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PHYS 1441 – Section 004 Lecture # Vectors

PHYS 1441 – Section 004 Lecture # Vectors. Monday, Sept. 8, 2008. Vectors and Scalars Properties of vectors Vector operations Components and unit vectors. Needs for Standards and Units. Basic quantities for physical measurements Length, Mass, and Time

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PHYS 1441 – Section 004 Lecture # Vectors

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  1. PHYS 1441 – Section 004Lecture # Vectors Monday, Sept. 8, 2008 • Vectors and Scalars • Properties of vectors • Vector operations • Components and unit vectors

  2. Needs for Standards and Units • Basic quantities for physical measurements • Length, Mass, and Time • Need a language that everyone can understand • Consistency is crucial for physical measurements • The same quantity measured by one must be comprehendible and reproducible by others • Practical matters contribute (appropriate size, etc.) • A system of units called SI (SystemInternationale) established in 1960 • Length in meters (m) • Mass in kilo-grams (kg) • Time in seconds (s)

  3. SI Units Definitions 1 m(Length) = 100 cm One meter is the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second. 1 kg (Mass) = 1000 g It is equal to the mass of the international prototype of the kilogram, made of platinum-iridium in International Bureau of Weights and Measure in France: could be defined in terms of electron mass 1 s (Time) One second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cesium 133 (Cs133) atom. Definition of Base Units • There are prefixes that scale the units larger or smaller for convenience (see text) • Units for other quantities, such as Kelvin for temperature etc.

  4. deci (d): 10-1 centi (c): 10-2 milli (m): 10-3 micro (m): 10-6 nano (n): 10-9 pico (p): 10-12 femto (f): 10-15 atto (a): 10-18 Prefixes, expressions and their meanings • deca (da): 101 • hecto (h): 102 • kilo (k): 103 • mega (M): 106 • giga (G): 109 • tera (T): 1012 • peta (P): 1015 • exa (E): 1018

  5. International Standard Institutes • International Bureau of Weights and Measure http://www.bipm.fr/ • Base unit definitions: http://www.bipm.fr/enus/3_SI/base_units.html • Unit Conversions: http://www.bipm.fr/enus/3_SI/ • US National Institute of Standards and Technology (NIST) http://www.nist.gov/

  6. How do we convert quantities from one unit to another? Unit 1 = Conversion factor X Unit 2

  7. Examples 1.3 & 1.4 • Ex 1.3: A silicon chip has an area of 1.25in2. Express this in cm2. What do we need to know? • Ex 1.4: Where the posted speed limit is 65 miles per hour (mi/h or mph), what is this speed (a) in meters per second (m/s) and (b) kilometers per hour (km/h)? (a) (b)

  8. Estimates & Order-of-Magnitude Calculations • Estimate = Approximation • Useful for rough calculations to check more accurate computer calculations which may contain program errors • Usually done with simplifying assumptions • Might require modification of assumptions, if higher precision is desired • Order of magnitude estimate: Estimates that yield about one significant figure and the power of 10 • Rapid estimating • Three orders of magnitude: 103=1,000 • Round up for Order of magnitude estimate; 8x107 ~ 108 • Similar terms: “Ball-park-figures”, “guesstimates”, etc

  9. r h Example 1.5 Estimate how much water is in a lake in the figure which is roughly circular, about 1km across, and you guess it to have an average depth of about 10m. simplify Volume of a cylinder What is the radius of the circle? Half the distance across… 1km/2=1000m/2=500m

  10. Some Fundamentals • Kinematics: Description of Motion without understanding the cause of the motion • Dynamics: Description of motion accompanied with understanding the cause of the motion • Vector and Scalar quantities: • Scalar: Physical quantities that require magnitude but no direction • speed, length, mass, etc • Vector: Physical quantities that require both magnitude and direction • velocity, acceleration, force, momentum • It does not make sense to say “I ran with velocity of 10miles/hour” if the runner is in a two or three dimensional space • Objects can be treated as point-like if their sizes are much smaller than the scale in the observer or reference object • Earth can be treated as a point like object (or a particle)in celestial problems • Any other examples?

  11. Vector and Scalar Vector quantities have both magnitude (size) and direction Force, displacement, momentum Normally denoted in BOLD letters, F, or a letter with arrow on top Their sizes or magnitudes are denoted with normal letters, F, or absolute values: Scalar quantities are specified completely with a value(+ or -) and units of measure Energy, heat, mass, charge Usually denoted in normal letters, E Both have units!!!

  12. y D F A B x E C Properties of Vectors sizes directions • Two vectors are the same if their and the are the same, no matter where they are on a coordinate system. Which are the same vectors? A=B=E=D Why aren’t the others? C: The same magnitude but opposite direction: C=-A:A negative vector F: The same direction but different magnitude

  13. A+B A+B A B A+B B B A A A-B A -B A B=2A Vector Operations • Addition: • Triangular Method: One can add vectors by connecting the head of one vector to the tail of the other (head-to-tail) • Parallelogram method: Connect the tails of the two vectors and extend • Addition is commutative: Changing order of operation does not affect the results A+B=B+A, A+B+C+D+E=E+C+A+B+D OR = • Subtraction: • The same as adding a negative vector:A - B = A + (-B) Since subtraction is the equivalent to adding a negative vector, subtraction is also commutative! True or false? • Multiplication by a scalar is increasing the magnitude A, B=2A

  14. y Ay A q x Ax • Unit vectors are dimensionless vectors with magnitudes exactly 1 • Unit vectors are usually expressed in i, j, k or • Vectors can be expressed using components and unit vectors Components and Unit Vectors }Components (+,+) Coordinate systems are useful in expressing vectors in their components (Ax,Ay) } Magnitude (-,+) (-,-) (+,-) So the above vector Acan be written as

  15. N Bsinq Bcosq E B 60o 20 q r A Example of Vector Addition A car travels 20.0km due north followed by 35.0km in a direction 60.0o west of north. Find the magnitude and direction of resultant displacement. Find other ways to solve this problem…

  16. Examples of Vector Operations Find the resultant vector which is the sum of A=(2.0i+2.0j) and B =(2.0i-4.0j) Find the resultant displacement of three consecutive displacements: d1=(15i+30j +12k)cm, d2=(23i+14j -5.0k)cm, and d3=(-13i+15j)cm Magnitude

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