Transcript Chap. 1
Convert 20 kilometers to METERS: 1000 m 4 20 km = 20 km ´ = 20 ´1000 m = 2.0 ´10 m 1 km Convert 20 miles to METERS: 1609 m 20 mi = 20 mi ´ = 20 ´1609 m = (32180 m) = 3.2 ´10 4 m 1 mi Convert 1.5 minutes to SECONDS: 60 s 1.5 min 1.5 min 1.5 60 s 90 s 1 min What is the length of the yellow bar? cm 0 1 2 3 4 5 6 7 8 9 10 11 cm 12 13 14 15 Length = 9.7 cm It makes NO sense to write Length = 9.73 cm, for example. The significant digits of a measurement are all those digits that we know for sure, plus one more digit. This last uncertain digit is the result of a careful estimate. With respect to significant digits, remember: 1. Zeros to the left of the first number different than zero are NOT significant digits. Example: 0.0000071 has two significant digits (7 and 1). 2. Zeros to the right of a significant digit ARE significant. Examples: 230.0 has four significant digits; (0.05600 ± 0.00005) has four significant digits. A = 125.391 Number A has 6 significant digits, and is the most precise of the numbers. B = 12.7 C = 2.17 B and C have 3 significant digits, but C is more precise than B. Sum and subtraction We keep the number of decimals of the least precise quantity. A+B+C = 140.261 140.3 Product and division We keep the number of significant digits of the least precise quantity. A x B = 1592.4657 159 x 101 We will adopt the international system of units which is the METRIC SYSTEM. Instead of miles, feet, inches ---- meters Instead of pounds, ounces PREFIX giga G mega M kilo k SCIENTIFIC NOTATION 109 IN FIGURES ---- kilograms IN WORDS 1 000 000 000 billion 106 1 000 000 million 103 1 000 thousand 100 1 one deci d 10-1 0.1 centi c 10-2 0.01 hundredth milli m 10-3 0.001 thousandth micro u 10-6 0.000 001 millionth nano n 10-9 0.000 000 001 billionth tenth It is to write numbers in terms of powers of 10 Examples: number 234.37 0.02 0.00430 written in scientific notation how many significant digits? 2.3437 ×102 five significant digits 2 ×10-2 one significant digit 4.30 ×10-3 three significant digits Discussion about significant digits and scientific notation in your textbook: Section 1.4 Let’s CHANGE THE UNITS of these measurements: L = 23 km L = _______ 2.3 x 104 m M = 10.3 kg M = _______ 1.03 x 104g g L = 224 m L = _______ 0.224 km km M = 23 g M = _______ 2.3 x 10-2 kg kg Notice that we have to preserve the number of significant digits!!! How to write RELATIVE ERRORS or UNCERTAINTIES: error % error or % uncertaint y 100 % measuremen t You can express a measurement both ways: measuremen t error measuremen t (% error) Example: (200 ± 5) cm 200 cm ± 2.5% VERY useful relation in physics: I call it “the rule of 3”: X1 X2 Y1 ? X1· ? = X2·Y1 ? = X2·Y1 ____X1 Mary eats 3 apples per day. How many apples will she have eaten in a week? 3 apples 1 day ? apples 7 days 3 · 7 = ? ·1 ? = 21 apples A year has 365 days. How many years do I have in 10 000 days? SAME UNITS!!! 1 year SAME UNITS!!! → 365 days x years → 10 000 days Two important words in a lab: In the fields of science, engineering, industry and statistics, accuracy is the degree of closeness of a measured or calculated quantity to its actual (true) value. How do you check the accuracy of a measurement? By using different tools and methods of measurement. Precision is also called reproducibility or repeatability, it is the degree to which further measurements or calculations show the same or similar results. How do you improve the precision of a measurement? By repeating the same measurement several times. High accuracy, but low precision High precision, but low accuracy Experimental errors arise in two forms: Random errors – Affect the PRECISION of the measurement. Various sources: judgment in reading a measurement instrument, fluctuations in the conditions of the experiment; poorly defined quantity such as an uneven side of a block, etc. How do we lessen the uncertainty from random errors? By repeating the measurements several times. Systematic errors – Affect the ACCURACY of the measurement. They are usually the same size of error in all measurements in a series: systematic error in the calibration of the measuring device, a flaw in the experiment such as the constant presence of friction, different temperature or pressure conditions, etc. How do we estimate the systematic errors? By using a different experimental design and comparing the results. Pre-requisite for PHY101: Fundamentals of Pre-Calculus I (MAT124) This is what you have learned in MA124 and will need again now: • intermediate algebra (appendix A.3) • trigonometry (appendix A.5) For this course you are required to demonstrate adequate mathematical background. NOT a mathematics course Math will be used as a tool that you already know We sympathize that math can be hard, therefore sometimes we will show problems in slow steps. But do not expect that always. You must then catch up at home, tutoring or office hours. • intermediate algebra appendix A.3 at the end of your book a) Some basic rules 8x = 32 x+2=8 b) Powers x2x4 = x6 x7 / x3 = x4 c) Factoring ax + ay + az = a(x + y + z) d) Quadratic equations 3x2 + 8x – 10 = 0 e) Linear equations f) Solving simultaneous linear equations x/5=9 DO the Extra Credit assignment #1 !!!!!!!!!!!! plot y = ax + b, where a is the slope of the line and b is the y-intercept. 5x + y = –8 and 2x – 2y = 4 ; solve for y and x. • trigonometry appendix A.5 at the end of your book definitions of sin, cos, tan are in Chapter 1 (Section 1.8) sin θ = side opposite θ hypotenuse sin2 θ + cos2 θ = 1 cos θ = side adjacent to θ hypotenuse sin 2θ = 2 sinθ cosθ tan θ = side opposite θ side adjacent to θ cos 2θ = cos2θ – sin2θ The following relationships apply to ANY triangle: 180 a b c Law of cosines: a b c 2bc cos 2 2 2 b a c 2ac cos 2 2 2 c 2 a 2 b 2 2ab cos Law of sines: a b c sin sin sin Which one is a RIGHT TRIANGLE? ONLY FOR THE RIGHT TRIANGLE: a b c 2 2 2 opposite side c sin θ hypotenuse a adjacent b cos θ hypotenuse a a c . b θ is an angle in degrees θ sin θ opposite c tan θ cosθ adjacent b