Chapter 7 Work and Kinetic Energy

2 Work Done by a Constant Force The definition of work, when the force is parallel to the displacement: SI unit: newton-meter (N·m) = joule, J 2

3 Convenient notation: the dot product The work can also be written as the dot product of the force and the displacement: vector “dot” operation: project one vector onto the other

4 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 Play Ball!

5 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 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 (Δr)cos  ), because  = 180 º, then W < 0. Note that the work done on the ball is negative, and its speed decreases. Play Ball! Follow-up: What about the work done by the ball on the catcher?

6 Tension and Work a) tension does no work at all b) tension does negative work c) tension does positive work A ball tied to a string is being whirled around in a circle with constant speed. What can you say about the work done by tension?

7 Tension and Work a) tension does no work at all b) tension does negative work c) tension does positive work A ball tied to a string is being whirled around in a circle with constant speed. What can you say about the work done by tension? v T No work is done because the force acts in a perpendicular direction to the displacement. Or using the definition of work (W = F (Δr)cos   ),  = 90 º, then W = 0. Follow-up: Is there a force in the direction of the velocity?

8 Work by gravity FgFg a h A ball of mass m drops a distance h. What is the total work done on the ball by gravity? N FgFg A ball of mass m rolls down a ramp of height h at an angle of 45 o. What is the total work done on the ball by gravity? h a θ F gx = F g sinθ h = L sinθ W = Fd = F gx L = (F g sinθ) (h / sinθ) W = F g h = mgh W = Fd = F gx h W = mgh Path doesn’t matter when asking “how much work did gravity do?” Only the change in height!

9 Motion and energy When positive work is done on an object, its speed increases; when negative work is done, its speed decreases.

10 Kinetic Energy As a useful word for the quantity of work we have done on an object, thereby giving it motion, we define the kinetic energy:

11 Work-Energy Theorem Work-Energy Theorem: The total work done on an object is equal to its change in kinetic energy. (True for rigid bodies that remain intact)

12 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 Kinetic Energy I

13 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 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. Kinetic Energy I

14 Work Done by a Variable Force We can interpret the work done graphically:

15 Work Done by a Variable Force We can then approximate a continuously varying force by a succession of constant values.

16 Work Done by a Variable Force The force needed to stretch a spring an amount x is F = kx. Therefore, the work done in stretching the spring is

Potential Energy and Conservation of Energy Chapter 8

18 Recall The total work done on an object is equal to its change in kinetic energy: Work is the force directed along a displacement: Lets wrap up our discussion of work and kinetic energy...

19 Force and Work a) one force b) two forces c) three forces d) four forces e) no forces are doing work A box is being pulled up a rough incline by a rope connected to a pulley. How many forces are doing work on the box?

20 Force and Work N f T mg displacement Any force not perpendicular to the motion will do work: N does no work T does positive work f does negative work mg does negative work a) one force b) two forces c) three forces d) four forces e) no forces are doing work A box is being pulled up a rough incline by a rope connected to a pulley. How many forces are doing work on the box?

21 Free Fall I 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?

22 Consider the work done by gravity to make the stone fall distance d:  KE = W net = F d cos   KE = mg d Thus, the stone with the greater mass has the greater KE, which is twice as big for the heavy stone. Free Fall I 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?

23 Power Power is a measure of the rate at which work is done: SI unit: J/s = watt, W 1 horsepower = 1 hp = 746 W if work is energy, then you would think of “energy flow”

24 Power

25 Power If an object is moving at a constant speed in the face of friction, gravity, air resistance, and so forth, the power exerted by the driving force can be written: Question: what is the total work per unit time done on this object (by all forces)? This expression, P = Fv, gives the instantaneous power applied, even if the object is not moving at constant speed

26 a) energy b) power c) current d) voltage e) none of the above Electric Bill When you pay the electric company by the kilowatt-hour, what are you actually paying for?

27 We have defined: Power = energy / time So we see that: Energy = power × time This means that the unit of power × time (watt- hour) is a unit of energy !! Electric Bill When you pay the electric company by the kilowatt-hour, what are you actually paying for? a) energy b) power c) current d) voltage e) none of the above

28 A block rests on a horizontal frictionless surface. A string is attached to the block, and is pulled with a force of 45.0 N at an angle above the horizontal, as shown in the figure. After the block is pulled through a distance of 1.50 m, its speed is 2.60 m/s, and 50.0 J of work has been done on it. (a) What is the angle (b) What is the mass of the block?

