Reading and Review

1. Reading and Review

2. Question 8.5Springs and Gravity A mass attached to a vertical spring causes the spring to stretch and the mass to move downwards. What can you say about the spring’s potential energy (PEs) and the gravitational potential energy (PEg) of the mass? a) both PEs and PEg decrease b) PEs increases and PEg decreases c) both PEs and PEg increase d) PEs decreases and PEg increases e) PEs increases and PEg is constant

3. Question 8.5Springs and Gravity A mass attached to a vertical spring causes the spring to stretch and the mass to move downwards. What can you say about the spring’s potential energy (PEs) and the gravitational potential energy (PEg) of the mass? a) both PEs and PEg decrease b) PEs increases and PEg decreases c) both PEs and PEg increase d) PEs decreases and PEg increases e) PEs increases and PEg is constant The spring is stretched, so its elastic PE increases, because PEs = kx2. The mass moves down to a lower position, so its gravitational PE decreases, because PEg = mgh.

4. A cart starting from rest rolls down a hill and at the bottom has a speed of 4 m/s. If the cart were given an initial push, so its initial speed at the top of the hill was 3 m/s, what would be its speed at the bottom? Question 8.9Cart on a Hill a)4 m/s b) 5 m/s c) 6 m/s d) 7 m/s e) 25 m/s

5. A cart starting from rest rolls down a hill and at the bottom has a speed of 4 m/s. If the cart were given an initial push, so its initial speed at the top of the hill was 3 m/s, what would be its speed at the bottom? Question 8.9Cart on a Hill a)4 m/s b) 5 m/s c) 6 m/s d) 7 m/s e) 25 m/s When starting from rest, thecart’s PE is changed into KE: DPE =DKE = m(4)2 When starting from 3 m/s, the final KE is: KEf= KEi +DKE = m(3)2 + m(4)2 = m(25) = m(5)2 Speed is not the same as kinetic energy

6. 8-4 Work Done by Nonconservative Forces In the presence of nonconservative forces, the total mechanical energy is not conserved: Solving, (8-9)

7. 8-4 Work Done by Nonconservative Forces In this example, the nonconservative force is water resistance:

8. Question 8.10aFalling Leaves You see a leaf falling to the ground with constant speed. When you first notice it, the leaf has initial total energy PEi + KEi. You watch the leaf until just before it hits the ground, at which point it has final total energy PEf + KEf. How do these total energies compare? a) PEi + KEi > PEf + KEf b) PEi + KEi = PEf + KEf c) PEi + KEi < PEf + KEf d) impossible to tell from the information provided

9. Question 8.10aFalling Leaves You see a leaf falling to the ground with constant speed. When you first notice it, the leaf has initial total energy PEi + KEi. You watch the leaf until just before it hits the ground, at which point it has final total energy PEf + KEf. How do these total energies compare? a) PEi + KEi > PEf + KEf b) PEi + KEi = PEf + KEf c) PEi + KEi < PEf + KEf d) impossible to tell from the information provided As the leaf falls, air resistance exerts a force on it opposite to its direction of motion. This force does negative work, which prevents the leaf from accelerating. This frictional force is a nonconservative force, so the leaf loses energy as it falls, and its final total energy is less than its initial total energy. Follow-up: What happens to leaf’s KE as it falls? What net work is done?

10. 8-5 Potential Energy Curves and Equipotentials The curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy:

11. 8-5 Potential Energy Curves and Equipotentials The potential energy curve for a spring:

12. 8-5 Potential Energy Curves and Equipotentials Contour maps are also a form of potential energy curve:

13. Lecture 11 Linear Momentum

14. Linear Momentum Momentum is a vector; its direction is the same as the direction of the velocity.

15. p p Going Bowling I a) the bowling ball b) same time for both c) the Ping-Pong ball d) impossible to say A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. Which one of the two has the greater kinetic energy?

16. p p Going Bowling I a) the bowling ball b) same time for both c) the Ping-Pong ball d) impossible to say A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. Which one of the two has the greater kinetic energy? Momentum is p = mv so the ping-pong ball must have a much greater velocity Kinetic Energy is KE = 1/2 mv2 so (for a single object): KE = p2 / 2m

17. Momentum and Newton’s Second Law Newton’s second law, as we wrote it before: is only valid for objects that have constant mass. Here is a more general form, also useful when the mass is changing:

18. Change in Momentum • Change in momentum: • mv • 2mv

19. Momentum and Force A net force of 200 N acts on a 100-kg boulder, and a force of the same magnitude acts on a 130-g pebble. How does the rate of change of the boulder’s momentum compare to the rate of change of the pebble’s momentum? a) greater than b) less than c) equal to

20. Momentum and Force A net force of 200 N acts on a 100-kg boulder, and a force of the same magnitude acts on a 130-g pebble. How does the rate of change of the boulder’s momentum compare to the rate of change of the pebble’s momentum? a) greater than b) less than c) equal to The rate of change of momentum is, in fact, the force. Remember that F =  p/t. Because the force exerted on the boulder and the pebble is the same, then the rate of change of momentum is the same.

