Aug. 29, 2013 כ " ג אלול תשע " ג

The metric system Length

The metric system Mass

The metric system Time

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)

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, wave-lenghts coming from Isotope Kripton-86 Red-Orange emission.

Units – Kilogram Definition of 1 kilogram

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?

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.

Conversion of units 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?

Example 1 Which is higher, meter or 2700 ft? 1 ft = m (cover of book) 1 = m/ft 2700 ft * 1 = 2700 ft * m/ft = m Not higher!!!

Example 2 – Derived units The speed limit in the USA is 65 Mph (or Miles/hour). In Israel it is 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

Solution A 1 mile = 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) = m/sec > m/sec In the USA we drive faster!!!

Solution B 1 mile/hour = m/sec (book cover) 1 = (m/sec) (mile/hour) 65 mile/hour * 1 = 65 mile/hour * (m/sec) = (mile/hour) = m/sec

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)?

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 ?

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)

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

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

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

Example 1 Attached PDF -

Trigonometry Mathematical tools to help describe how the physical world works. 3 function: sinus – sin(x) cosinus – cos(x) tangent – tan(x)

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.

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)

Example 1 At 10AM the building casts a 50 m long shadow. At 2PM the angle between the sun’s 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?

Solution H X = 30 0

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 X+5 – the angle at 11AM Tan(x+5) = H/A1 = H/50 = tan(35)  H = 50 * tan(35) = m Tan(x) = H/A2 = tan(30) => A2= H/tan(30) =50*tan(35)/tan(30)= m

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.

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?

Solution

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) = 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.

Additional Trigonometric Relations Sinus relation CoSinus relation

Pythagorean Theorem H 2 = A 2 + O 2

Additional Practice Attached PDF

Summary Topics covered: Units and dimensions Unit conversion Linear functions and graph Trigonometry Next meeting: Scalars and vectors

The End

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.