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30 The pulley system shown is used to lift a 52 kg crate. Note that one chain connects the upper pulley to the ceiling and a second chain connects the lower pulley to the crate. Assuming the masses of the chains, pulleys, and ropes are negligible, determine (a) the force F required to lift the crate with constant speed, and (b) the tension in two chains

31 (a) the force F required to lift the crate with constant speed, and (b) the tension in two chains (a) constant velocity, a=0, so net force =0. 2T - (52kg)(9.8m/s2) = 0 T = 250 N F = -250 Ny (b) massless pully! Upper: T ch - 2T rope = 0 T ch = 500 N Lower: T ch -2T rope =0 T ch = 500 N Mechanical Advantage!

32 (a) how much power is applied to the box by the chain? (b) how much power is applied on the rope by the applied force? What about work? T rope = 250 N T chain = 500 N F = -250 Ny (a) P = Fv = 500 N * v box (b) P = Fv = 250 N * v hand v hand = 2 v box !

33 You can explain what is about to happen, in terms of force (elastic band is about to pull on the rock, accelerating it toward our camera lens)......or using work/energy: work has been done on the elastic band, and it now “contains” energy. Energy wants to be “free” - in fact, physics can be described in terms of the rules that govern stored energy Also: energy really is a “thing”. E = mc 2 relates mass to energy, and stored energy counts... it’s not just an accounting rule!

34 Work by gravity FgFg a h A ball of mass m drops a distance h. What is the total work done on the ball by gravity? N FgFg A ball of mass m rolls down a ramp of height h at an angle of 45 o. What is the total work done on the ball by gravity? h a θ F gx = F g sinθ h = L sinθ W = Fd = F gx L = (F g sinθ) (h / sinθ) W = F g h = mgh W = Fd = F gx h W = mgh Path doesn’t matter when asking “how much work did gravity do?” Only the change in height!

35 Application: ball on a track how high must I place the ball so that it can complete a loop? Condition: F cp > mg at top of loop F cp = mv 2 /r = mgv 2 = gr KE = mv 2 / 2 = mgr/2 Gravity must provide this energy W g = mgh = KE h = r/2 above the top of the loop! 35

36 Conservative and Nonconservative Forces Conservative force: - the work it does is stored in the form of energy that can be released at a later time -the work done by a conservative force moving an object around a closed path is zero -Force depends upon position only Example of a conservative force: gravity Example of a nonconservative force: friction

37 Work done by gravity on a closed path is zero

38 Work done by friction on a closed path is not zero

39 The work done by a conservative force is zero on any closed path So the work must be reversible (opposite when taking the same path) AND path independent (same amount of work for any two different paths connecting two points) Go A-B on path 1, the back B-A. W t = W 1 + -W 1 Go A-B on path 1, the B-A on path 2. W t = W 1 + -W 2

40 Potential Energy If we pick up a ball and put it on the shelf, we have done work on the ball. We can get that energy back if the ball falls back off the shelf (gravity does positive work on the ball, “releasing” the work that we put in before). Until that happens, we say the energy is stored as potential energy.

41 Potential Energy Consider the process in which the book goes from h=0 to h=0.50 m Work done by gravity: W = - (mg)h = J For the book to go up against gravity, another force must be applied to overcome the weight. This other force did a (minimum) work of 13.5 J If I lft the book steadily, the “external force” is provided by my hand with F~mg, work done by me: W=(mg)h = 13.5 J The book’s potential energy changed by: 13.5 J

42 Potential Energy The work done against a conservative force is stored in the form of (potential) energy that can be released at a later time. Note the minus sign: positive W c (work by the conservative force) is negative potential energy (energy is released) negative W c is positive potential energy (another force as done work against the conservative force)

43 Gravitational Potential Energy Q: What does “U G = 0” mean?

44 Work Done by a Variable Force The force needed to stretch a spring an amount x is F = kx. Therefore, the work done in stretching (or compressing) the spring is on the spring with positive work applied leading to a positive change in potential: W = U f - U i

45 Potential energy in a spring The corresponding conservative force is the force of the spring acting on the hand: positive work by the spring releases potential energy W c = - ΔU So, taking U=0 at x=0:

46 Up the Hill a) the same b) twice as much c) four times as much d) half as much e) you gain no PE in either case Two paths lead to the top of a big hill. One is steep and direct, while the other is twice as long but less steep. How much more potential energy would you gain if you take the longer path?