21. Impulse The same change in momentum may be produced by a large force acting for a short time, or by a smaller force acting for a longer time. Impulse quantifies the overall change in momentum Impulse is a vector, in the same direction as the average force.

22. Impulse We can rewrite as So we see that The impulse is equal to the change in momentum.

23. Why we don’t dive into concrete The same change in momentum may be produced by a large force acting for a short time, or by a smaller force acting for a longer time.

24. p p Going Bowling II a) the bowling ball b) same time for both c) the Ping-Pong ball d) impossible to say A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longertimeto bring to rest?

25.  p p = F  t av p Going Bowling II a) the bowling ball b) same time for both c) the Ping-Pong ball d) impossible to say A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longertimeto bring to rest? We know: so p = Favt Here,Fandp are thesamefor both balls! It will take thesame amount of timeto stop them.

26. p p Going Bowling III a) the bowling ball b) same distance for both c) the Ping-Pong ball d) impossible to say A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, for which is the stopping distance greater?

27. p p Going Bowling III a) the bowling ball b) same distance for both c) the Ping-Pong ball d) impossible to say A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, for which is the stopping distance greater? Use the work-energy theorem:W = KE. The ball withless masshas thegreater speed,and thus thegreater KE.In order to remove that KE, work must be done, whereW = Fd. Because the force is thesamein both cases, the distance needed to stop theless massive ballmust bebigger.

28. With no net force: Conservation of Linear Momentum The net force acting on an object is the rate of change of its momentum: If the net force is zero, the momentum does not change! • A vector equation • Works for each coordinate separately

29. Internal Versus External Forces Internal forces act between objects within the system. As with all forces, they occur in action-reaction pairs. As all pairs act between objects in the system, the internal forces always sum to zero: Therefore, the net force acting on a system is the sum of the external forces acting on it.

30. With no net external force: Momentum of components of a system Internal forces cannot change the momentum of a system. However, the momenta of components of the system may change. An example of internal forces moving components of a system:

31. Kinetic Energy of a System Another example of internal forces moving components of a system: The initial momentum equals the final (total) momentum. But the final Kinetic Energy is very large

32. Opposite case: Two identical cars travelling at identical speeds in opposite directions collide head on. BUT: VERY inelastic collision!

33. 1 2 Nuclear Fission I a) the heavy one b) the light one c) both have the same momentum d) impossible to say A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the greater momentum?

34. 1 2 Nuclear Fission I a) the heavy one b) the light one c) both have the same momentum d) impossible to say A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the greater momentum? The initial momentum of the uranium was zero, so the final total momentum of the two fragments must also be zero.Thus the individual momenta are equal in magnitude and opposite in direction.

35. 1 2 Nuclear Fission II a) the heavy one b) the light one c) both have the same speed d) impossible to say A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the greater speed?

36. 1 2 Nuclear Fission II a) the heavy one b) the light one c) both have the same speed d) impossible to say A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the greater speed? We have already seen that the individual momenta are equal and opposite. In order to keep the magnitude of momentum mv the same, the heavy fragment has the lower speed and thelight fragment has the greater speed.

37. Systems with Changing Mass: Rocket Propulsion If a mass of fuel Δm is ejected from a rocket with speed v, the change in momentum of the rocket is: The force, or thrust, is

38. A plate drops onto a smooth floor and shatters into three pieces of equal mass. Two of the pieces go off with equal speeds v along the floor, but at right angles to one another. Find the speed and direction of the third piece. We know that px=0, py = 0 in initial state and no external forces act in the horizontal

39. An 85-kg lumberjack stands at one end of a 380-kg floating log, as shown in the figure. Both the log and the lumberjack are at rest initially. (a) If the lumberjack now trots toward the other end of the log with a speed of 2.7 m/s relative to the log, what is the lumberjack’s speed relative to the shore? Ignore friction between the log and the water. (b) If the mass of the log had been greater, would the lumberjack’s speed relative to the shore be greater than, less than, or the same as in part (a)? Explain.