Introduction to Mechanics Unit Conversions Order of Magnitude
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1 Introduction to Mechanics Unit Conversions Order of Magnitude Lana Sheridan De Anza College Sept 28, 2017
2 Last time symbols for scaling units scientific notation precision and accuracy dimensional analysis unit conversions (non-si units)
3 Overview unit conversions (non-si units) order of magnitude calculations (how to solve problems?)
4 Unit Conversion Example: what is 9 inches (in) in feet (ft)? 3/4 of a foot, or 0.75 feet. 12 in = 1 ft. ( ) 1 foot (9 inches) 12 = 3 inches 4 ft
5 Unit Conversion Examples To solve that problem, we multiplied the value we wished to convert by 1. ( ) 1 foot (9 inches) = 0.75 ft 12 inches }{{} 1 Any number times 1 remains unchanged. The value remains the same, but the units change, in this case, from inches to feet.
6 Unit Conversion Examples The distance between two cities is 100 mi. What is the number of kilometers between the two cities? A smaller than 100 B larger than 100 C equal to 100
7 Unit Conversion Examples It may be necessary to change units several times to get to the unit you need. Example: how many seconds are there in a day?
8 Unit Conversion Examples What is 60.0 mi/hr in m/s? (mi is miles, hr is hours) 1 mi = km
9 Order of Magnitude Calculation One way to get a hypothesis what an answer should be: do an Order of Magnitude Calculation. This is a useful tool for estimating the answer. The goal is just to get an idea of how big the answer should be.
10 Order of magnitude examples About how many times does your heart beat during your life?
11 Order of magnitude examples About how many times does your heart beat during your life? Your heart rate?
12 Order of magnitude examples About how many times does your heart beat during your life? Your heart rate? Call it 100 (10 2 ) beats per minute for simplicity. How many minutes in a life...? years in a life minutes in a year beats in a minute
13 Order of magnitude examples About how many times does your heart beat during your life? Your heart rate? Call it 100 (10 2 ) beats per minute for simplicity. How many minutes in a life...? years in a life minutes in a year beats in a minute Years in a life?
14 Order of magnitude examples About how many times does your heart beat during your life? Your heart rate? Call it 100 (10 2 ) beats per minute for simplicity. How many minutes in a life...? years in a life minutes in a year beats in a minute Years in a life? Optimistic: 100 = Minutes in a year: = 500, 000 = min/year
15 Order of magnitude examples About how many times does your heart beat during your life? Total heart beats in your life: years in a life minutes in a year beats in a minute (10 2 years) ( min/year) (10 2 beats/min) = beats = 5 billion beats 1 vendian.org
16 Order of magnitude examples What is the radius of the Earth?
17 Order of magnitude examples What is the radius of the Earth? If you fly across the United States, how many time zones do you cross?
18 Order of magnitude examples What is the radius of the Earth? If you fly across the United States, how many time zones do you cross? Answer: 3.
19 Order of magnitude examples What is the radius of the Earth? If you fly across the United States, how many time zones do you cross? Answer: 3. What is the average distance across the US?
20 Order of magnitude examples What is the radius of the Earth? If you fly across the United States, how many time zones do you cross? Answer: 3. What is the average distance across the US? Answer: about 3000 miles. On average, there are about 1000 miles of distance traveled per time zone. How many time zones are around the Earth?
21 Order of magnitude examples What is the radius of the Earth? If you fly across the United States, how many time zones do you cross? Answer: 3. What is the average distance across the US? Answer: about 3000 miles. On average, there are about 1000 miles of distance traveled per time zone. How many time zones are around the Earth? There must be 24 time zones around the earth in all since there are 24 hours in the day.
22 Order of magnitude examples What is the circumference of the Earth? 1 maa.org
23 Order of magnitude examples What is the circumference of the Earth? Answer: about 24,000 miles. The circumference of a circle is c = 2πr where r is the radius. Take 2π 6. The radius of the Earth: r = c 24, 000 mi = 4, 000 mi 2π 6 1 maa.org
24 Order of magnitude examples What is the circumference of the Earth? Answer: about 24,000 miles. The circumference of a circle is c = 2πr where r is the radius. Take 2π 6. The radius of the Earth: 1 mi 1.6 km Radius of the Earth in meters: r = c 24, 000 mi = 4, 000 mi 2π 6 4, 000 mi 1600 m/mi = 6, 400, 000 m = m 1 maa.org
25 Order of magnitude examples What is the circumference of the Earth? Answer: about 24,000 miles. The circumference of a circle is c = 2πr where r is the radius. Take 2π 6. The radius of the Earth: 1 mi 1.6 km Radius of the Earth in meters: r = c 24, 000 mi = 4, 000 mi 2π 6 4, 000 mi 1600 m/mi = 6, 400, 000 m = m Actual answer: m Pretty close! 1 maa.org
26 How to solve problems Solving physics problems is often not simple. To get into good habits for future work in physics, we will follow a set process. This process is similar to the process that physicists and engineers go through solving problems, sometimes only mentally, sometimes explicitly. (Also have a look at the similar process and examples on page 12 of the textbook.)
