Chapter 10 Energy Chapter Goal: To introduce the concept of energy and the basic energy model. В© 2013 Pearson Education, Inc. Slide 10-2 Chapter 10 Preview В© 2013 Pearson Education, Inc. Slide 10-3 Chapter 10 Preview В© 2013 Pearson Education, Inc. Slide 10-4 Chapter 10 Preview В© 2013 Pearson Education, Inc. Slide 10-6 Chapter 10 Preview В© 2013 Pearson Education, Inc. Slide 10-7 Kinetic Energy K п‚§ Kinetic energy is the energy of motion. п‚§ All moving objects have kinetic energy. п‚§ The more massive an object or the faster it moves, the larger its kinetic energy. В© 2013 Pearson Education, Inc. Slide 10-20 Potential Energy U п‚§ Potential energy is stored energy associated with an objectвЂ™s position. п‚§ The roller coasterвЂ™s gravitational potential energy depends on its height above the ground. В© 2013 Pearson Education, Inc. Slide 10-21 Thermal Energy Eth п‚§ Thermal energy is the sum of the microscopic kinetic and potential energies of all the atoms and bonds that make up the object. п‚§ An object has more thermal energy when hot than when cold. В© 2013 Pearson Education, Inc. Slide 10-22 The Basic Energy Model п‚§ Within a system, energy can be transformed from one type to another. п‚§ The total energy of the system is not changed by these transformations. п‚§ This is the law of conservation of energy. п‚§ Energy can also be transferred from one system to another. п‚§ The mechanical transfer of energy to a system via forces is called work. В© 2013 Pearson Education, Inc. Slide 10-23 Kinetic Energy and Gravitational Potential Energy п‚§ The figure shows a before-and-after representation of an object in free fall. п‚§ One of the kinematics equations from Chapter 2, with ay = пЂg, is: п‚§ Rearranging: п‚§ Multiplying both sides by ВЅm: В© 2013 Pearson Education, Inc. Slide 10-24 Kinetic Energy and Gravitational Potential Energy Define kinetic energy as an energy of motion: Define gravitational potential energy as an energy of position: The sum K + Ug is not changed when an object is in free fall. Its initial and final values are equal: В© 2013 Pearson Education, Inc. Slide 10-25 Kinetic Energy and Gravitational Potential Energy В© 2013 Pearson Education, Inc. Slide 10-26 QuickCheck 10.1 A child is on a playground swing, motionless at the highest point of his arc. What energy transformation takes place as he swings back down to the lowest point of his motion? A. K п‚® Ug B. Ug п‚® K C. Eth п‚® K D. Ug п‚® Eth E. K п‚® Eth В© 2013 Pearson Education, Inc. Slide 10-27 QuickCheck 10.1 A child is on a playground swing, motionless at the highest point of his arc. What energy transformation takes place as he swings back down to the lowest point of his motion? A. K п‚® Ug B. Ug п‚® K C. Eth п‚® K D. Ug п‚® Eth E. K п‚® Eth В© 2013 Pearson Education, Inc. Slide 10-27 Example 10.1 Launching a Pebble В© 2013 Pearson Education, Inc. Slide 10-31 Example 10.1 Launching a Pebble В© 2013 Pearson Education, Inc. Slide 10-32 Example 10.1 Launching a Pebble В© 2013 Pearson Education, Inc. Slide 10-33 Energy Bar Charts п‚§ A pebble is tossed up into the air. п‚§ The simple bar charts below show how the sum of K + Ug remains constant as the pebble rises and then falls. В© 2013 Pearson Education, Inc. Slide 10-34 QuickCheck 10.4 Rank in order, from largest to smallest, the gravitational potential energies of the balls. A. 1 > 2 = 4 > 3 B. 1 > 2 > 3 > 4 C. 3 > 2 > 4 > 1 D. 3 > 2 = 4 > 1 В© 2013 Pearson Education, Inc. Slide 10-38 QuickCheck 10.4 Rank in order, from largest to smallest, the gravitational potential energies of the balls. A. 1 > 2 = 4 > 3 B. 1 > 2 > 3 > 4 C. 3 > 2 > 4 > 1 D. 3 > 2 = 4 > 1 В© 2013 Pearson Education, Inc. Slide 10-39 The Zero of Potential Energy п‚§ Amber and