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Energy and Its Conservation Learn that energy is a property of an object that can change the object’s position, motion, or its environment. Learn that energy changes from one form to another, and that the total amount of energy in a closed system remains constant. Chapter 11 In this chapter you will:

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Table of Contents Chapter 11: Energy and Its Conservation Section 11.1: The Many Forms of Energy Section 11.2: Conservation of Energy Chapter 11 Assignments Read Chapter 11. HW 11.A: p.307: 54,55,57,59,62,63,65 – 68. HW 11.B: p.308: 73-77, 79, 82. Energy Study Guide is due before the test.

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The Many Forms of Energy Identify different forms of energy Define kinetic energy and gravitational potential energy. Explain the relationship between work and kinetic energy. Solve problems with potential and kinetic energy. In this section you will: Section 11.1

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The Many Forms of Energy The word energy is used in many different ways in everyday speech. Some fruit-and-cereal bars are advertised as energy sources. Athletes use energy in sports. Companies that supply your home with electricity, natural gas, or heating fuel are called energy companies. Section 11.1 Scientists and engineers use the term energy much more precisely. energy – the ability to produce a change in itself or its surroundings. The SI unit of energy is the joule (J).

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The Many Forms of Energy Work causes a change in the energy of a system. That is, work transfers energy between a system and the external world. Work-Energy Theorem – the work done on an object is equal to its change in kinetic energy. W = KE = KE f – KE i Section 11.1

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The Many Forms of Energy Kinetic Energy Section 11.1 The kinetic energy is proportional to the object’s mass. A 7.26-kg shot put thrown through the air has much more kinetic energy than a kg baseball with the same velocity, because the shot put has a greater mass. where m is the mass of the object and v is the magnitude of its velocity. kinetic energy (KE) – the energy of motion Anything that is moving (v 0) has KE. KE = ½ mv 2

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The Many Forms of Energy The kinetic energy of an object is also proportional to the square of the object’s velocity. A car speeding at 20 m/s has four times the kinetic energy of the same car moving at 10 m/s. Kinetic energy also can be due to rotational motion. If you spin a toy top in one spot, does it have kinetic energy? Kinetic Energy Section 11.1

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The Many Forms of Energy You might say that it does not because the top is not moving anywhere. However, to make the top rotate, someone had to do work on it. Therefore, the top has rotational kinetic energy. This is one of the several varieties of energy. The kinetic energy of a spinning object is called rotational kinetic energy. Rotational Kinetic Energy Section 11.1 Rotational kinetic energy depends on an object’s moment of inertia and the object’s angular velocity.

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The Many Forms of Energy A diver does work as she is diving off the diving board. This work produces both linear and rotational kinetic energies. When the diver’s center of mass moves as she leaps, linear kinetic energy is produced. When she rotates about her center of mass, rotational kinetic energy is produced. Because she is moving toward the water and rotating at the same time while in the tuck position, she has both linear and rotational kinetic energy. When she slices into the water, she has linear kinetic energy. Kinetic Energy Section 11.1

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The Many Forms of Energy Kinetic Energy Throw a ball through the air. Your hand does work on the ball because you apply a force through a distance. The work done on the ball results in the ball’s kinetic energy. Section 11.1 What happens when you catch a ball? Before hitting your hands or glove, the ball is moving, so it has kinetic energy. In catching it, you exert a force on the ball in the direction opposite to its motion. Therefore, you do negative work on it, causing it to stop. Now that the ball is not moving, it has no kinetic energy.

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The Many Forms of Energy Imagine a group of boulders high on a hill. These boulders have been lifted up by geological processes against the force of gravity; thus, they have stored energy. In a rock slide, the boulders are shaken loose. They fall and pick up speed as their stored energy is converted to kinetic energy. In the same way, a small, spring-loaded toy, such as a jack-in-the-box, has stored energy, but the energy is stored in a compressed spring. While both of these examples represent energy stored by mechanical means, there are many other means of storing energy. Stored Energy Section 11.1

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The Many Forms of Energy Automobiles, for example, carry their energy stored in the form of chemical energy in the gasoline tank. Energy is made useful or causes motion when it changes from one form to another. There are different ways energy can be stored as potential energy. Stored Energy Section 11.1 potential energy (PE) – stored energy; energy of position

