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## Chapter 1

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**Chapter 1**Units, Physical Quantities, and Vectors**Three KEYS for Chapter 1**• Fundamental quantities in physics (length, mass, time) • Units (meters, kilograms, seconds...) • Dimensional Analysis • Force = kg meter/sec2 • Power = Force x Velocity = kg m2/sec3**Three KEYS for Chapter 1**Fundamental quantities in physics (length, mass, time) Units (meters, kilograms, seconds...) Dimensional Analysis Significant figures in calculations 6.696 x 104 miles/hour 67,000 miles hour**Three KEYS for Chapter 1**Fundamental quantities in physics (length, mass, time) Units (meters, kilograms, seconds...) Dimensional Analysis Significant figures in calculations Vectors (magnitude, direction, units) 5 m/s at 45°**What you MUST be able to do…**Vectors & Vector mathematics vector componentsEx: v = velocity; vx = v cosqis the “x” component unit vectors (indicating direction only) Adding, subtracting, & multiplying vectors**Standards and units**• Length, mass, and time = three fundamental quantities (“dimensions”) of physics. • The SI (Système International) is the most widely used system of units. • Meeting ISO standards are mandatory for some industries. Why? • In SI units, length is measured inmeters, mass inkilograms, and time inseconds.**Unit consistency and conversions**• An equation must bedimensionally consistent. Terms to be added or equated must always have the same units. (Be sure you’re adding “apples to apples.”) • OK: 5 meters/sec x 10 hours =~ 2 x 102 km • (distance/time) x (time) = distance**Unit consistency and conversions**• An equation must bedimensionally consistent. Terms to be added or equated must always have the same units. (Be sure you’re adding “apples to apples.”) • OK: 5 meters/sec x 10 hours =~ 2 x 102 km • 5 meters/sec x 10 hour x(3600 sec/hour) • = 180,000 meters = 180 km = ~ 2 x 102 km**Unit consistency and conversions**• An equation must bedimensionally consistent. Terms to be added or equated must always have the same units. (Be sure you’re adding “apples to apples.”) • OK: 5 meters/sec x 10 hours =~ 2 x 102 km • NOT: 5 meters/sec x 10 kg = 50 Joules • (velocity) x (mass) = (energy)**Unit prefixes**• Table 1.1 shows some larger and smaller units for the fundamental quantities. • Learn these – and prefixes like Mega, Tera, Pico, etc.! • Skip Ahead to Slide 24 – Sig Fig Example**Measurement & Uncertainty**No measurement is exact; there is always some uncertainty due to limited instrument accuracy and difficulty reading results.**Measurement & Uncertainty**• The precision – and also uncertainty - of a measured quantity is indicated by its number of significant figures. • Ex: 8.7 centimeters • 2 sig figs • Specific rules for significant figures exist • In online homework, sig figs matter!**Significant Figures**• Number of significant figures = number of “reliably known digits” in a number. • Often possible to tell # of significant figures by the way the number is written: • 23.21 cm = four significant figures. • 0.062 cm = two significant figures (initial zeroes don’t count).**Significant Figures**• Numbers ending in zero are ambiguous. Does the last zero mean uncertainty to a factor of 10, or just 1? • Is 20 cm precise to 10 cm, or 1? We need rules! • 20 cm = one significant figure (trailing zeroes don’t count w/o decimal point) • 20. cm = two significant figures (trailing zeroes DO count w/ decimal point) • 20.0 cm = three significant figures**Rules for Significant Figures**• When multiplying or dividing numbers, or using functions, result has as many sig figs as term with fewest (the least precise). • ex: 11.3 cm x 6.8 cm = 77 cm. • When adding or subtracting, answer is no more precise than least precise number used. • ex: 1.213 + 2 = 3, not 3.213!**Significant Figures**• Calculators will not give right # of sig figs; usually give too many but sometimes give too few (especially if there are trailing zeroes after a decimal point). • top image: result of 2.0/3.0 • bottom image: result of 2.5 x 3.2**Scientific Notation**• Scientific notation is commonly used in physics; it allows the number of significant figures to be clearly shown. • Ex: cannot easily tell how many significant figures in “36,900”. • Clearly 3.69 x 104 has three; and 3.690 x 104 has four.**Measurement & Uncertainty**No measurement is exact; there is always some uncertainty due to limited instrument accuracy and difficulty reading results. Photo illustrates this – it would be difficult to measure the width of this board more accurately than ± 1 mm.**Uncertainty and significant figures**Every measurement has uncertainty Ex: 8.7 cm (2 sig figs) “8” is (fairly) certain 8.6? 8.8? 8.71? 8.69? Good practice – include uncertainty with every measurement! 8.7 0.1 meters**Uncertainty and significant figures**Uncertainty should match measurement in the least precise digit: 8.7 0.1 centimeters 8.70 0.10 centimeters 8.709 0.034 centimeters 8 1 centimeters Not… 8.7 +/- 0.034 cm**Relative Uncertainty**• Relative uncertainty: a percentage, the ratio of uncertainty to measured value, multiplied by 100. • ex. Measure a phone to be 8.8 ± 0.1 cmWhat is the relative uncertainty in this measurement?**Uncertainty and significant figures**Physics involves approximations; these can affect the precision of a measurement.**Uncertainty and significant figures**As this train mishap illustrates, even a small percent error can have spectacular results!**Conceptual Example: Significant figures**Using a protractor, you measure an angle to be 30°. (a) How many significant figures should you quote in this measurement?**Conceptual Example: Significant figures**Using a protractor, you measure an angle to be 30°. (a) How many significant figures should you quote in this measurement? What uncertainty? 