27 How to solve problems 1 Draw a diagram, sketch, or graph showing the situation in the question.
28 How to solve problems 1 Draw a diagram, sketch, or graph showing the situation in the question. 2 Make a hypothesis or estimate of what the answer will be.
29 How to solve problems 1 Draw a diagram, sketch, or graph showing the situation in the question. 2 Make a hypothesis or estimate of what the answer will be. 3 Solve the question or problem: a If it s a question - i Explanation or proof; make sure that the principles used are clearly stated.
30 How to solve problems 1 Draw a diagram, sketch, or graph showing the situation in the question. 2 Make a hypothesis or estimate of what the answer will be. 3 Solve the question or problem: a If it s a question - i Explanation or proof; make sure that the principles used are clearly stated. b If it s a problem - i Write out quantities given in question and quantity asked for. ii Write out the equation(s) you will use. (Start from equations we have discussed in class.) iii Do any required algebra. iv Plug in givens and solve. v Check units.
31 How to solve problems 1 Draw a diagram, sketch, or graph showing the situation in the question. 2 Make a hypothesis or estimate of what the answer will be. 3 Solve the question or problem: a If it s a question - i Explanation or proof; make sure that the principles used are clearly stated. b If it s a problem - i Write out quantities given in question and quantity asked for. ii Write out the equation(s) you will use. (Start from equations we have discussed in class.) iii Do any required algebra. iv Plug in givens and solve. v Check units. 4 Analyze answer as appropriate. a Compare answer to hypothesis - if it is not the same try to explain why. b Is your answer reasonable? / Compare to other things your are familiar with. c Consider limits or special cases.
32 How to solve problems Example question: What is the volume of a sphere in terms of the radius of the sphere? You should use dimensional analysis to check your answer.
33 How to solve problems Example question: What is the volume of a sphere in terms of the radius of the sphere? You should use dimensional analysis to check your answer. Sketch:
34 How to solve problems Example question: What is the volume of a sphere in terms of the radius of the sphere? You should use dimensional analysis to check your answer. Sketch: Hypothesis / guess: I can t remember, but I think it s something like 4πr 2.
35 How to solve problems What is the volume of a sphere in terms of the radius of the sphere? You should use dimensional analysis to check your answer. Solve: (The volume of a sphere can be found using calculus, but this is not a calculus based course.) Looked up the formula in math notes: 4 3 πr 3.
36 How to solve problems What is the volume of a sphere in terms of the radius of the sphere? You should use dimensional analysis to check your answer. Solve: (The volume of a sphere can be found using calculus, but this is not a calculus based course.) Looked up the formula in math notes: 4 3 πr 3. The dimensions of the formula are [L] 3, since the radius is a length and 4 3π is a constant (dimensionless quantity). If r is measured in meters, the units of the volume answer would be m 3.
37 How to solve problems Continued: Analyze answer: The answer is different from my guess. My guess is wrong because it has the wrong dimensions: dimensions of area [L] 2, not volume [L] 3. Actually, 4πr 2 is the surface area of a sphere. 4 3 πr 3 is reasonable because the dimensions are correct.
38 How to solve problems Can say even more... Analyze answer cont d: Also, 4 3π 4, a sphere would fit in a cube that has side length 2r. That cube would have volume 8r 3. The volume of the cube must be greater than the sphere, and 8 > 4, so this equation would agree. Moreover, the sphere can be inscribed in a cylinder of radius r, height 2r. The cylinder s volume is πr 2 h = 2πr 3, and must be greater than the sphere s. 2 > 4/3.
39 Summary unit conversions order of magnitude calculations (how to solve problems?) Homework unit conversion worksheet, due Mon, Oct 2 Not collected Homework Walker Physics: Ch 1, onward from page 14. Probs: 37, 39
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