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The Many Forms of Energy A pole-vaulter runs with a flexible pole and plants its end into the socket in the ground. When the pole-vaulter bends the pole, as shown in the figure, some of the pole- vaulter’s kinetic energy is converted to elastic potential energy. Section 11.1 Elastic Potential Energy

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The Many Forms of Energy When a string on a bow is pulled, work is done on the bow, storing energy in it. Thus, the energy of the system increases. Identify the system as the bow, the arrow, and Earth. When the string and arrow are released, energy is changed into kinetic energy. The stored energy in the pulled string is called elastic potential energy, which is often stored in rubber balls, rubber bands, slingshots, and trampolines. Elastic Potential Energy Section 11.1

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The Many Forms of Energy Albert Einstein recognized yet another form of potential energy: mass itself. He said that mass, by its very nature, is energy. This energy, E 0, is called rest energy and is represented by the following famous formula. Mass Section 11.1 Rest Energy E 0 = mc 2 The rest energy of an object is equal to the object’s mass times the speed of light squared.

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The Many Forms of Energy gravitational potential energy – energy stored as a result of the gravitational attraction between an object and the earth. Gravitational potential energy is usually referred to simply as potential energy, or PE. reference level - the place in a system where the PE is defined to be zero. examples: the bottom of a mountain; a pendulum’s lowest point It is convenient to pick a reference level so the PE is always positive; however it is possible for it to be negative. Section 11.1

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The Many Forms of Energy The formula for gravitational potential energy is: PE = mgh or mg h where m is mass in kilograms g is the acceleration due to gravity (9.8 m/s 2 ) h is the height above the reference level in meters example: In an experiment, Galileo dropped a rock off the top of the Leaning Tower of Pisa. The rock had a mass of 0.5 kg and the tower is 20 m high. How much potential energy did the rock have before it was dropped? PE = mgh = (0.5 kg) ( 9.8 m/s 2 ) ( 20 m) PE = 98 J Section 11.1

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The Many Forms of Energy Section 11.1 Potential Energy at Varying Locations

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The Many Forms of Energy Section 11.1 Potential Energy at Varying Locations

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The Many Forms of Energy Gravitational Potential Energy You lift a 7.30-kg bowling ball from the storage rack and hold it up to your shoulder. The storage rack is m above the floor and your shoulder is 1.12 m above the floor. a. When the bowling ball is at your shoulder, what is the bowling ball’s gravitational potential energy relative to the floor? b. What is the bowling ball’s gravitational potential energy relative to the storage rack? c. How much work was done by gravity as you lifted the ball from the rack to shoulder level? Section 11.1

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Gravitational Potential Energy Sketch the situation. The Many Forms of Energy Section 11.1 Choose a reference level.

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The Many Forms of Energy Gravitational Potential Energy Section 11.1 Known: m = 7.30 kg h s rel r = m (relative to the floor) h s rel f = 1.12 m (relative to the floor) g = 9.80 m/s 2 Unknown: PE s rel f = ? PE s rel r = ? Identify the known and unknown variables.

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Gravitational Potential Energy Set the reference level to be at the floor. Solve for the potential energy of the ball at shoulder level. The Many Forms of Energy Section 11.1 Substitute m = 7.30 kg, g = 9.80 m/s 2, h s rel f = 1.12 m PE s rel f = (7.30 kg)(9.80 m/s 2 )(1.12 m) PE s rel f = mgh s rel f PE s rel f = 80.1 J Set the reference level to be at the rack height. Solve for the height of your shoulder relative to the rack. h = h s – h r

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Gravitational Potential Energy Solve for the potential energy of the ball. The Many Forms of Energy Section 11.1 PE s rel r = mgh Substitute h = h s – h r PE = mg (h s – h r ) Substitute m = 7.3 kg, g = 9.80 m/s 2, h s = 1.12 m, h r = m PE = (7.30 kg)(9.80 m/s 2 )(1.12 m – m) PE = 36.5 J

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The work done by gravity is the weight of the ball times the distance the ball was lifted. The Many Forms of Energy Section 11.1 Because the weight opposes the motion of lifting, the work is negative. W = Fd W = –(mg)h W = – (mg) (h s – h r ) Gravitational Potential Energy Substitute m = 7.30 kg, g = 9.80 m/s 2, h s = 1.12 m, h r = m W = -(7.30 kg)(9.80 m/s 2 )(1.12 m – m) W = J

