Physics 8 Monday, September 12, 2011

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1 Physics 8 Monday, September 12, 2011 If you didn t pick up second half of textbook (starting with Chapter 8) on Friday, you can pick it up now. The bookstore is expecting more clickers to arrive soon (tomorrow? next week). We will use the clickers today in class, but not having a clicker in class will not start to affect your grade until the bookstore has sufficient stock for everyone.

2 Clarify a few things from Friday why mole useful The most straightforward way to see the use of a mole is to compute the mass of a mole of protons (or neutrons): ( kg ) ( ) = kg/mol mol So a mole of protons or neutrons weighs 1 gram (± 1%). That s why a mole of carbon atoms weighs grams (or exactly 12 grams if it s pure 12 C).

3 Clarify a few things from Friday estimation Idea of order-of-magnitude estimation is to make a decent guess (good to a factor of 10) of some quantity. Useful as a reality check of a more detailed calculation, e.g. a computer model of your skyscraper bending in the wind. Floor area of Comcast Center: 60 floors (50 m)2 floor 10 5 m ft 2 The textbook s evaluate result and develop a feel involve making some kind of reality check on answers you calculate. If you really enjoy this sort of calculation, check out Guesstimation:

4 Clarify a few things from Friday dimensional analysis Here are some of your own words about converting a quantity from one unit to another. To convert a quantity given in one unit to the same quantity in a different unit, you need to multiply the first quantity by a ratio that is equal to 1. For example, if you would like to covert a quantity in inches to centimeters, you would multiply the quantity in inches by the ratio 2.54 cm/1 in. Because 2.54 cm/1 in is equal to 1, you are simply converting the original quantity from inches to centimeters without changing the quantity at all.

5 Clarify a few things from Friday dimensional analysis Here are some of your own words about converting a quantity from one unit to another. The simplest way to convert from one unit to another is to write the conversation factor for the two units as a ratio. For example, if I am trying to find out how many minutes are in 2 hours, I would use the conversation ratio of 1 hour : 60 minutes. Since 1 hour = 60 minutes, I can multiply 2 hours by 60 minutes / 1 hour since this is essentially like multiplying by 1. The units cancel out and I am left with 120 minutes

6 Clarify a few things from Friday dimensional analysis Here are some of your own words about converting a quantity from one unit to another. The simplest way to convert from one unit to another is to write the conversion factor for the two units as a ratio. Because multiplying a number by 1 doesn t change the number, you can multiply these ratios to convert units. The best way is to use ratios. First, determine the ratio that represents the relationship between the units (for example, 1 yard equals about 0.92 meters). Then, use that ratio to write an equation that will allow you to convert the units. (3 yds. = [3 yds.][0.92 m/1 yd.] = 3[0.92]m = 2.76 m. It is important to write the units into the equation because some will cancel out and leave you with the correct unit for the answer.

7 Today s responses in your own words A vector is a mathematical object that has both a direction and a magnitude. In physics it is very useful to model forces and motion. A vector is a quantity that defines the magnitude and direction in space. It is good for understanding and calculating the displacement of objects. A vector has both direction and magnitude. It is a measuring tool that is useful for figuring out where exactly something is. If a person walks 13 feet north and then 6 feet south, it is easy to express his displacement of 7 feet by using vectors. A unit vector is a simplified vector that has a magnitude of 1 (whereas regular vectors can have a magnitude of any number) and is used to describe direction in space. A vector is denoted by placing a small arrow above a letter. Average velocity and displacement are both examples of vectors.