Bill use coordinate systems with different origins to determine the potential energy of a rock. п‚§ No matter where the rock is, AmberвЂ™s value of Ug will be equal to BillвЂ™s value plus 9.8 J. п‚§ If the rock moves, both will calculate exactly the same value for пЃ„Ug. п‚§ In problems, only пЃ„Ug has physical significance, not the value of Ug itself. В© 2013 Pearson Education, Inc. Slide 10-40 Example 10.2 The Speed of a Falling Rock ASSESS The figure below shows energy bar charts for Amber and Bill. despite their disagreement over the value of Ug, Amber and Bill arrive at the same value for vf and their Kf bars are the same height. You can place the origin of your coordinate system, and thus the вЂњzero of potential energy,вЂќ wherever you choose and be assured of getting the correct answer to a problem. В© 2013 Pearson Education, Inc. Slide 10-44 Gravitational Potential Energy on a Frictionless Surface п‚§ The total mechanical energy for a particle moving along any frictionless smooth surface is conserved, regardless of the shape of the surface. В© 2013 Pearson Education, Inc. Slide 10-48 QuickCheck 10.5 Starting from rest, a marble first rolls down a steeper hill, then down a less steep hill of the same height. For which is it going faster at the bottom? A. Faster at the bottom of the steeper hill. B. Faster at the bottom of the less steep hill. C. Same speed at the bottom of both hills. D. CanвЂ™t say without knowing the mass of the marble. В© 2013 Pearson Education, Inc. Slide 10-49 QuickCheck 10.5 Starting from rest, a marble first rolls down a steeper hill, then down a less steep hill of the same height. For which is it going faster at the bottom? A. Faster at the bottom of the steeper hill. B. Faster at the bottom of the less steep hill. C. Same speed at the bottom of both hills. D. CanвЂ™t say without knowing the mass of the marble. В© 2013 Pearson Education, Inc. Slide 10-50 Example 10.3 The Speed of a Sled В© 2013 Pearson Education, Inc. Slide 10-53 Example 10.3 The Speed of a Sled В© 2013 Pearson Education, Inc. Slide 10-54 Example 10.3 The Speed of a Sled В© 2013 Pearson Education, Inc. Slide 10-55 QuickCheck 10.7 Three balls are thrown from a cliff with the same speed but at different angles. Which ball has the greatest speed just before it hits the ground? A. Ball A. B. Ball B. C. Ball C. D. All balls have the same speed. В© 2013 Pearson Education, Inc. Slide 10-57 QuickCheck 10.7 Three balls are thrown from a cliff with the same speed but at different angles. Which ball has the greatest speed just before it hits the ground? A. Ball A. B. Ball B. C. Ball C. D. All balls have the same speed. В© 2013 Pearson Education, Inc. Slide 10-58 Restoring Forces and HookeвЂ™s Law п‚§ The figure shows how a hanging mass stretches a spring of equilibrium length L0 to a new length L. п‚§ The mass hangs in static equilibrium, so the upward spring force balances the downward gravity force. В© 2013 Pearson Education, Inc. Slide 10-61 Restoring Forces and HookeвЂ™s Law п‚§ The figure shows measured data for the restoring force of a real spring. п‚§ пЃ„s is the displacement from equilibrium. п‚§ The data fall along the straight line: п‚§ The proportionality constant k is called the spring constant. п‚§ The units of k are N/m. В© 2013 Pearson Education, Inc. Slide 10-62 HookeвЂ™s Law п‚§ One end of a spring is attached to a fixed wall. п‚§ (Fsp)s is the force produced by the free end of the spring. п‚§ пЃ„s = s вЂ“ se is the displacement from equilibrium. п‚§ The negative sign is the mathematical indication of a restoring force. В© 2013 Pearson Education, Inc. Slide 10-63 QuickCheck 10.9 The restoring force of three springs is measured as they are stretched. Which spring has the