Copyright © 2010 Pearson Education, Inc. Chapter 7 Work and Kinetic Energy

Copyright © 2010 Pearson Education, Inc. Reading and Review

Copyright © 2010 Pearson Education, Inc. 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?

Copyright © 2010 Pearson Education, Inc. 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?

Copyright © 2010 Pearson Education, Inc. 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?

Copyright © 2010 Pearson Education, Inc. 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? Follow-up: How do the initial values of gravitational PE compare?

Copyright © 2010 Pearson Education, Inc. In the previous question, just before hitting the ground, what is the final speed of the heavy stone compared to the light one? a) quarter as much b) half as much c) the same d) twice as much e) four times as much Free Fall II

Copyright © 2010 Pearson Education, Inc. In the previous question, just before hitting the ground, what is the final speed of the heavy stone compared to the light one? a) quarter as much b) half as much c) the same d) twice as much e) four times as much All freely falling objects fall at the same rate, which is g. Because the acceleration is the same for both, and the distance is the same, then the final speeds will be the same for both stones. Free Fall II

Copyright © 2010 Pearson Education, Inc. 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

Copyright © 2010 Pearson Education, Inc. Application: work by a spring Hooke’s Law: F = - kx k = (3kg)(9.8 m/s 2 ) / (3.9 cm) k = 760 N/m Loaded spring: W = kx 2 /2 = (760 N/m) (0.04m) 2 / 2 W = 0.61 J How fast?: v = d/t = (0.020 m) (0.020 s) = 1 m/s KE = mv 2 /2 = (1kg)(1m/s) 2 / 2 KE = 0.55 J Kinetic Energy:

Copyright © 2010 Pearson Education, Inc. 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

Copyright © 2010 Pearson Education, Inc. Power

Copyright © 2010 Pearson Education, Inc. 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 the object?

Copyright © 2010 Pearson Education, Inc. 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?

Copyright © 2010 Pearson Education, Inc. 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

Copyright © 2010 Pearson Education, Inc. 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?

Copyright © 2010 Pearson Education, Inc.

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

Copyright © 2010 Pearson Education, Inc. (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) upper pulley doesn’t move: T ch - 2T rope = 0 T ch = 500 N lower pulley has constant acceleration T ch -2T rope =0 T ch = 500 N Mechanical Advantage!

Copyright © 2010 Pearson Education, Inc. (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 hand moves twice as fast hand moves twice as far

Copyright © 2010 Pearson Education, Inc. Chapter 8 Potential Energy and Conservation of Energy

Copyright © 2010 Pearson Education, Inc. Units of Chapter 8 Conservative and Nonconservative Forces Potential Energy and the Work Done by Conservative Forces Conservation of Mechanical Energy Work Done by Nonconservative Forces Potential Energy Curves and Equipotentials

Copyright © 2010 Pearson Education, Inc. 8-1 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 Example of a conservative force: gravity Example of a nonconservative force: friction Also: the work done by a conservative force moving an object around a closed path is zero; this is not true for a nonconservative force

Copyright © 2010 Pearson Education, Inc. 8-1 Conservative and Nonconservative Forces Work done by gravity on a closed path is zero:

Copyright © 2010 Pearson Education, Inc. 8-1 Conservative and Nonconservative Forces Work done by friction on a closed path is not zero:

Copyright © 2010 Pearson Education, Inc. 8-1 Conservative and Nonconservative Forces The work done by a conservative force is zero on any closed path:

Copyright © 2010 Pearson Education, Inc. 8-2 The Work Done by Conservative Forces 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; in the meantime, we say the energy is stored as potential energy. (8-1)

Copyright © 2010 Pearson Education, Inc. 8-2 The Work Done by Conservative Forces Gravitational potential energy:

Copyright © 2010 Pearson Education, Inc. Is it possible for the gravitational potential energy of an object to be negative? a) yes b) no Sign of the Energy II

Copyright © 2010 Pearson Education, Inc. 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 II

Copyright © 2010 Pearson Education, Inc. 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 Question 8.2 KE and PE A) skier’s PE B) skier’s change in PE C) skier’s final KE

Copyright © 2010 Pearson Education, Inc. 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. Question 8.2 KE and PE A) skier’s PE B) skier’s change in PE C) skier’s final KE Follow-up: Does anything change physically by the choice of y = 0?

Copyright © 2010 Pearson Education, Inc. 8-2 The Work Done by Conservative Forces Springs: (8-4)

Copyright © 2010 Pearson Education, Inc. 8-3 Conservation of Mechanical Energy Definition of mechanical energy: (8-6) Using this definition and considering only conservative forces, we find: Or equivalently:

Copyright © 2010 Pearson Education, Inc. 8-3 Conservation of Mechanical Energy Energy conservation can make kinematics problems much easier to solve:

Copyright © 2010 Pearson Education, Inc. Example: A mass m slides down a 2 m long smooth ramp which makes an angle of 30 o with the top of a table which is 1 m above the floor. The end of the ramp is at the edge of the table. At what horizontal distance from the edge of the table does the mass hit the floor?

Copyright © 2010 Pearson Education, Inc. 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

Copyright © 2010 Pearson Education, Inc. 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. KE and PE A) skier’s PE B) skier’s change in PE C) skier’s final KE Follow-up: Does anything change physically by the choice of y = 0?

Copyright © 2010 Pearson Education, Inc. Example: Two water slides are shaped differently, but start at the same height h and are of equal length. Two rides, Paul and Kathy, start from rest at the same time on different slides. a)Which is travelling faster at the bottom? b)Which makes it to the bottom first?

Copyright © 2010 Pearson Education, Inc. 8-4 Work Done by Nonconservative Forces In the presence of nonconservative forces, the total mechanical energy is not conserved: Solving, (8-9)

Copyright © 2010 Pearson Education, Inc. 8-4 Work Done by Nonconservative Forces In this example, the nonconservative force is water resistance:

Copyright © 2010 Pearson Education, Inc. 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:

Copyright © 2010 Pearson Education, Inc. 8-5 Potential Energy Curves and Equipotentials The potential energy curve for a spring:

Copyright © 2010 Pearson Education, Inc. 8-5 Potential Energy Curves and Equipotentials Contour maps are also a form of potential energy curve:

Copyright © 2010 Pearson Education, Inc. Summary of Chapter 8 Conservative forces conserve mechanical energy Nonconservative forces convert mechanical energy into other forms Conservative force does zero work on any closed path Work done by a conservative force is independent of path Conservative forces: gravity, spring

Copyright © 2010 Pearson Education, Inc. Summary of Chapter 8 Work done by nonconservative force on closed path is not zero, and depends on the path Nonconservative forces: friction, air resistance, tension Energy in the form of potential energy can be converted to kinetic or other forms Work done by a conservative force is the negative of the change in the potential energy Gravity: U = mgy Spring: U = ½ kx 2

Copyright © 2010 Pearson Education, Inc. Summary of Chapter 8 Mechanical energy is the sum of the kinetic and potential energies; it is conserved only in systems with purely conservative forces Nonconservative forces change a system’s mechanical energy Work done by nonconservative forces equals change in a system’s mechanical energy Potential energy curve: U vs. position