8 Today s responses in your own words Speed is distance traveled divided by time. Velocity is speed with a direction. Speed is scalar and velocity is vector, similar to how distance is scalar and displacement is vector. Distance and displacement are similar because distance is the actual amount of moving that was made while displacement is the difference between where you started and where you ended up. You could have walked in a circle and ended up where you started- and that would have a positive distance but a zero displacement

9 Common questions about today s reading It takes a while to get used to the textbook s vector notation What is a unit vector? Why do you add & subtract vectors? How do you compute instantaneous velocity? What is the difference between average and instantaneous velocity? Spell out this chapter s concepts with clear definitions Please work out example 2.9 Maybe work through some instantaneous-velocity problems, preferably without using calculus. Perhaps talk about the impossible motion shown in graph

10 Definitions position r is vector (x, y, z) pointing from the origin (0, 0, 0) to the object s location in space displacement r = r f r i

11 Definitions position r is vector (x, y, z) pointing from the origin (0, 0, 0) to the object s location in space displacement r = r f r i velocity v = d r dt = (v x, v y, v z ) = ( dx dt, dy dt, dz dt ) speed v = v is magnitude (scalar) of velocity (vector)

12 Definitions position r is vector (x, y, z) pointing from the origin (0, 0, 0) to the object s location in space displacement r = r f r i velocity v = d r dt = (v x, v y, v z ) = ( dx dt, dy dt, dz dt ) speed v = v is magnitude (scalar) of velocity (vector) distance d = t f t i v dt is (formally) what you get when you integrate the (scalar) speed from time t i to time t f less formally, distance is what you get by breaking up your trajectory into segments (ending each time you change direction), and adding up r for all segments

13 Definitions position r is vector (x, y, z) pointing from the origin (0, 0, 0) to the object s location in space displacement r = r f r i velocity v = d r dt = (v x, v y, v z ) = ( dx dt, dy dt, dz dt ) speed v = v is magnitude (scalar) of velocity (vector) distance d = t f t i v dt is (formally) what you get when you integrate the (scalar) speed from time t i to time t f less formally, distance is what you get by breaking up your trajectory into segments (ending each time you change direction), and adding up r for all segments vectors are very useful on a 2D map ((x, y) or geocode) or in a 3D CAD model (x, y, z); in the 1D problems of Chapter 2, the use of vectors is a little bit contrived but we want to get into the physics before reviewing too much math

Example 2.9: Average velocity and average speed Consider my motion between frames 1 and 8 in Figure 2.1. Use the values listed in Table 2.1 to determine the answers to the following questions.

14 Example 2.9: Average velocity and average speed Consider my motion between frames 1 and 8 in Figure 2.1. Use the values listed in Table 2.1 to determine the answers to the following questions. (a) What is my average speed over this time interval? (b) What is the x component of my average velocity over this time interval? (c) Write the average velocity in terms of the unit vector î.

15 Vectors and vector notation The most common source of confusion in today s reading was the textbook s vector notation x components of vectors, unit vectors, etc. To give everyone a chance to get used to vector notation, let s look at the position and displacement vectors that describe the placement of floor joists in the back porch my dad helped my wife and me to build over the summer. Vectors turn out to be just as useful in computer graphics as they are in physics. I ll show you my first humble attempt to use Google Sketchup Maybe one homework session you can teach me to use Revit

16 Position and displacement vectors in computer graphics

17 Simplified plan view, ignoring finite joist thickness

18 Position and displacement vectors on simplified plan view displacement between neighboring joists is vector D = (1, 0). D x = 1, D y = 0. position of center of joist n (n = 1, 2,...) is vector C n = (n, 0) C n+1 = C n + D C n+1 C n = D position of end of joist n is E n = (n, L) E n,x = n, E n,y = L E n = n (1, 0) + L (0, 1) = n î + L ĵ î is just a name for the unit vector (1, 0), and ĵ is just a name for (0, 1). But until chapter 10, we will consider only the x coordinate.

19 Displacement vectors on a map

20 Displacement vectors on a map let V = S P. V x = 0 V y = +3 V = +3 ĵ V = (0, +3) To go from pool P to supermarket S, you displace yourself by the vector V = S P. You walk 3 blocks west.

21 Displacement vectors on a map let V = H M. V x = 2 V y = 0 V = 2 î V = ( 2, 0) To go from Meyerson M to hospital H, you displace yourself by the vector V = H M. You walk 2 blocks south.

22 Displacement vectors on a map let V = A D. V x = +4 blocks V y = 3 blocks V = 4 î 3 ĵ blocks V = (+4, 3) blocks What if I start at DRL, walk 4 blocks north, then walk 3 blocks west? What if I start at DRL, walk 3 blocks west, then walk 4 blocks north?