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The Many Forms of Energy Are the units correct? The potential energy and work are both measured in joules. Is the magnitude realistic? The ball should have a greater potential energy relative to the floor than relative to the rack, because the ball’s distance above the reference level is greater. Section 11.1 Gravitational Potential Energy

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The Many Forms of Energy Gravitational Potential Energy Section 11.1 Ch11_1_movanim

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The Many Forms of Energy Consider the energy of a system consisting of an orange used by the juggler plus Earth. The energy in the system exists in two forms: kinetic energy and gravitational potential energy. Kinetic Energy and Potential Energy of a System Section 11.1

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The Many Forms of Energy At the beginning of the orange’s flight, all the energy is in the form of kinetic energy, as shown in the figure. On the way up, as the orange slows down, energy changes from kinetic energy to potential energy. At the highest point of the orange’s flight, the velocity is zero. Thus, all the energy is in the form of gravitational potential energy. Kinetic Energy and Potential Energy of a System Section 11.1

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The Many Forms of Energy On the way back down, potential energy changes back into kinetic energy. The sum of kinetic energy and potential energy is constant at all times because no work is done on the system by any external forces. Section 11.1 Kinetic Energy and Potential Energy of a System

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The Many Forms of Energy In the figure, the reference level is the juggler’s hand. That is, the height of the orange is measured from the juggler’s hand. Thus, at the juggler’s hand, h = 0 m and PE = 0 J. You can set the reference level at any height that is convenient for solving a given problem. In general, pick a reference level so your numbers stay positive. Section 11.1 Reference Levels

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The Many Forms of Energy Practice Problems p. 287: 1-3; p.291: 5-8 Section 11.1

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Section Check A boy running on a track doubles his velocity. Which of the following statements about his kinetic energy is true? Question 1 Section 11.1 A. Kinetic energy will be doubled. B. Kinetic energy will reduce to half. C. Kinetic energy will increase by four times. D. Kinetic energy will decrease by four times.

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Answer: C Answer 1 Reason: The kinetic energy of an object is equal to half times the mass of the object multiplied by the speed of the object squared. Section 11.1 Section Check Kinetic energy is directly proportional to the square of velocity. Since the mass remains the same, if the velocity is doubled, kinetic energy will increase by four times.

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Section Check If an object moves away from the Earth, energy is stored in the system as the result of the force between the object and the Earth. What is this stored energy called? Question 2 Section 11.1 A. Rotational kinetic energy B. Gravitational potential energy C. Elastic potential energy D. Linear kinetic energy

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Answer: B Answer 2 Reason: The energy stored due to the gravitational force between an object and the Earth is called gravitational potential energy. Section 11.1 Section Check Gravitational potential energy is represented by the equation, PE = mgh. That is, the gravitational potential energy of an object is equal to the product of its mass, the acceleration due to gravity, and the distance from the reference level. Gravitational potential energy is measured in joules (J).

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Section Check Two girls, Sarah and Susan, having same masses are jumping on a floor. If Sarah jumps to a greater height, what can you say about the gain in their gravitational potential energy? Question 3 Section 11.1 A. Since both have equal masses, they gain equal gravitational potential energy. B. Gravitational potential energy of Sarah is greater than that of Susan. C. Gravitational potential energy of Susan is greater than that of Sarah. D. Neither Sarah nor Susan possesses gravitational potential energy.

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Answer: B Answer 3 Reason: The gravitational potential energy of an object is equal to the product of its mass, the acceleration due to gravity, and the distance from the reference level. Section 11.1 Section Check Gravitational potential energy is directly proportional to height. As the masses of Sarah and Susan are the same, the one jumping to a greater height (Sarah) will gain greater potential energy. PE = mgh

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Section 11.1 Type of EnergyDefinitionExample energy of motion energy that depends on an object’s moment of inertia and its angular velocity vibration of air particles due to breaking bonds between atoms in a molecule release of photons fusion – combining of atoms fission – splitting of atoms energy of position; stored energy energy stored in an Earth-object system as a result of gravitational attraction between the object and Earth movement of electrons energy stored in an object as a result of a change in its shape makes temperature rise; usually due to friction Einstein’s famous equation; potential energy of mass itself

