1. PHYS 1110 Lecture 5 Professor Stephen Thornton September 11, 2012

2. Reading Quiz Three balls of equal mass start from rest and roll down different ramps. All ramps have the same height. Which ball has the greater speed at the bottom of its ramp? A D) same speed for all balls B C

3. same initial gravitational PE same height same final KEsame speed All of the balls have the same initial gravitational PE, since they are all at the same height (PE = mgh). Thus, when they get to the bottom, they all have the same final KE, and hence the same speed (KE = 1/2 mv 2 ). Three balls of equal mass start from rest and roll down different ramps. All ramps have the same height. Which ball has the greater speed at the bottom of its ramp? A D) same speed for all balls B C Follow-up: Which ball takes longer to get down the ramp?

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6. Conservative Forces Gravity Springs Nonconservative Forces Friction Tension

7. Potential Energy When we do work, say to lift a box off the floor, then we give the box energy. We call that energy potential energy. Potential energy, in a sense, has potential to do work. It is like stored energy. However, it only works for conservative forces.

8. Do potential energy demo. Burn string and let large mass drop.

9. Notes on potential energy  Potential energy is part of the work- energy theorem. Potential energy can be changed into kinetic energy.  Think about gravity for a good example to use.  There is no single “equation” to use for potential energy.  Remember that it is only useful for conservative forces.

10. Definition of potential energy We will use a subscript on W c to remind us about conservative forces. This doesn’t work for friction. SI unit is the joule (still energy).

11. Remember gravity The work done by a conservative force is equal to the negative of the change in potential energy. Hold a box up. It has potential energy. Drop the box. Gravity does positive work on the box. The change in the gravitational potential energy is negative. The box has less potential energy when it is on the floor.

12. Gravity Is a Conservative Force: Kinetic energy, potential energy, and speed are the same at points A and D.

13. Gravitational Potential Energy Boy does +mgy work to climb up to y. (Gravity does negative work, -mgy). He has potential energy mgy. Gravity does work on boy to bring him down. The potential energy is converted into kinetic energy.

14. More potential energy (PE) notes  Gravitational potential energy = mgh  Only change in potential energy  U is important.  There is no absolute value of PE.  We choose the zero of PE to be at the most convenient position to solve problem.

15. Gravitational potential energy Because we can choose the “zero” of potential energy anywhere we want, it might be convenient to place it at y = 0 (but not always!). y

16. Where might we choose the zero of potential energy to be here?

17. Do demos Loop the loop Bowling ball

18. Example 3-4 The water flowing through Hoover Dam’s turbines is about 1.1 x 10 10 m 3 each year. The water falls on average 160 m from the water intake system down to the turbine. How much work does gravity do each year when the water drops? What is the potential energy loss? Strategy If we know how much water mass passes over Hoover Dam each year, and we know the height of the water drop, we can find the work done by using Equation (3-19), (3-20), or (3-21). We can convert the 4.2 billion kWh into joules to determine the efficiency.

20. is B.

21. Springs The work required to compress a spring is. The potential energy of springs is

22. Conservation of mechanical energy Mechanical energy E is defined to be the sum of K + U. E = K + U Mechanical energy is conserved. Only happens for conservative forces.

23. Solving a Kinematics Problem Using Conservation of Energy E = mgh E = 0

26. Ball rolling on a frictionless track

27. Gravitational potential energy vs position for the previous track. See also kinetic and total energy.

28. A Mass on a Spring

29. Bath County, Virginia, pumped storage facility electrical power plant. Day – water flows down from upper reservoir producing electricity. Night – use power from other plants to pump water back up.

30. L

31. Case 1Case 2 Coordinate system originPowerhouse (y = 0)Upper reservoir (y = 0) Potential energy zero y = 0y = 0 Potential energy at top Potential energy at bottom Kinetic energy at top Kinetic energy at bottom Energy at top Energy at bottom In both cases the energy E has to be conserved, and in both cases we must have

32. Conceptual Quiz: Two unequal masses are hung from a string that pass over an ideal pulley. What is true about the gravitational potential energy U and the kinetic energy K of the system after the masses are released from rest? A)  U > 0 and  K 0 and  K > 0. C)  U > 0 and  K = 0. D)  U = 0 and  K = 0. E)  U 0.

