1. Chapter C2 Vectors Fall 2008 Problems C2B.1, B2, B4, B7, B8, B9, C2S.1 Due Thursday.

2. Scalars Most physical quantities can be completely described by a single number. Some examples are: Most physical quantities can be completely described by a single number. Some examples are: Mass Mass Volume Volume Time Time Charge Charge Height Height Score on test Score on test Physical quantities that can be described by a single number are called scalars. Physical quantities that can be described by a single number are called scalars.

3. Vectors Certain physical quantities need a direction as well as a size (magnitude) to completely describe them. Some examples are: Certain physical quantities need a direction as well as a size (magnitude) to completely describe them. Some examples are: Velocity Velocity Force Force Displacement Displacement Momentum Momentum Acceleration Acceleration These are called vectors. These are called vectors.

4. Vectors

5. adding Vectors You may move a vector anyplace as long as the magnitude and the direction remain unchanged You may move a vector anyplace as long as the magnitude and the direction remain unchanged When adding vectors, 1) move the second vector, putting the tail of the second vector on the head of the first vector. When adding vectors, 1) move the second vector, putting the tail of the second vector on the head of the first vector. 2) Begin the tail of the resultant vector on the tail of the first vector and put the head of the resultant on the head of the last vector. 2) Begin the tail of the resultant vector on the tail of the first vector and put the head of the resultant on the head of the last vector. 3) Be certain to draw the head on the resultant vector! 3) Be certain to draw the head on the resultant vector! To subtract vectors, change the direction of the vector with the minus sign and add. To subtract vectors, change the direction of the vector with the minus sign and add.

6. adding Vectors

7. Uncertainty revisited What you are expected to know. Be able to estimate the uncertainty in a particular measurement. Be able to estimate the uncertainty in a particular measurement. Be able to estimate the uncertainty of the result when quantities containing uncertainties are used in calculations. Be able to estimate the uncertainty of the result when quantities containing uncertainties are used in calculations.

8. Example What is the uncertainty in the area of circle whose radius is 20 paces as determined by my “pacing it off” if I know my normal pace is.8 m? What is the uncertainty in the area of circle whose radius is 20 paces as determined by my “pacing it off” if I know my normal pace is.8 m? Step 1 – What is the uncertainty in my pace? Step 1 – What is the uncertainty in my pace? There is no single correct answer to this, give an answer you can defend) There is no single correct answer to this, give an answer you can defend) I will guess the uncertainty in my pace is 5 cm/pace. I will guess the uncertainty in my pace is 5 cm/pace.

9. The 5 cm represents 5/80 = 6% The 5 cm represents 5/80 = 6% The radius of the circle is.8 x 20 = 16 m The radius of the circle is.8 x 20 = 16 m The area of the circle is πr 2 = π16 2 =804 m 2 The area of the circle is πr 2 = π16 2 =804 m 2 Uncertainty is (0.06 2 + 0.06 2 ) ½ = 8% Uncertainty is (0.06 2 + 0.06 2 ) ½ = 8% 8% of 804 is 64 m 2 8% of 804 is 64 m 2 Final answer 804±64 m 2 Final answer 804±64 m 2 It would probably be better to write it as 800±60 m 2 It would probably be better to write it as 800±60 m 2

10. The best review for the Chapter 1-2 test is the old tests that are on the web page. The best review for the Chapter 1-2 test is the old tests that are on the web page. Chapter C1 problems should now be in the box on the front table. Chapter C1 problems should now be in the box on the front table. Test will be Monday. Test will be Monday. C2 problems are due Thursday in lab. C2 problems are due Thursday in lab. Thursday we will finish the vectors lab. Thursday we will finish the vectors lab.

11. Vector Components θ This is the length or the magnitude of the vector. The choice of a coordinate system is arbitrary. In physical problems there is often a particular choice that makes the problem easier.

12. Vector Components θ u y =u sinθ u x =u cosθ tanθ=u y /u x

13. In class exercises On your paper draw:

14. In class exercises On your paper draw: On your paper draw:

15. In class exercises Do on your paper: Do on your paper:

16. In class exercises

17. On your paper, draw the following vectors.

18. Unit vectors are unit vectors Note that u x,u y and u z are not vectors

19. In class exercises Find the components of the vector a below if its magnitude is 30 and the angle with respect to the x axis is 35º. Raise your hand when you have the equation for the vector written in the form. Find the components of the vector a below if its magnitude is 30 and the angle with respect to the x axis is 35º. Raise your hand when you have the equation for the vector written in the form. Find the components of the vector B below if its magnitude is 20 and the angle with respect to the y axis is 75º. Raise your hand when done. Find the components of the vector B below if its magnitude is 20 and the angle with respect to the y axis is 75º. Raise your hand when done. A ayay axax θ 75º B

20. Same problem, continued Add the two vectors Add the two vectors Give the components of the resultant Give the components of the resultant Give the magnitude of the resultant Give the magnitude of the resultant Give the angle of the result with the x axis Give the angle of the result with the x axis Draw a figure showing your results. Draw a figure showing your results.

21. C θ C = 29.9 Θ =35.9° Φ=54.1° -24 -17 Now calculate the magnitude and direction of the vector B-2A.

22. Test will be Monday. If you wish more time you may start early or stay after class. Test will be Monday. If you wish more time you may start early or stay after class. Hand in the vector lab (even if the last part is not complete. Hand in the vector lab (even if the last part is not complete. Any questions on the problems? All problems should be handed in. Any questions on the problems? All problems should be handed in. After a brief discussion of uncertainty and the unit multiplier methods, we will discuss any questions you have about the old tests. After a brief discussion of uncertainty and the unit multiplier methods, we will discuss any questions you have about the old tests.

23. Use of the unit operator Given that there are 2.21 lbs/kg, calculate the number of nanograms in a ton (2000 lb). Given that there are 2.21 lbs/kg, calculate the number of nanograms in a ton (2000 lb). lbs  ng lbs  ng

24. On Wednesday we will review for the test over chapters 1 and 2. On Wednesday we will review for the test over chapters 1 and 2. Ask any question – ask to see any type of example worked. Ask any question – ask to see any type of example worked. Hint for test: You not only need to know how to do the problem, you must clearly show how you arrived at your answer. Hint for test: You not only need to know how to do the problem, you must clearly show how you arrived at your answer. Thursday – Finish vectors lab Thursday – Finish vectors lab