 = 8.84

A lesson learned : Since the last significant digit is only one representative from a range of possible values, this amount of discrepancies is expected.

Conversion of Units

• Some typical unit conversions 1 mile = 1,609 m = 1.609 km, 1 ft = 0.3048 m = 30.48 cm

1 m = 39.37 in. = 3.281 ft, 1 in. = 0.0254 m = 2.54 cm

• Example 1.4 28.0 m/s = ? mi/h

mi/s 1074.1m 1609

mi 00.1

s

m 28.0 m/s 0.28 2

Step 1: Conversion from m/s to mi/s:

Step 2: Conversion from mi/s to mi/h:

mi/h 6.62h

min0.60

min

s0.60

s

mi101.74 mi/s 1074.1 2-2

Since we use more than one unit for the same quantity, it is often necessary to convert one unit to another

Estimates and Order-of Magnitude

• Examples

- 75 kg ~ 102 kg (~ means “is on the order of” or “is approximately”)

- = 3.1415…~1 (~3 for less crude estimate)

For many problems, knowing the approximate value of a quantity within a factor of 10 or so is quite useful. This approximate value is called an order-of-magnitude estimate.

Estimates and Order-of Magnitude

• Example 1.6 : How much gasoline do we use?Estimate the number of gallons of gasoline used by all cars in theU.S. each year

Step 1: Number of cars cars 10~n)cars/perso (0.5people) 1000.3( 88

Step 2: Number of gallons used by a car per year

car

gal/yr10

galmi

10

carmi/yr10

mi/gal

driven mi of#

car

gal/yr# 3

4

Step 3: Number of gallons consumed per year

gal/yr 10)car

gal/yr(10cars) (10~gal# 1138

Estimates and Order-of Magnitude (cont’d)

• Example 1.8 : Number of galaxies in the UniverseInformation given: Observable distance = 10 billion light year (1010 ly)

14 galaxies in our local galaxy group 2 million (2x106) ly between local groups

1 ly = 9.5 x 1015 m

Volume of the local group of galaxies: 318363lg 10)10(~

3

4lylyrV

Number of galaxies per cubic ly:

317

318lg

310

10

10~

galaxies of #galaxies of #

ly

galaxies

ly

galaxies

Vly

Volume of observable universe:3303103 10)10(~

3

4lylyrVu

Number of galaxies in the Universe:

galaxieslyly

galaxiesV

ly u 10)10(10galaxies of #

~ galaxies of # 133303

173

Coordinate Systems

• Cartesian coordinate system

A point in the two dimensional Cartesian system is labeled with the coordinate (x,y)

Locations in space need to be specified by a coordinate system

Coordinate Systems (cont’d)

• Polar coordinate system

A point in the two dimensional polar system is labeled with the coordinate (r

Trigonometry

• sin, cos, tan etc.

hypotenuse

side opposite

side adjacent

Pythagorean theorem:

222 yxr

x

y

x

yr

x

r

xr

y

r

y

1

1

1

tantan

coscos

sinsin

Inverse functions:

Trigonometry (cont’d)

• Example 1.9 : Cartesian and polar coordinates

216714.0tan

714.0m 50.3

m 50.2

x

ytan

m 30.4

m) 50.2(m) 50.3(

1

22

22

yxrCartesian to polar: (x, y)=(-3.50,-2.50) m

Polar to Cartesian: (r)=(5.00 m, 37.0o)

m 01.30.37sinm) 00.5(sin

m 99.30.37cosm) 00.5(cos

ry

rx

Trigonometry (cont’d)

• Example 1.10 : How high is the building

m 37.3m) 0.46)(810.0(

m) 0.46)(0.39(tanheightm 46.0

height0.39tan

What is the height of the building?

What is the distance to the roof top?

m 2.59

m) 0.46(m) 3.37( 22

22

yxr

r, hyp

otenuse