2 sig figs! (30. +/- 1 degrees or 3.0 x 101 +/- 1 degrees)**Conceptual Example: Significant figures**Using a protractor, you measure an angle to be 30°. (b) What result would a calculator give for the cosine of this result? What should you report?**Conceptual Example: Significant figures**Using a protractor, you measure an angle to be 30°. (b) What result would a calculator give for the cosine of this result? What should you report? 0.866025403, but to two sig figs, 0.87!**1-3 Accuracy vs. Precision**• Accuracy is how close a measurement comes to the true value. • ex. Acceleration of Earth’s gravity = 9.81 m/sec2 Your experiment produces 10 ± 1 m/sec2 • You were accurate! How accurate? Measured by ERROR. • |Actual – Measured|/Actual x 100% • | 9.81 – 10 | / 9.81 x 100% = 1.9% Error**Accuracy vs. Precision**• Accuracy is how close a measurement comes to the true value • established by % error • Precision is a measure of repeatability of the measurement using the same instrument. • established by uncertainty in a measurement • reflected by the # of significant figures**Accuracy vs. Precision Example**• Example:You measure the acceleration of Earth’s gravitational force in the lab, which is accepted to be 9.81 m/sec2 • Your experiment produces 8.334 m/sec2 • Were you accurate? Were you precise?**Accuracy vs. Precision**Accuracy is how close a measurement comes to the true value. (established by % error) ex. Your experiment produces 8.334 m/sec2for the acceleration of gravity (9.81 m/sec2) Accuracy: (9.81 – 8.334)/9.81 x 100% = 15% error Is this good enough? Only you (or your boss/customer) know for sure! **Accuracy vs. Precision**Precision is the repeatability of the measurement using the same instrument. ex. Your experiment produces 8.334 m/sec2for the acceleration of gravity (9.81 m/sec2) Precision indicated by 4 sig figs Seems (subjectively) very precise – and precisely wrong!**Accuracy vs. Precision**Better Technique: Include uncertainty Your experiment produces 8.334 m/sec2 +/- 0.077 m/sec2 Your relative uncertainty is .077/8.334 x 100% = ~1% But your error was ~ 15% NOT a good result!**Accuracy vs. Precision**Better Technique: Include uncertainty Your experiment produces 8.3 m/sec2 +/- 1.2 m/sec2 Your relative uncertainty is 1.2 / 8.3 x 100% = ~15% Your error was still ~ 15% Much more reasonable a result!**Accuracy vs. Precision**• Precision is a measure of repeatability of the measurement using the same instrument. • established by uncertainty in a measurement • reflected by the # of significant figures • improved by repeated measurements! • Statistically, if each measurement is independent • make n measurements (and n> 10) • Improve precision by √(n-1) • Make 10 measurements, % uncertainty ~ 1/3**1-6 Order of Magnitude: Rapid Estimating**• Quick way to estimate calculated quantity: • round off all numbers in a calculation to one significant figure and then calculate. • result should be right order of magnitude • expressed by rounding off to nearest power of 10 • 104 meters • 108 light years**Order of Magnitude: Rapid Estimating**Example: Volume of a lake Estimate how much water there is in a particular lake, which is roughly circular, about 1 km across, and you guess it has an average depth of about 10 m.**Order of Magnitude: Rapid Estimating**Example: Volume of a lake Volume = p x r2 x depth = ~ 3 x 500 x 500 x 10 = ~75 x 105 = ~ 100 x 105 = ~ 107 cubic meters**Order of Magnitude: Rapid Estimating**Example: Volume of a lake Volume = p x r2 x depth = 7,853,981.634 cu. m~ 107 cubic meters**1-6 Order of Magnitude: Rapid Estimating**Example: Thickness of a page. Estimate the thickness of a page of your textbook. (Hint: you don’t need one of these!)**Solving problems in physics**The textbook offers a systematic problem-solving strategy with techniques for setting up and solving problems efficiently and accurately.**Solving problems in physics**Step 1: Identify relevant concepts, variables, what is known, what is needed, what is missing.**Solving problems in physics**Step 2: Set up the Problem – MAKE a SKETCH, label it, act it out, model it, decide what equations might apply. What units should the answer have? What value?**Solving problems in physics**Step 3: Execute the Solution, and EVALUATE your answer! Are the units right? Is it the right order of magnitude? Does it make SENSE?**Solving problems in physics**Good problems to gauge your learning “Test your Understanding” Questions throughout the book Conceptual “Clicker” questions linked online “Two dot” problems in the chapter Good problems to review before exams BRIDGING Problem @ end of each chapter *****Vectors and scalars**• A scalar quantity can be described by a single number, with some meaningful unit • 4 oranges • 20 miles • 5 miles/hour • 10 Joules of energy • 9 Volts**Vectors and scalars**A scalar quantity can be described by a single number with some meaningful unit A vector quantity has a magnitude and a direction in space, as well as some meaningful unit. 5 miles/hour North 18 Newtons in the “x direction” 50 Volts/meter down**Vectors and scalars**A scalar quantity can be described by a single number with some meaningful unit A vector quantity has a magnitude and a direction in space, as well as some meaningful unit. To establish the direction, you MUST first have a coordinate system! Standard x-y Cartesian coordinates common Compass directions (N-E-S-W)**y**5 m/sec at 30 degrees from the x axis towards y in the xy plane x z Drawing vectors • Draw a vector as a line with an arrowhead at its tip. • The length of the line shows the vector’s magnitude. • The direction of the line shows the vector’s directionrelativeto a coordinate system (that should be indicated!)**Drawing vectors**• Vectors can be identical in magnitude, direction, and units, but start from different places…