largest spring constant? В© 2013 Pearson Education, Inc. Slide 10-64 QuickCheck 10.9 The restoring force of three springs is measured as they are stretched. Which spring has the largest spring constant? Steepest slope. Takes lots of force for a small displacement. В© 2013 Pearson Education, Inc. Slide 10-65 Example 10.5 Pull Until It Slips В© 2013 Pearson Education, Inc. Slide 10-66 Example 10.5 Pull Until It Slips В© 2013 Pearson Education, Inc. Slide 10-67 Example 10.5 Pull Until It Slips В© 2013 Pearson Education, Inc. Slide 10-68 Example 10.5 Pull Until It Slips В© 2013 Pearson Education, Inc. Slide 10-69 Example 10.5 Pull Until It Slips В© 2013 Pearson Education, Inc. Slide 10-70 Example 10.5 Pull Until It Slips В© 2013 Pearson Education, Inc. Slide 10-71 Elastic Potential Energy п‚§ Springs and rubber bands store potential energy that can be transformed into kinetic energy. п‚§ The spring force is not constant as an object is pushed or pulled. п‚§ The motion of the mass is not constant-acceleration motion, and therefore we cannot use our old kinematics equations. п‚§ One way to analyze motion when spring force is involved is to look at energy before and after some motion. В© 2013 Pearson Education, Inc. Slide 10-73 Elastic Potential Energy п‚§ The figure shows a beforeand-after situation in which a spring launches a ball. п‚§ Integrating the net force from the spring, as given by HookeвЂ™s Law, shows that: п‚§ Here K = ВЅ mv2 is the kinetic energy. п‚§ We define a new quantity: В© 2013 Pearson Education, Inc. Slide 10-74 Elastic Potential Energy п‚§ An object moving without friction on an ideal spring obeys: where п‚§ Because пЃ„s is squared, Us is positive for a spring that is either stretched or compressed. п‚§ In the figure, Us has a positive value both before and after the motion. В© 2013 Pearson Education, Inc. Slide 10-75 QuickCheck 10.10 A spring-loaded gun shoots a plastic ball with a launch speed of 2.0 m/s. If the spring is compressed twice as far, the ballвЂ™s launch speed will be A. B. C. D. E. 1.0 m/s. 2.0 m/s. 2.8 m/s 4.0 m/s. 16.0 m/s. В© 2013 Pearson Education, Inc. Slide 10-76 QuickCheck 10.10 A spring-loaded gun shoots a plastic ball with a launch speed of 2.0 m/s. If the spring is compressed twice as far, the ballвЂ™s launch speed will be A. B. C. D. E. 1.0 m/s. 2.0 m/s. 2.8 m/s 4.0 m/s. 16.0 m/s. В© 2013 Pearson Education, Inc. Conservation of energy: Double пЃ„x пѓћdouble v Slide 10-77 Example 10.6 A Spring-Launched Plastic Ball В© 2013 Pearson Education, Inc. Slide 10-80 Example 10.6 A Spring-Launched Plastic Ball В© 2013 Pearson Education, Inc. Slide 10-81 Example 10.6 A Spring-Launched Plastic Ball В© 2013 Pearson Education, Inc. Slide 10-82 Example 10.6 A Spring-Launched Plastic Ball В© 2013 Pearson Education, Inc. Slide 10-83 Example 10.6 A Spring-Launched Plastic Ball В© 2013 Pearson Education, Inc. Slide 10-84 Energy Diagrams п‚§ Potential energy is a function of position. п‚§ Functions of position are easy to represent as graphs. п‚§ A graph showing a systemвЂ™s potential energy and total energy as a function of position is called an energy diagram. п‚§ Shown is the energy diagram of a particle in free fall. п‚§ Gravitational potential energy is a straight line with slope mg and zero y-intercept. п‚§ Total energy is a horizontal line, since mechanical energy is conserved. В© 2013 Pearson Education, Inc. Slide 10-85 Energy Diagrams п‚§ Shown is the energy diagram of a mass on a horizontal spring. п‚§ The potential energy (PE) is the parabola: Us = ВЅk(x вЂ“ xe)2 п‚§ The PE curve is determined by the spring constant; you canвЂ™t change it. п‚§ You can set the total energy (TE) to any height you wish simply by stretching the spring to the proper