Transcript of Aug. 29, 2013 כ " ג אלול תשע " ג. The metric system Length.
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- Aug. 29, 2013 " "
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- The metric system Length
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- The metric system Mass
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- The metric system Time
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- Units SI Units 3 base units: meter (m) as the unit of length
kilogram (kg) as the unit of mass second (sec) as the unit of time
CGS centimeter (cm), gram (gr), second (sec) BE (British
Engineering) foot (ft), slug (sl), second (sec)
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- Units Meter 1 meter was defined 4 times by different means base
on available technology4 times Current definition: (Oct, 14 th
1960) The length of 1,650,763.73 wave-lenghts coming from Isotope
Kripton-86 Red-Orange emission.
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- Units Kilogram Definition of 1 kilogram
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- Units Second Several definitions that were changed due to
inaccuracy and technology advances Current definition: The time
needed for 9,192,631,770 waves cycles of Cesium- 133 to occur in an
Atomic clock. How do you tell the time in Italy?
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- Units cont. Derived units units used to define a physical
property which are made from the base units, e.g. velocity is
defined by length and time. Large or small quantities of units are
presented as multiples of 10.
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- Conversion of units 829.8 meters The tallest building in the
world today is in Dubai. In China they plan on a taller building
that will be 2700 ft high Will it really be higher than the Burj
Khalifa?
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- Example 1 Which is higher, 829.8 meter or 2700 ft? 1 ft =
0.3048 m (cover of book) 1 = 0.3048 m/ft 2700 ft * 1 = 2700 ft *
0.3048 m/ft = 822.96 m Not higher!!!
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- Example 2 Derived units The speed limit in the USA is 65 Mph
(or Miles/hour). In Israel it is 27.78 m/sec. Where can we drive
faster? Solution A: Convert miles to meters, and hours to seconds
Solution B: Use conversion factor 1 for speed
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- Solution A 1 mile = 1.609 km = 1609 m 1 = 1609 m/mile 1 hour =
3600 sec 1 = 3600 sec/hour 65 Mph*1/1 = 65miles/hour*(1609m/miles)
(3600 sec/hour) = 29.05 m/sec > 27.78 m/sec In the USA we drive
faster!!!
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- Solution B 1 mile/hour = 0.4470 m/sec (book cover) 1 = 0.4470
(m/sec) (mile/hour) 65 mile/hour * 1 = 65 mile/hour * 0.4470
(m/sec) = (mile/hour) = 29.05 m/sec
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- Units summary challenge We are defining a new physical
dimension called Force. The dimension of force equals to the
multiplication of the body mass with the body acceleration (change
in speed during a fix time period). Denote the force with the
letter F 1. What are the force dimensions? 2. What are the force
units (in SI units)?
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- Units summary challenge 3. A force that is applied on a body
having a mass of 1kg cause acceleration of 1 m/sec 2, is defined as
having magnitude of 1N (Newton). What will be the force applied on
a body having a mass of 2kg that will cause acceleration of 5 m/sec
2 ?
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- Linear Function Y = AX + B or F(X) = AX + B A the function
slope B the Y value of the function were it crosses the Y axis
(X=0)
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- Graph Y=2X + 4 A = 2 B = 4 P1(X1,Y1) = (0,4) P2(X2,Y2) = (-2,0)
A = (Y2-Y1)/(X2-X1)= (0-4)/(-2-0) = 2 P1 P2
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- Direct and inverse relationships If I manufacture more bikes
the total revenue will increase. Direct relationship if we increase
the value of one parameter, the value of the other parameter will
be increased as well, and vice versa
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- Direct and inverse relationships If I smoke more cigarettes per
day my life expectancy will be lower Inverse relationship if we
increase the value of one parameter, the other parameter will be
decreased, and vice versa
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- Example 1 Attached PDF -
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- Trigonometry Mathematical tools to help describe how the
physical world works. 3 function: sinus sin(x) cosinus cos(x)
tangent tan(x)
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- Function Definitions Sin(x) = O/H Cos(x) = A/H Tan(x) = O/A O
the length of the side opposite to the angle x A the length of the
side adjacent to the angle x H the length of the hypotenuse of a
right triangle Sin(), Cos() and Tan() are values without units as
they are the ratio between lengths.
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- Additional relations O = A * tan(x)O = H * sin(x) A = O /
tan(x)A = H * cos(x) H = A / cos(x)H = O / sin(x)
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- Example 1 At 10AM the building casts a 50 m long shadow. At 2PM
the angle between the suns ray and the ground is 30 0, and it is
smaller than the angle of 10AM by 5 0 Calculate the height of the
building and the length of the shadow at 2PM?
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- Solution H X = 30 0
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- Solution cont. H the height of the building A1 the length of
shadow at 11AM A2 the length of the shadow at 2PM X the angle at
2PM - 30 0 X+5 the angle at 11AM Tan(x+5) = H/A1 = H/50 = tan(35) H
= 50 * tan(35) = 35.01 m Tan(x) = H/A2 = tan(30) => A2=
H/tan(30) =50*tan(35)/tan(30)= 60.64 m
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- Calculation of the angle There are cases were we know the
lengths of the triangle sides, and need the angle. Each
trigonometric function has an inverse function that gives the angle
for a given length ratio. X = sin -1 (O/H) X = cos -1 (A/H) X= tan
-1 (O/A) The -1 in the exponent does not mean it is a reciprocal.
This is the notation.
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- Example 2 I want to buy a ladder to climb on my house roof for
fixing. The height of the roof is 4.5 m. In the store I found a 6m
long ladder. On the web I found some tips on working with ladders,
suggesting that it must have at least 35 0 from the ground to
maintain stability. Will the ladder from the store maintain its
stability?
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- Solution
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- O = 4.5 m height of house H = 6 m ladder length* O/H = sin(x) X
= sin -1 (O/H) = sin -1 (4.5/6) = 48.59 0 The ladder will maintain
stability !!! * If the ladder will be put against the roof as shown
in the picture we will have to use a shorter length of H in the
calculation, and still we will see that the resulting angle will be
bigger.
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- Additional Trigonometric Relations Sinus relation CoSinus
relation
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- Pythagorean Theorem H 2 = A 2 + O 2
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- Additional Practice Attached PDF
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- Summary Topics covered: Units and dimensions Unit conversion
Linear functions and graph Trigonometry Next meeting: Scalars and
vectors
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- The End
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- Definition of 1 meter In 1791 The French Academy of Science
defined it as the distance between the North pole and the equator
along the line passing through Paris divided by 10,000,000 In 1889
definition as a distance between two marks on a Platinum-Iridium
bar kept in a temperature of 0 (zero) degrees Celsius 1 meter is
defined as the distance the light travels in a vacuum in a time of
1/299,792,458 seconds.