1. Classical Mechanics Lecture 8
Midterm 2 will be held on March 13. Covers units 4-9
Classical Mechanics Lecture 8
Today’s Examples
a) Work and Kinetic Energy Problems
Today's Concepts:
a) Potential Energy
b) Mechanical Energy

2. Main Points

3. Main Points

4. Main Points

5. Work done by a Spring: Conservative Force!
Net Work over closed path = 0
Conservative Force
On the last page we calculated the work done by a constant force when the orientation of the path relative to the force was changing. On this page we will calculate the work done by a spring in one dimension. In this case the orientation of the path relative to the force is simple, but the magnitude of the force changes as we move.
If the coordinate system is chosen such that x=0 is the relaxed length of the spring as shown, then the force exerted by the spring on some object attached to its end as function of position is given by Hookes law [F = -kx]. As we move the object between two positions x1 and x2, the force on the object clearly changes. Breaking the movement into tiny steps we see that the work done by the spring along each step will depend on the position : [dW = F(x)*dx = -kxdx]. To find the total work done we need to integrate this expression between x1 and x2 [show].
The mathematics of doing this is straightforward, but its very important to realize what it means conceptually. Evaluating a one dimensional integral of some function F(x) between two points x1 and x2 is really just finding the area under the curve between these points [show] . Evaluating the above integral to find the work done is therefore just finding the area under the force versus position plot between the points x1 and x2. [show]
Notice that the formula for the work done by a spring that we just derived depends only on the endpoints of the motion x1 and x2, hence we see that this is a conservative force also. 5

6. Work done by Gravitational Force

7. Work done by Gravitational Force

8. Work done by Gravitational Force

9. Work done by Gravitational Force

10. Clicker Question
In Case 1 we send an object from the surface of the earth to a height above the earth surface equal to one earth radius.
In Case 2 we start the same object a height of one earth radius above the surface of the earth and we send it infinitely far away.
In which case is the magnitude of the work done by the Earth’s gravity on the object biggest?
A) Case B) Case C) They are the same
Case 2
Case 1
Mechanics Lecture 7, Slide 10

11. Clicker Question Solution
Same!
Case 1:
Case 2: RE
Case 2
Case 1
2RE

12. Potential Energy

13. Checkpoint Clicker Question
In Case 1 we release an object from a height above the surface of the earth equal to 1 earth radius, and we measure its kinetic energy just before it hits the earth to be K1.
In Case 2 we release an object from a height above the surface of the earth equal to 2 earth radii, and we measure its kinetic energy just before it hits the earth to be K2.
Compare K1 and K2.
wrong
A) K2 = 2K1
B) K2 = 4K1
C) K2 = 4K1/3
D) K2 = 3K1/2
Case 1
Case 2

14. A) B) C) RE Clicker Question GMem GMem GMem +U0 Usurface = Usurface =
For gravity:
What is the potential energy of an object of mass m on the earths surface:
GMem A)
Usurface =
GMem B)
Usurface = RE
GMem C)
Usurface =
2RE RE
Mechanics Lecture 8, Slide 14

15. RE Clicker Question A) B) C) Case 1 Case 2
What is the potential energy of a object starting at the height of Case 1? A) B) C)
Case 1
Case 2 RE

16. RE Clicker Question A) B) C) Case 1 Case 2
What is the potential energy of a object starting at the height of Case 2? A) B) C)
Case 1
Case 2 RE

17. What is the change in potential in Case 1?B)
Case 1
Case 2 RE 17

18. RE A) B) What is the change in potential in Case 2? Case 1 Case 218

19. A) 2
B) 4
C) 4/3
D) 3/2
What is the ratio
Case 1
Case 2

20. Work on Two Blocks

21. Work on Two Blocks

22. Work on Two Blocks

23. Work on Two Blocks 2

24. Work on Two Blocks 2 or

25. Block Sliding

26. Block Sliding

27. Block Sliding

28. Block Sliding 2

29. Block Sliding 2

30. Block Sliding 2

31. Block Sliding 2

32. Potential & Mechanical Energy

33. Work is Path Independent for Conservative Forces

34. Work is Path Independent for Conservative Forces

35. Conservative Forces. Work is zero over closed path

36. Potential Energy

37. Gravitational Potential Energy

38. Mechanical Energy

39. Mechanical Energy

40. Conservation of Mechanical Energy

41. Relax. There is nothing new here
It’s just re-writing the work-KE theorem:
everything except gravity
and springs
If other forces aren't doing work

