PHYS 407S: General Physics I Hybrid Studio. Studio Activities. Kai Germaschewski Michael Briggs

Transcription

1 PHYS 407S: General Physics I Hybrid Studio Studio Activities Kai Germaschewski Michael Briggs

2 ii

3 1 FUNDAMENTALS 1 Fundamentals Name: 1.1 Numbers, Units & Significant Figures Units In science and engineering, we seek to describe the world not only qualitatively ( he was going pretty fast. ) but rather quantitatively ( he was going 76 mph when he was pulled over ). Physical quantities are properties that can be measured. Most quantities we use in science and engineering are meaningless without units. If you ask, How far away do you live?, an answer of 5. really isn t all the useful. A 5 minute walk? 5 miles away? A 5 hour drive? Of course we often skip indicating units in daily life if someone gives you the answer 60 to the question What s the speed limit here?, it s clear that it s 60 miles per hour, (though a Canadian might think it s 60 kilometers per hour.) Still, 60 by itself isn t meaningful only by the (maybe implied) unit does it become a meaningful quantity. In physics and engineering, we aim not to be sloppyto rely on others to guess what unit we meant, but we include the unit explicitly. Not only is it good style, keeping units in your calculations also provides a way to avoid errors and sanity check your results and it would have avoided the Gimli glider incident. Another classic example of why units are important is NASAâĂŹs 1999 Mars Climate Orbiter. The intricate maneuvering required to place the craft in orbit around Mars was being done by two separate teams. Lockheed Martin, who designed the craft, used English units (feet, pounds, etc.), and their control interface assumed such units were being used (but did not indicate units to make that clear). The flight engineering team at the Jet Propulsion Laboratory uses metric units, as input on how to fire the thrusters. The ground software was supposed to provide commands using metric units as well, but actually provided the data in imperial units. The result of this mismatch was that the $125 million craft was steered too close to the planet and was torn up in Mars atmosphere. and sent hurtling towards the sun. Unit conversions are done by multiplying a quantity by a ratio that has a value of 1. To clarify this, let s find how many seconds are in 3 days. We can do this by using equalities relating seconds, minutes, hours and days: 1 day = 24 h 1 h = 60 min 1 min = 60 s We can make conversion factors from these equalities. For example, to convert 3 days into a number of hours, we need to make a fraction with days in the denominator and hours in the numerator. If we divide the leftmost equation above by 1 day, we get 1 = 24 h 1 day This fraction has a value of 1, so if multiply another number by it, we are not changing the value of that number but we can use the fraction to change the units of another number. So, we can convert 3 days to hours like this: 3 days = 3 days 24 h 1 day = 72 h 1

4 1.1 Numbers, Units & Significant Figures 1 FUNDAMENTALS The same thing can be done to convert hours to minutes, and then minutes to seconds: 3 days = 3 days 24 h 60 min 60 s 1 day 1 h 1 min = s Each of these conversion factors is a fraction of magnitude 1, which can be used to convert from one unit to another. The process of unit conversion can often involve multiple conversion factors, as in the case of converting from days to seconds above. To do a unit conversion, you need to first find an equality relating one quantity to another (e.g., 1 inch = 2.54 cm, 1 mile = 1760 yards), and turn that equality into a fraction that will accomplish the conversion you want. Which side of the equality should be the numerator and which the denominator depends on the direction you are converting you want the unit you want get rid of to cancel out, so that you ll be left with just the new unit! Notice that in the conversion process from days to seconds, the conversion factors were created such that the units in the denominator of each conversion factor cancelled out the units left by the last conversion, turning the units into the units given in the numerator. Now it is your turn to get some practice (show your work!). 1. You are building a deck on your house, and want the deck to be 16 8 long. But, your only measuring tape is a metric one. How many meters long should the deck be? 2. The Chinese philosopher Confucius said a journey of a thousand miles begins with a single step. Estimate how long an average step is in inches, then calculate how many steps you would have to take to go a thousand miles. 2