23 Vector notation Suppose the x axis points north, and I am walking north at 2 m/s. Using (more familiar?) notation, my velocity vector is v = (+2 m/s, 0, 0) = 2 m/s (1, 0, 0) So v x = +2 m/s, v y = 0, v z = 0. The unit vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1) are often given the names î, ĵ, and ˆk, respectively but for now we will use only î, and we will use only the x coordinate (not yet y or z). Since î is the unit vector in the x direction, we can write v = +2 m/s î or if you prefer, you can just write v x = +2 m/s

24 Drawing position (or displacement) vs. time Which statement best describes the motion depicted by this graph? (A) I walk 1.0 m/s forward for 10 s. Then I rest 10 s. Then I walk 1.0 m/s backward for 10 s. (B) I walk 0.5 m/s forward for 10 s. Then I rest 10 s. Then I walk 1.0 m/s forward for 10 s. (C) I walk 0.5 m/s forward for 10 s. Then I rest 10 s. Then I walk 0.5 m/s forward for 10 s. (D) I walk 1.0 m/s forward for 10 s. Then I rest 10 s. Then I walk 0.5 m/s forward for 10 s.

25 Average velocity What is my average velocity v av during the 30 second interval shown on this graph? (A) +1.0 m/s î (B) m/s î (C) +0.5 m/s î (D) m/s î

26 Instantaneous velocity What is my instantaneous velocity v at time t = 5 s? What is v at time t = 15 s? (A) +1.0 m/s î and 0 m/s î, respectively (B) +0.5 m/s î and +1.0 m/s î, respectively (C) +1.0 m/s î and +0.5 m/s î, respectively (D) +0.5 m/s î and +0.5 m/s î, respectively

27 Slope of the x(t) curve The slope of the curve in the position vs. time graph for an object s motion gives (A) the object s speed (B) the object s acceleration (C) the object s average velocity (D) the object s instantaneous velocity (E) not covered in today s material

Representing motion as x vs. t A person initially at point P in the illustration stays there a moment and then moves along the axis to Q and stays there a moment.

28 Representing motion as x vs. t A person initially at point P in the illustration stays there a moment and then moves along the axis to Q and stays there a moment. She then runs quickly to R, stays there a moment, and then strolls slowly back to P. Which of the position vs. time graphs below correctly represents this motion?

29 At t = 15 s in the position-vs-time graph below, am I (a) not moving (b) moving at constant speed (c) speeding up (d) slowing down

30 At t = 15 s in the position-vs-time graph below, am I (a) not moving (b) moving at constant speed (c) speeding up (d) slowing down

31 At t = 15 s in the position-vs-time graph below, is v x (the x component of velocity) (a) zero (b) not changing (c) increasing (d) decreasing

32 At t = 15 s in the position-vs-time graph below, is v x (the x component of velocity) (a) zero (b) not changing (c) increasing (d) decreasing

33 An object goes from one point in space to another. After it arrives at its destination, its displacement is: (a) either greater than or equal to (b) always greater than (c) always equal to (d) either smaller than or equal to (e) always smaller than (f) either smaller or larger than the distance it traveled.

A puzzle! A marathon runner runs at a steady 15 km/hr. When the runner is 7.5 km from the finish, a bird begins flying from the runner to the finish at 30 km/hr.

34 A puzzle! A marathon runner runs at a steady 15 km/hr. When the runner is 7.5 km from the finish, a bird begins flying from the runner to the finish at 30 km/hr. When the bird reaches the finish line, it turns around and flies back to the runner, and then turns around again, repeating the back-and-forth trips until the runner reaches the finish line. How many kilometers does the bird travel? (a) 10 km (b) 15 km (c) 20 km (d) 30 km

Chapter 2 Motion in One Dimension. Slide 2-1

Chapter 2 Motion in One Dimension Slide 2-1 MasteringPhysics, PackBack Answers You should be on both by now. MasteringPhysics first reading quiz Wednesday PackBack should have email & be signed up 2014