42. Conservation of Mechanical Energy
Upper reservoir
Energy “battery” using conservation of mechanical energy Dh
Pumped Storage Hydropower: Store energy by pumping water into upper reservoir at times when demand for electric power is low….Release water from upper reservoir to power turbines when needed ...

43. Gravitational Potential Energy

44. Earth’s Escape Velocity
What is the escape velocity at the Schwarzchild radius of a black hole?

45. Potential Energy Function
U0 : can adjust potential by an arbitrary constant…that must be maintained for all calculations for the situation.

46. Finding the potential energy change:
Use formulas to find the magnitude
Check the sign by understanding the problem…

47. Clicker Question

48. Spring Potential Energy: Conserved

49. Vertical Springs
Massless spring

50. Vertical Spring in Gravitational Field
Formula for potential energy of vertical spring in gravitational field has same form as long as displacement is measured w.r.t new equilibrium position!!!

51. Vertical Spring in Gravitational Field

52. Non-conservative Forces
Work performed by non-conservative forces depend on exact path.

53. Summary Lecture 7 Work – Kinetic Energy theorem Lecture 8
For springs & gravity (conservative forces)
Total Mechanical Energy E = Kinetic + Potential
Work done by any force other than gravity and springs will change E

54. Summary

55. Spring Summary
kx2 M x

56. Checkpoint
Three balls of equal mass are fired simultaneously with equal speeds from the same height h above the ground. Ball 1 is fired straight up, ball 2 is fired straight down, and ball 3 is fired horizontally. Rank in order from largest to smallest their speeds v1, v2, and v3 just before each ball hits the ground. 1 2 3 h
87% correct
A) v1 > v2 > v3
B) v3 > v2 > v1
C) v2 > v3 > v1
D) v1 = v2 = v3

57. CheckPoint 1 2 3 h
A) v1 > v2 > v3 B) v3 > v2 > v1 C) v2 > v3 > v1 D) v1 = v2 = v3
87% correct
They begin with the same height, so they have the same potential energy and change therein, resulting in the same change in kinetic energy and therefore the same speed when they hit the ground.
The total mechanical energy of all 3 masses are equal. When they reach the same height their kinetic energies must be equal, hence equal velocities.

58. Clicker Question
Which of the following quantities are NOT the same for the three balls as they move from height h to the floor: 1 2 3 h
A) The change in their kinetic energies
B) The change in their potential energies
C) The time taken to hit the ground

59. Clicker Checkpoint x A) B) C)
A box sliding on a horizontal frictionless surface runs into a fixed spring, compressing it a distance x1 from its relaxed position while momentarily coming to rest.
If the initial speed of the box were doubled, how far x2 would the spring compress?
90% correct
A) B) C) x

60. CheckPoint x
A) B) C)

62. Clicker Question
A block attached to a spring is oscillating between point x (fully compressed) and point y (fully stretched). The spring is un-stretched at point o. At which point is the acceleration of the block zero? x o y
A) At x
B) At o
C) At y

63.
A=0 at x=0!

64. Integrals as Area Under a curve

65. Homework Average=85% Average=86%
Average=82% including 2 who did not do the homework

66. Example Test Problem
1. What is the work done by gravity during its slide to the bottom of the ramp?
2. What is the work done by friction during one pass through the rough spot?
3. What is the maximum distance the spring is compressed by the box?
4. What is the maximum height to which the box returns on the ramp?

67. Lecture Thoughts