Physics 168 Luis Anchordoqui The Nature of Science Ø  Observation: important first step toward scientific theory! Requires imagination to tell what is important! Ø  Theories: created to explain observations ! Will make predictions! Ø  Observations will tell if the prediction is accurate, and the cycle goes on! Planetary Orbits How does a new theory get accepted? •  Predictions agree better with data! •  Explains a greater range of phenomena! Models, Theories & Laws Models are useful in understanding phenomena A model creates mental pictures: care must be taken to understand ! the limits of the model and not take it too seriously! A theory is detailed and can give testable predic:ons A law is a brief descrip:on of how nature behaves in a broad set of circumstances A principle is similar to a law, but applies to a narrower range of phenomena Units, Standards & SI System Quan:ty Unit Standard Length Meter Length of the path traveled by light in 1/299,792,458 second. Time Second Time required for 9,192,631,770 periods of radia:on emiMed by cesium atoms Mass Kilogram Pla:num cylinder in Interna:onal Bureau of Weights and Measures, Paris Measurement & Uncertainty Significant Figures No measurement is exact; there is always some uncertainty due to limited instrument accuracy and difficulty reading results! For example, it would be difficult to measure the width of this table to better than a millimeter! Measurement & Uncertainty Significant Figures (cont’d) Estimated uncertainty is written with a ± sign! 8.8 ± 0.1cm Percent uncertainty is the ratio of the uncertainty to the measured value, multiplied by 100:! 0.1 ⇥ 100% ⇡ 1% 8.8 Measurement & Uncertainty Significant Figures (cont’d) The number of significant figures is the number of reliably known digits in a number! It is usually possible to tell the number of significant figures ! by the way the number is written:! •  23.21 cm has 4 significant figures! •  0.062 cm has 2 significant figures(the initial zeroes don’t count)! •  80 km is ambiguous- it could have 1 or 2 significant figures! If it has 2, it should be written 80. km! Measurement & Uncertainty Significant Figures (cont’d) When multiplying or dividing numbers, the result has as many significant figures as the number used in the calculation with the fewest significant figures! Example: 11.3 cm ⇥ 6.8 cm = 77cm When adding or subtracting, the answer is no more accurate than the least accurate number used! Global positioning satellites (GPS) can be used to determine positions with great accuracy System works by determining distance between observer and each of several satellites orbiting Earth! If one of satellites is at a distance of 20,000 km from you, what percent accuracy in distance is required if we desired a 2 m uncertainty? ! How many significant figures do we need to have in that distance? ! percentage accuracy ! 10 5 % 8 significant figures are needed ! Order of Magnitude: Rapid Estimating A quick way to estimate a calculated quantity is to round off all numbers to one significant figure and then calculate Your result should at least be the right order of magnitude this can be expressed by rounding it off to the nearest power of 10! Diagrams are also very useful in making estimations! 1.5 m x 13 = )x= m 2 m 18 m 2 Finally you add your eye height to find the building is about 15 m tall Prefixes for Powers of 10 Universe by Orders of Magnitude Laser beam from Macdonald Observatory used to measure distance to moon ! Distance can be measured within a few centimeters by measuring time required for beam to go to moon and back after reflecting off a mirror place on moon by Apollo 14 astronauts! (measured time 2.5 seconds) dEarth moon 5 = 3.8 ⇥ 10 km Hold a pencil in front of your eye at a position where its blunt end just blocks out Moon ! Make appropriate measurements to estimate diameter of Moon! Moon diameter = 3500 km ! Dimensions & Dimensional Analysis Dimensions of a quantity are the base units that make it up ! they are generally written using square brackets! Example: Speed = distance / :me Dimensions of speed: [L/T] Quantities that are being added or subtracted must have same dimensions ! Quantity calculated as solution to a problem should have correct dimensions! Dimensions of Physical Quantities Quantity! Area ! Volume! Speed ! Acceleration! Force! Pressure! Density! Energy! Power ! Symbol ! A V v a F P ⇢ E P Dimension! 2 L 3 L L/T 2 L/T 2 mL/T m/LT2 m/L3 2 2 mL /T 2 3 mL /T Vectors Quantities that have magnitude and direction à vectors ! Quantities with magnitude but no associated direction à scalars ! Examples Vectors are equal if their magnitudes and directions are same ! Addition of Vectors Parallelogram method of vector addition ! Vector addition is associative ! Subtraction of Vectors Components of Vectors q ~ = A2x + A2y |A| Ay tan ✓ = Ax Unit Vectors A unit vector is a dimenionless vector with magnitude exactly equal to 1! ~ Example A = Axˆi + Ay ˆj + Az kˆ Properties of Vectors