2. Work and Energy

3. Kinetic Energy, K Classically, the only type of energy in a system is kinetic energy. Potential energy is the energy an object or system has due to its position relative to another object or system.

4. Mechanical Energy Potential Energy Kinetic Energy U G = mgh U G = -GMm/r U S = ½ kx 2 K = ½ mv 2

5. Equations Kinetic energy: K= ½ mv 2 Gravitational potential energy: U grav = mgh Gravitational potential energy: U grav = -GMm/r Work: W = Fs cos  = F·s Power: P = W/t = E/t Power: P = Fv cos  =F·v Potential energy of a spring: U s = ½ kx 2

6. Work and energy are scalar quantities that can be measured in joules or ergs 1 J = 1 kg-m 2 /s 2 1 erg = 1 g-cm 2 /s 2 1 J = 10 7 ergs

7. If this weight lifter lifts a total mass of 8.0 kg a vertical distance of 0.5 meters ten times: (a) How much positive work is done by the woman on the weights? (b) What is the potential energy of the weights each time they are lifted?

8. This man can’t lift the 10 kg dumbbells, so he gives them a push across the floor, causing them to move at 2.0 m/s. How much work has he done?

9. A coconut falls a vertical distance of 10 meters onto the top of the car. If the coconut has a mass of 2.5 kg, how much work does it do on the car when it hits the roof and comes to a stop? Where is the energy after the coconut lands?

10. Saipan, Marianas Islands, July 2005

11. Though work is a scalar quantity, it can be calculated as a product of vectors: W = F  s

12. This guy is “mowing” by pushing on a handle that is angled 60 degrees above horizontal with a force of 100 N. (a)What is the work he does in pushing his mower a horizontal distance of 5.0 meters? (b) If he pushes the mower at constant speed, what is the work done by friction?

13. Conservation of Energy: 1. If there is no work done on a system, its total mechanical energy remains constant. 2. Work done on a system will change the energy of a system. Positive work done on a system will increase the mechanical energy of the system, and negative work done on a system will decrease the mechanical energy. 3. Friction forces do negative work on systems. 4. Work done by friction ends up as thermal energy of molecules in and around the system.

14. The total mechanical energy of the oscillating spring remains constant in the absence of friction.

15. Remember: The spring force is a variable force, with spring force increasing as displacement increases (for a linear spring). Therefore, resist the temptation to use F=ma to find a constant acceleration. The force varies, as does the acceleration.

16. The total mechanical energy of the ball remains constant as it falls, assuming no air friction.

18. Potential energy of a pendulum: mgh or mg(l - lcos  ) h l lcos  

19. Potential energy of a spring: ½ kx 2

20. Determining work by graphing: *The work done on an object is equal to its change in mechanical energy.

21. What can we conclude from the plots of“Kinetic Energy vs. Time” and “Potential Energy vs. Time” for a falling object? Their sum is constant…so mechanical energy is conserved!

22. Work done in changing mechanical energy: Doing positive work on a spring at equilibrium can increase its potential energy or its kinetic energy. Explain. Doing positive work on a ball sitting on the floor can increase its potential energy or its kinetic energy. Explain. Doing positive work on the bowling ball as it hangs at rest…..Get it? Doing negative work on a ball as it rolls across the floor decreases its mechanical energy. Explain. Can negative work be done on an oscillating spring or pendulum? When does an applied force do no work?

23. So….does friction do positive work or negative work? Yee..ouch!

24. The top of the slide is 2.0 meters above the ground. (a)Without friction, how fast is the child moving at the bottom of the slide? (b) Suppose the child is only moving 5.0 m/s at the bottom. Calculate the work done by friction on the child.

25. Potential Energy and Work Conservative/Non-conservative Forces

26. Conservative forces are those forces that act on a body or system without changing the total energy in that system. Example: Gravitational force acting on a box at the top of a frictionless ramp will cause the box to slide down the ramp—converting gravitational potential energy to kinetic energy without changing the total mechanical energy in the box-ramp system.

27. Non-conservative forces are those forces that act on a body or system causing the total energy in that system to change. Example: Friction force acting on a box at the top of a rough-surfaced ramp will cause part of the total mechanical energy in the box- ramp system to be “lost” to thermal energy in the ramp, air around the ramp, etc. Thus, the total mechanical energy of the system is not conserved.

28. A box of mass m is pushed by a force F to the top of a ramp of length d and height h. Determine the work done by the force in pushing the box to the top of the ramp (a) neglecting friction, and (b) including friction, with coefficient . h d F

29. First, add the forces on the box. Then use d and h to determine the angle . F N mg 

30. Construct the components for mg. F N mg  mgcos  mgsin 

31. F N mg  mgcos  mgsin  Method 1: Energy method for determining work with no friction Work = Change in Potential Energy W =  U = mgh

32. Method 2: Force method for determining work with no friction. W = F  s where F = mgsin  and s = d W = mgsin  d BUT dsin  = ______ SO: W = _______ F N mg  mgcos  mgsin 

33. Now we consider the same situation with friction. F N mg  mgcos  mgsin  FfFf

34. Method 1: Energy method with friction W = Change in Potential Energy + Loss to Thermal Energy due to Friction W = mgh + F f d W = mgh +  mgcos  F N mg  mgcos  mgsin  FfFf

35. Method 2: Force method of determining work with friction W= F  s where F = F f + mgsin  and s = d W =  mgcos  d + mgsin  d W = ____ + _______ F N mg  mgcos  mgsin  FfFf

36. A Combination Pendulum Motion

37. A pendulum of length L is released from angle . When it swings to vertical, it hits a rod that is perpendicular to the plane of the swing (i.e., the rod projects out of the page) and positioned at ½ L. 

38. Find the angle to which the pendulum will swing after hitting the bar. 

39. ?  

40. Consider the initial potential energy of the pendulum bob and the final potential energy of the bob. They should be equal.   What seemed like a tough problem becomes easier when you use energy methods.

41. mg(L - Lcos  ) = mg(½ L – ½ Lcos  ) 1 - cos  = (1/2) (1 - cos  ) ½ = cos  - ½ cos   