Chapter 7 Work and Kinetic Energy

Chapter 7 Work and Kinetic Energy http://people.virginia.edu/~kdp2c/downloads/WorkEnergySelections.html

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

Work done by gravity on a closed path is zero

Work done by friction on a closed path is not zero

The work done by a conservative force is zero on any closed path Go A-B on path 1, the back B-A. Wt = W1 + -W1 Go A-B on path 1, the B-A on path 2. Wt = W1 + -W2 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)

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.

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 = -13.5 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

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 Wc (work by the conservative force) is negative potential energy (energy is released) • negative Wc is positive potential energy (another force as done work against the conservative force)

Gravitational Potential Energy Q: What does “UG = 0” mean?

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

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 Wc = - ΔU So, taking U=0 at x=0:

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?

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? Because your vertical position (height) changes by the same amount in each case, the gain in potential energy is the same. Follow-up: How much more work do you do in taking the steeper path? Follow-up: Which path would you rather take? Why?

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

Sign of the Energy 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.

KE and PE 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 A) skier’s PE B) skier’s change in PE C) skier’s final KE

KE and PE 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 A) skier’s PE B) skier’s change in PE C) skier’s final KE 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.

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) It is useful to define the mechanical energy: Then: The work done by all non-conservative forces is the change in the mechanical energy of a body

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:

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...

Runaway Truck a)half the height b) the same height c) < 2 times the height d) twice the height e) four times the height 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?

Runaway Truck a)half the height b) the same height c) < 2 times the height d) twice the height e) four times the height 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? Use energy conservation: • initial energy: Ei = PEg = mgH • final energy: Ef = KE= mv2 Conservation of Energy: Ei = mgH= Ef = mv2 therefore: gH = v2 So if v doubles, H quadruples!

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

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

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?

Potential Energy for a Spring

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

Question 8.5 Springs 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

Question 8.5 Springs 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.

8-4 Work Done by Nonconservative Forces In this example, the nonconservative force is water resistance: 30

Chapter 9 Linear Momentum

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

Momentum is a vector Change in momentum: • mv • 2mv

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?

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

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):

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

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 = Δp/Δt. Because the force exerted on the boulder and the pebble is the same, then the rate of change of momentum is the same.

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.

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

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.

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?

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.

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?

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.

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

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.

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

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

Birth of the neutrino Beta decay fails momentum conservation? First detection 1956 Pauli “fixes” it with a new ghost-like, undetectable particle Bohr scoffs

Chapter 7 Work and Kinetic Energy