47 Because your vertical position (height) changes by the same amount in each case, the gain in potential energy is the same. Up the Hill a) the same b) twice as much c) four times as much d) half as much e) you gain no PE in either case Two paths lead to the top of a big hill. One is steep and direct, while the other is twice as long but less steep. How much more potential energy would you gain if you take the longer path? Follow-up: How much more work do you do in taking the steeper path? Follow-up: Which path would you rather take? Why?

48 Is it possible for the gravitational potential energy of an object to be negative? a) yes b) no Sign of the Energy

49 Is it possible for the gravitational potential energy of an object to be negative? a) yes b) no Gravitational PE is mgh, where height h is measured relative to some arbitrary reference level where PE = 0. For example, a book on a table has positive PE if the zero reference level is chosen to be the floor. However, if the ceiling is the zero level, then the book has negative PE on the table. Only differences (or changes) in PE have any physical meaning. Sign of the Energy

50 You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on? a) only B b) only C c) A, B, and C d) only A and C e) only B and C KE and PE A) skier’s PE B) skier’s change in PE C) skier’s final KE

51 You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on? a) only B b) only C c) A, B, and C d) only A and C e) only B and C The gravitational PE depends upon the reference level, but the difference  PE does not! The work done by gravity must be the same in the two solutions, so  PE and  KE should be the same. A) skier’s PE B) skier’s change in PE C) skier’s final KE KE and PE

52 Mechanical Energy It is useful to define the mechanical energy: Consider the total amount of work done on a body by the conservative and the non-conservative forces. This is the change in kinetic energy (work-energy theorem) Then: The work done by all non-conservative forces is the change in the mechanical energy of a body

53 Conservation of Mechanical Energy The work done by all non-conservative forces is the change in the mechanical energy of a body If there are only conservative forces doing work during a process, we find:

54 Work-Energy Theorem vs. Conservation of Energy? Work-Energy Theorem total work done (by both conservative and non- conservative forces) = change in kinetic energy Conservation of mechanical energy total work done by non-conservative forces = change in mechanical energy These two are completely equivalent. The difference is only how to treat conservative forces. Do NOT use both potential energy AND work by the conservative force... that’s double-counting! In general, energy conservation makes kinematics problems much easier to solve...

55 Runaway Truck A truck, initially at rest, rolls down a frictionless hill and attains a speed of 20 m/s at the bottom. To achieve a speed of 40 m/s at the bottom, how many times higher must the hill be? a) half the height b) the same height c)  2 times the height d) twice the height e) four times the height

56 Runaway Truck A truck, initially at rest, rolls down a frictionless hill and attains a speed of 20 m/s at the bottom. To achieve a speed of 40 m/s at the bottom, how many times higher must the hill be? a) half the height b) the same height c)  2 times the height d) twice the height e) four times the height Use energy conservation:  initial energy: E i = PE g = mgH  final energy: E f = KE = mv 2 Conservation of Energy: E i = mgH = E f = mv 2 therefore: gH = v 2 So if v doubles, H quadruples!

57 Cart on a Hill 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? a) 4 m/s b) 5 m/s c) 6 m/s d) 7 m/s e) 25 m/s

58 Cart on a Hill When starting from rest, the cart’s PE is changed into KE:  PE =  KE = m(4) 2 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? a) 4 m/s b) 5 m/s c) 6 m/s d) 7 m/s e) 25 m/s When starting from 3 m/s, the final KE is: KE f = KE i +  KE = m(3) 2 + m(4) 2 = m(25) = m(5) 2

59 Potential Energy Curves The curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy: Q: at what point is speed maximized? Q: where might apparent weight be minimized? 59

60 Potential Energy for a Spring

61 Potential Energy Curves and Equipotentials Contour maps are also a form of potential energy curve: Each contour is an equal height, and so an “equipotential” for gravitational potential energy

62 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 (PE s ) and the gravitational potential energy (PE g ) of the mass? a) both PE s and PE g decrease b) PE s increases and PE g decreases c) both PE s and PE g increase d) PE s decreases and PE g increases e) PE s increases and PE g is constant Question 8.5 Springs and Gravity

63 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 (PE s ) and the gravitational potential energy (PE g ) of the mass? a) both PE s and PE g decrease b) PE s increases and PE g decreases c) both PE s and PE g increase d) PE s decreases and PE g increases e) PE s increases and PE g is constant The spring is stretched, so its elastic PE increases, because PE s = kx 2. The mass moves down to a lower position, so its gravitational PE decreases, because PE g = mgh. Question 8.5 Springs and Gravity

Work Done by Nonconservative Forces In this example, the nonconservative force is water resistance: 64