PHY121 Summer Session I, 2006
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Transcript of PHY121 Summer Session I, 2006
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: IntroductionStandards 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 MKSA 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 mDistance 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 kgObservable 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 00.1sm 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
min0.60min
s0.60s
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
cargal/yr10
galmi10
carmi/yr10
mi/galdriven mi of#
cargal/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 mVolume of the local group of galaxies: 318363
lg 10)10(~34 lylyrV
Number of galaxies per cubic ly:
317
318lg
3 1010 10~galaxies of #galaxies of #
lygalaxies
lygalaxies
Vly
Volume of observable universe: 3303103 10)10(~34 lylyrVu
Number of galaxies in the Universe:
galaxieslyly
galaxiesVly u 10)10(10galaxies of # ~ galaxies of # 13330
317
3
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
xy
xy
rx
rx
ry
ry
1
1
1
tantan
coscos
sinsin
Inverse functions:
Trigonometry (cont’d)• Example 1.9 : Cartesian and polar coordinates
216714.0tan
714.0m 50.3m 50.2
xytan
m 30.4m) 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(sinm 99.30.37cosm) 00.5(cos
ryrx
Trigonometry (cont’d)• Example 1.10 : How high is the building
m 37.3m) 0.46)(810.0(m) 0.46)(0.39(tanheight
m 46.0height0.39tan
What is the height of the building?
What is the distance to the roof top?
m 2.59m) 0.46(m) 3.37( 22
22
yxr
r, hyp
otenuse