length at the beginning of the motion. В© 2013 Pearson Education, Inc. Slide 10-87 A Four-Frame Movie of a Mass Oscillating on a Spring В© 2013 Pearson Education, Inc. Slide 10-88 Elastic Collisions п‚§ During an inelastic collision of two objects, some of the mechanical energy is dissipated inside the objects as thermal energy. п‚§ A collision in which mechanical energy is conserved is called a perfectly elastic collision. п‚§ Collisions between two very hard objects, such as two billiard balls or two steel balls, come close to being perfectly elastic. В© 2013 Pearson Education, Inc. Slide 10-106 A Perfectly Elastic Collision п‚§ Consider a head-on, perfectly elastic collision of a ball of mass m1 and initial velocity (vix)1, with a ball of mass m2 initially at rest. п‚§ The ballsвЂ™ velocities after the collision are (vfx)1 and (vfx)2. п‚§ Momentum is conserved in all isolated collisions. п‚§ In a perfectly elastic collision in which potential energy is not changing, the kinetic energy must also be conserved. В© 2013 Pearson Education, Inc. Slide 10-107 A Perfectly Elastic Collision п‚§ Simultaneously solving the conservation of momentum equation and the conservation of kinetic energy equations allows us to find the two unknown final velocities. п‚§ The result is: В© 2013 Pearson Education, Inc. Slide 10-108 A Perfectly Elastic Collision: Special Case 1 п‚§ Consider a head-on, perfectly elastic collision of a ball of mass m1 and initial velocity (vix)1, with a ball of mass m2 initially at rest. п‚§ Case 1: m1 = m2. п‚§ Equations 10.42 give vf1 = 0 and vf2 = vi1. п‚§ The first ball stops and transfers all its momentum to the second ball. В© 2013 Pearson Education, Inc. Slide 10-109 A Perfectly Elastic Collision: Special Case 2 п‚§ Consider a head-on, perfectly elastic collision of a ball of mass m1 and initial velocity (vix)1, with a ball of mass m2 initially at rest. п‚§ Case 2: m1 >> m2. п‚§ Equations 10.42 give vf1 п‚» vi1 and vf2 п‚» 2vi1. п‚§ The big first ball keeps going with about the same speed, and the little second ball flies off with about twice the speed of the first ball. В© 2013 Pearson Education, Inc. Slide 10-110 A Perfectly Elastic Collision: Special Case 3 п‚§ Consider a head-on, perfectly elastic collision of a ball of mass m1 and initial velocity (vix)1, with a ball of mass m2 initially at rest. п‚§ Case 3: m1 << m2. п‚§ Equations 10.42 give vf1 в‰€ в€’vi1 and vf2 в‰€ 0. п‚§ The little first rebounds with about the same speed, and the big second ball hardly moves at all. В© 2013 Pearson Education, Inc. Slide 10-111 Perfectly Elastic Collisions: Using Reference Frames п‚§ Equations 10.42 assume ball 2 is at rest. п‚§ What if you need to analyze a head-on collision when both balls are moving before the collision? п‚§ You could solve the simultaneous momentum and energy equations, but there is an easier way. В© 2013 Pearson Education, Inc. Slide 10-112 Using Reference Frames: Quick Example A 200 g ball moves to the right at 2.0 m/s. It has a head-on, perfectly elastic collision with a 100 g ball that is moving toward it at 3.0 m/s. What are the final velocities of both balls? В© 2013 Pearson Education, Inc. Slide 10-113 Using Reference Frames: Quick Example п‚§ Figure (a) shows the situation just before the collision in the lab frame L. п‚§ Figure (b) shows the situation just before the collision in the frame M that is moving along with ball 2. В© 2013 Pearson Education, Inc. Slide 10-114 Using Reference Frames: Quick Example п‚§ We can use Equations 10.42 to find the post-collision velocities in the moving frame M: п‚§ Transforming back to the lab frame L: В© 2013 Pearson Education, Inc. Slide 10-115

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