5 1 FUNDAMENTALS 1.1 Numbers, Units & Significant Figures 3. Usain Bolt currently holds the world record for the 200 meter sprint (and the title of being the fastest human), with a time of seconds. For that 200 meters, what was his average speed in miles per hour? (1 mile = 5,280 ft) 4. Volumes can be measured in SI units in cubic meters (or cubic centimeters, etc.), or in liters. Liters were defined by setting 1 ml equal to 1 cm 3, to create a metric unit for volume. In US units, we measure volumes in gallons, quarts, etc. (1 gallon = 4 quarts = 3.79 liters). An Olympic size swimming pool is 164 feet long by 82 feet wide, and at least 6 7 deep. Assuming a pool slopes uniformly from 6 7 deep at the shallow end to 16 5" deep at the deep end, how many gallons of water would it take to fill the pool? Assess whether your answer seems reasonable (i.e., if you get 1.4 gallons, does that sound right?) 5. Actually, in metric countries the Olympic size swimming pools dimensions are defined to be at least 50 meters long by 25 meters wide and at least 2 meters deep. Let s say the pool slopes from 2 meters deep on the shallow end to 5 meters deep on the deep end. How many cubic meters of water would be needed to fill this pool? How many liters is that? Is this problem easier then the previous one? (Why / why not?) 3

6 1.1 Numbers, Units & Significant Figures 1 FUNDAMENTALS 6. Let s put you into the pilot seat of the Boing 767 that conducted the glider landing in Gimli. The pilot correctly calculated a quantity of 22,300 kg of fuel needed for their flight from Montreal to Edmonton. They used a dripstick to figure out the current fuel in the plane s tanks to be 7,682 liters. Their goal was to figure out how much fuel (in liters) to add so that they would have the desired 22,300 kg of fuel on board. The refueling paperwork showed a factor of 1.77 to convert between fuel weight and volume. They assumed it to mean that 1 liter of fuel weighs 1.77 kg, though in reality 1 liter of fuel weighs 1.77 pounds. How many of liters of fuel did they add to their tanks? What would have been the right amount? 7. The Voyager 1 and 2 spacecraft, launched in 1977 to study the planets of our solar system, are the most distant man-made objects from Earth. They are also the fastest moving manmade objects, moving at roughly km/year. How fast are these spacecraft moving in miles per hour? 8. In the space below (or on scrap paper), try to come up with a method to calculate the sum of all numbers between 1 and 1,000 (inclusive) that are divisible by 3 or 5, but not both. Do not use your calculator (the intent is to challenge you to figure out how to do it yourself.) 4

7 1 FUNDAMENTALS 1.1 Numbers, Units & Significant Figures Significant Figures When writing down values of physical quantities or constants, we need to pay attention to how accurate those numbers are. Joe wants to buy new carpeting and measures the length and width of his rectangular-shaped bedroom. He gets L = 172 and W = 103. Writing down his numbers like this (and assuming Joe took PHYS 407 before), the last digit of the number is assumed to be rounded, ie., the actual measurement for L might have been between and Sometimes it is useful to write the uncertainty explicitly, like this: L = (172 ± 0.5) The ±0.5 is referred to as the absolute error. It is often more useful to think in terms of a relative (or percent) error, i.e. in this case the relative error is 0.5/172 = %. 9. What are the absolute and relative errors in Joe s measurement for the width W? 10. How do the absolute errors compare between the two measurements? How do the relative errors compare? 11. What is the area of Joe s bedroom? Also calculate the absolute and relative errors in that quantity. How large is Joe s room in square feet? 5

8 1.1 Numbers, Units & Significant Figures 1 FUNDAMENTALS In this class, when we measure or calculate a quantity, we will generally try to write down the result with three significant figures. E.g., If you measure a weight to be grams, you would round this down to 49.0 grams it is important to keep the trailing zero to indicate that your measurement is exact to three rather than just two ( 49 grams ) significant figures. If you wanted to express this in kilograms, you would write grams leading zeros don t get counted as significant figures, just trailing ones do. 6