33. Answer: E Initially the system is at rest. Let the potential energy be zero at this point. Therefore the total mechanical energy is zero. If the system starts moving, then  K > 0. Since E = 0, then  U < 0.

35. In a baseball game, the catcher stops a 90-mph pitch. What can you say about the work done by the catcher on the ball? A) catcher has done positive work B) catcher has done negative work C) catcher has done zero work Conceptual Quiz Conceptual Quiz

36. In a baseball game, the catcher stops a 90-mph pitch. What can you say about the work done by the catcher on the ball? A) catcher has done positive work B) catcher has done negative work C) catcher has done zero work opposite in direction to the displacement of the ball, so the work is negative W = F d cos  = 180 º W < 0 The force exerted by the catcher is opposite in direction to the displacement of the ball, so the work is negative. Or using the definition of work (W = F d cos  ), because  = 180 º, then W < 0. Note that because the work done on the ball is negative, its speed decreases. Conceptual Quiz Conceptual Quiz Follow-up: What about the work done by the ball on the catcher?

37. A child on a skateboard is moving at a speed of 2 m/s. After a force acts on the child, her speed is 3 m/s. What can you say about the work done by the external force on the child? A)positive work was done B) negative work was done C) zero work was done Conceptual Quiz

38. A child on a skateboard is moving at a speed of 2 m/s. After a force acts on the child, her speed is 3 m/s. What can you say about the work done by the external force on the child? A)positive work was done B) negative work was done C) zero work was done The kinetic energy of the child increased because her speed increasedincrease in KE positive work being done KE f > KE i work W must be positive The kinetic energy of the child increased because her speed increased. This increase in KE was the result of positive work being done. Or, from the definition of work, because W =  KE = KE f – KE i and we know that KE f > KE i in this case, then the work W must be positive. Conceptual Quiz Follow-up: What does it mean for negative work to be done on the child?

39. Conceptual Quiz A) 20 m B) 30 m C) 40 m D) 60 m E) 80 m If a car traveling 60 km/hr can brake to a stop within 20 m, what is its stopping distance if it is traveling 120 km/hr? Assume that the braking force is the same in both cases.

40. F d = W net =  KE = 0 – mv 2, |F| d = mv 2. and thus, |F| d = mv 2. doubles Therefore, if the speed doubles, four times larger the stopping distance gets four times larger. Conceptual Quiz A) 20 m B) 30 m C) 40 m D) 60 m E) 80 m If a car traveling 60 km/hr can brake to a stop within 20 m, what is its stopping distance if it is traveling 120 km/hr? Assume that the braking force is the same in both cases.

41. By what factor does the kinetic energy of a car change when its speed is tripled? A) no change at all B) factor of 3 C) factor of 6 D) factor of 9 E) factor of 12 Conceptual Quiz Conceptual Quiz

42. By what factor does the kinetic energy of a car change when its speed is tripled? A) no change at all B) factor of 3 C) factor of 6 D) factor of 9 E) factor of 12 mv 2 speed increases by a factor of 3KE will increase by a factor of 9 Because the kinetic energy is mv 2, if the speed increases by a factor of 3, then the KE will increase by a factor of 9. Conceptual Quiz Conceptual Quiz Follow-up: How would you achieve a KE increase of a factor of 2?

43. Conceptual Quiz Conceptual Quiz A) quarter as much B) half as much C) the same D) twice as much E) four times as much Two stones, one twice the mass of the other, are dropped from a cliff. Just before hitting the ground, what is the kinetic energy of the heavy stone compared to the light one?

44. Consider the work done by gravity to make the stone fall distance d:  KE = W net = F d cos   KE = mg d greater massgreater Thus, the stone with the greater mass has the greater KEtwice KE, which is twice as big for the heavy stone. Conceptual Quiz Conceptual Quiz A) quarter as much B) half as much C) the same D) twice as much E) four times as much Two stones, one twice the mass of the other, are dropped from a cliff. Just before hitting the ground, what is the kinetic energy of the heavy stone compared to the light one? Follow-up: How do the initial values of gravitational PE compare?

