2. Homework Format

3. Cover Page NAME PHY 108.002 DateProblemsGrade Staple at 45 0

4. Harry Downing PHY 108.002 1-28-09 Ch 11 – 2, 6, 9, 16 Grade 5, 4, 5, 3 Cover Page, Example Pass out some example engineering pad paper and take pictures. Staple at 45 0

5. What are the x and y components of the force shown? Name Problem # x y x y

6. Chapter 1 Speed, Displacement, Velocity: An Introduction to Vectors

7. Imagine that you have a map that leads you to a buried treasure. Imagine that you have a map that leads you to a buried treasure. This map has instructions such as 15 This map has instructions such as 15 paces west northwest paces west northwest of the skull. The 15 paces is The 15 paces is a distance and west northwest is a direction. N

8. A quantity that has nothing to do with spatial direction. A quantity that has nothing to do with spatial direction. Examples of scalars in physics are Examples of scalars in physics are mass time distance or lengthdensity workenergy temperaturecharge A SCALAR QUANTITY

9. A distance traveled, a path length, etc. A distance traveled, a path length, etc. DISTANCE ( l )

10. AVERAGE SPEED Average Speed = distance/time Units - m/s, ft/s, etc.

11. Sorry, Ma’am, but you were doing 45 mph in a 30 mph zone. But I haven’t driven 45 miles yet. Okay, okay, would you believe that I haven’t been driving for an hour yet? Speeding Little Old Lady

12. From A to B you average 30 mph. Example of Average Speed You take a trip from A to B and back to A. You take a trip from A to B and back to A. You want to average 60 mph for the round trip A to B to A. You want to average 60 mph for the round trip A to B to A. A B 2 miles What is your average speed on the return trip from B to A? 30 mph ?

13. Average Speed

14. INSTANTANEOUS SPEED Instantaneous Speed is the speed you would read from a speedometer.

15. Slope of a distance versus time curve

16. A quantity that can be specified completely only if we provide both its magnitude (size) and direction. A quantity that can be specified completely only if we provide both its magnitude (size) and direction. Examples of vectors in physics are Examples of vectors in physics are displacementvelocity accelerationforce momentumangular momentum A VECTOR QUANTITY

17. The math associated with scalars is familiar to everyone. The math associated with scalars is familiar to everyone. The math associated with vectors is more involved. The math associated with vectors is more involved.

18. THE DISPLACEMENT When an object moves from one point in space to another, the displacement is the vector from the initial location to the final location. It is independent of the actual distance traveled.

19. VELOCITY Average Velocity = Displacement/time Average Velocity = Displacement/time Units - m/s, ft/s, etc. Instantaneous Velocity of an object is its instantaneous speed plus the direction it is traveling. Instantaneous Velocity of an object is its instantaneous speed plus the direction it is traveling. Velocity is a vector. Velocity is a vector.

20. Displacement and Average Velocity Distance traveled is the length of the path taken. Average velocity =

21. Average Velocity

22. INSTANTANEOUS VELOCITY

23. The resultant is the sum of a number of vectors of a particular type. Example: Force Vectors Displacement Vectors Displacement Vectors THE ADDITION OF VECTORS

24. Let’s use a treasure map again as an example of the addition of vectors. Let’s use a treasure map again as an example of the addition of vectors. Let’s imagine the instructions tell you to go 4 miles east then 3 miles north. Let’s imagine the instructions tell you to go 4 miles east then 3 miles north. THE TIP-TO-TAIL (OR POLYGON) METHOD

25. 4 miles 3 miles 5 miles 36.9 0

26. In this case you could have gone 3 miles north first and then 4 miles east next and still end up at the same location. In this case you could have gone 3 miles north first and then 4 miles east next and still end up at the same location. Your final position is 5 miles at 36.9 0 north of east. Your final position is 5 miles at 36.9 0 north of east. It would have saved time if that had been the one distance and one direction traveled in the first place. It would have saved time if that had been the one distance and one direction traveled in the first place.

27. We say that the 5 miles at 36.9 0 north of east is the vector sum of the 4 miles east vector and the 3 miles north vector. We say that the 5 miles at 36.9 0 north of east is the vector sum of the 4 miles east vector and the 3 miles north vector. The order of the addition does not matter. The order of the addition does not matter. Examples of addition of vectors follows. The method used will be the head-to-tail. Examples of addition of vectors follows. The method used will be the head-to-tail.

29. W E S N

30. 0 90 0 180 0 270 0 W E S N Resultant Equilibrant

31. PARALLELOGRAM METHOD

33. SUBTRACTION OF VECTORS

34.  THE TRIGONOMETRIC FUNCTIONS adjacent-  opposite-  hypotenuse A C B

35.  adjacent-  opposite-  hypotenuse A C B

36.  adjacent-  opposite-  hypotenuse A C B

37. A COMPONENT OF A VECTOR y x

38. y x

39. COMPONENT METHOD FOR ADDING VECTORS Resolve each vector into components along the axes used. Resolve each vector into components along the axes used. Remember that components along the negative direction of an axis will be considered negative in the following additions. Add all the x-components together. Add all the x-components together. This will be This will be Add all the y-components together. Add all the y-components together. This will be This will be Add all the z-components together. Add all the z-components together. This will be This will be Then Then In two dimensions

40. z y x

41. y x

42. y x UNIT VECTORS Consider a vector that has unit length along the x-axis direction. Call it the i unit vector. Consider a vector that has unit length along the y-axis direction. Call it the j unit vector. Then the x-component of R becomesThe y-component of R becomes In general a vector has components along three mutually perpendicular axes (a k unit vector along the z-axis direction) and thus can be written as