45. Conceptual Quiz Conceptual Quiz A) 0  30 mph B) 30  60 mph C) both the same A car starts from rest and accelerates to 30 mph. Later, it gets on a highway and accelerates to 60 mph. Which takes more energy, the 0  30 mph, or the 30  60 mph?

46. mv 2 velocity squared The change in KE ( mv 2 ) involves the velocity squared. m (30 2 − 0 2 ) = m (900) So in the first case, we have: m (30 2 − 0 2 ) = m (900) m (60 2 − 30 2 ) = m (2700) In the second case, we have: m (60 2 − 30 2 ) = m (2700) bigger energy changesecond case Thus, the bigger energy change occurs in the second case. Conceptual Quiz Conceptual Quiz A car starts from rest and accelerates to 30 mph. Later, it gets on a highway and accelerates to 60 mph. Which takes more energy, the 0  30 mph, or the 30  60 mph? A) 0  30 mph B) 30  60 mph C) both the same Follow-up: How much energy is required to stop the 60-mph car?

47. The work W 0 accelerates a car from 0 to 50 km/hr. How much work is needed to accelerate the car from 50 km/hr to 150 km/hr? Conceptual Quiz Conceptual Quiz A) 2 W 0 B) 3 W 0 C) 6 W 0 D) 8 W 0 E) 9 W 0

48. The work W 0 accelerates a car from 0 to 50 km/hr. How much work is needed to accelerate the car from 50 km/hr to 150 km/hr? A) 2 W 0 B) 3 W 0 C) 6 W 0 D) 8 W 0 E) 9 W 0 Let’s call the two speeds v and 3v, for simplicity. We know that the work is given by W =  KE = KE f – Ke i. v 2 0 2 v 2 Case #1: W 0 = m (v 2 – 0 2 ) = m (v 2 ) 3vv 2 9v 2 v 2 8v 2 Case #2: W = m ((3v) 2 – v 2 ) = m (9v 2 – v 2 ) = m (8v 2 ) = 8 W 0 Conceptual Quiz Conceptual Quiz Follow-up: How much work is required to stop the 150-km/hr car?

49. Conceptual Quiz Conceptual Quiz A) m 1 B) m 2 C) they will go the same distance Two blocks of mass m 1 and m 2 (m 1 > m 2 ) slide on a frictionless floor and have the same kinetic energy when they hit a long rough stretch (  > 0), which slows them down to a stop. Which one goes farther? m1m1 m2m2

50. same  KE same work forcelessm 2 distancegreater With the same  KE, both blocks must have the same work done to them by friction. The friction force is less for m 2 so stopping distance must be greater. Conceptual Quiz Conceptual Quiz A) m 1 B) m 2 C) they will go the same distance Two blocks of mass m 1 and m 2 (m 1 > m 2 ) slide on a frictionless floor and have the same kinetic energy when they hit a long rough stretch (  > 0), which slows them down to a stop. Which one goes farther? m1m1 m2m2 Follow-up: Which block has the greater magnitude of acceleration?

51. A golfer making a putt gives the ball an initial velocity of v 0, but he has badly misjudged the putt, and the ball only travels one-quarter of the distance to the hole. If the resistance force due to the grass is constant, what speed should he have given the ball (from its original position) in order to make it into the hole? A) 2 v 0 B) 3 v 0 C) 4 v 0 D) 8 v 0 E) 16 v 0 Conceptual Quiz Conceptual Quiz

52. A golfer making a putt gives the ball an initial velocity of v 0, but he has badly misjudged the putt, and the ball only travels one-quarter of the distance to the hole. If the resistance force due to the grass is constant, what speed should he have given the ball (from its original position) in order to make it into the hole? A) 2 v 0 B) 3 v 0 C) 4 v 0 D) 8 v 0 E) 16 v 0 four times the distance four times the workinitial KE must be four times greater increase in the initial speed by a factor of 2KE = mv 2 In traveling four times the distance, the resistive force will do four times the work. Thus, the ball’s initial KE must be four times greater in order to just reach the hole—this requires an increase in the initial speed by a factor of 2, because KE = mv 2. Conceptual Quiz Conceptual Quiz