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Section 11.1 Type of EnergyDefinitionExample kinetic (KE)energy of motionthrow a ball rotational kineticenergy that depends on an object’s moment of inertia and its angular velocity top soundvibration of air particlesspeakers chemicaldue to breaking bonds between atoms in a molecule batteries, photosynthesis lightrelease of photonslight bulb nuclearfusion – combining of atoms fission – splitting of atoms nuclear power plant, the Sun, nuclear weapons potentialenergy of position; stored energyyo-yo gravitational potential energy stored in an Earth-object system as a result of gravitational attraction between the object and Earth playground slide electricalmovement of electronswiring in a building elastic potentialenergy stored in an object as a result of a change in its shape bow & arrow, slingshot thermalmakes temperature rise; usually due to frictionrub hands together rest energy E 0 = mc 2 Einstein’s famous equation; potential energy of mass itself nuclear explosion

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Conservation of Energy Solve problems using the law of conservation of energy. Analyze collisions to find the change in kinetic energy. Define the Law of Conservation of Energy. In this section you will: Section 11.2

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Conservation of Energy Consider a ball near the surface of Earth. The sum of gravitational potential energy and kinetic energy in that system is constant. As the height of the ball changes, energy is converted from kinetic energy to potential energy, but the total amount of energy stays the same. Conservation of Energy Section 11.2 The law of conservation of energy states that in a closed, isolated system, energy can neither be created nor destroyed; rather, energy is conserved. closed system – no mass in or out isolated system – not forces in our out Under these conditions, energy changes from one form to another while the total energy of the system remains constant.

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Conservation of Energy The sum of the kinetic energy and gravitational potential energy of a system is called mechanical energy. Section 11.2 The mechanical energy of a system is equal to the sum of the kinetic energy and potential energy if no other forms of energy are present. In any given system, if no other forms of energy are present, mechanical energy is represented by the following equation. Mechanical Energy of a System E = KE+PE Law of Conservation of Energy

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Conservation of Energy Conservation of Mechanical Energy Section 11.2 ch11_3_movanim

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Conservation of Energy In the case of a roller coaster that is nearly at rest at the top of the first hill, the total mechanical energy in the system is the coaster’s gravitational potential energy at that point. Suppose some other hill along the track were higher than the first one. The roller coaster would not be able to climb the higher hill because the energy required to do so would be greater than the total mechanical energy of the system. Roller Coasters Section 11.2

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Conservation of Energy Suppose you ski down a steep slope. When you begin from rest at the top of the slope, your total mechanical energy is simply your gravitational potential energy. Once you start skiing downhill, your gravitational potential energy is converted to kinetic energy. Skiing Section 11.2

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Conservation of Energy As you ski down the slope, your speed increases as more of your potential energy is converted to kinetic energy. In ski jumping, the height of the ramp determines the amount of energy that the jumper has to convert into kinetic energy at the beginning of his or her flight. Skiing Section 11.2

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Conservation of Energy The simple oscillation of a pendulum also demonstrates conservation of energy. The system is the pendulum bob and Earth. Usually, the reference level is chosen to be the height of the bob at the lowest point, when it is at rest. If an external force pulls the bob to one side, the force does work that gives the system mechanical energy. Pendulums Section 11.2

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Conservation of Energy When the bob is at the lowest point, its gravitational potential energy is zero, and its kinetic energy is equal to the total mechanical energy in the system. Note that the total mechanical energy of the system is constant if we assume that there is no friction. Pendulums Section 11.2

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Conservation of Energy The oscillations of a pendulum eventually come to a stop, a bouncing ball comes to rest, and the heights of roller coaster hills get lower and lower. Where does the mechanical energy in such systems go? Any object moving through the air experiences the forces of air resistance. In a roller coaster, there are frictional forces between the wheels and the tracks. Loss of Mechanical Energy Section 11.2

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Conservation of Energy When a ball bounces off of a surface, all of the elastic potential energy that is stored in the deformed ball is not converted back into kinetic energy after the bounce. Some of the energy is converted into thermal energy and sound energy. As in the cases of the pendulum and the roller coaster, some of the original mechanical energy in the system is converted into another form of energy within members of the system or transmitted to energy outside the system, as in air resistance. Loss of Mechanical Energy Section 11.2 The total energy of the molecules is called thermal energy. Thermal energy causes the temperature of objects to rise slightly.

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Conservation of Energy During a hurricane, a large tree limb, with a mass of 22.0 kg and at a height of 13.3 m above the ground, falls on a roof that is 6.0 m above the ground. Section 11.2 a. Ignoring air resistance, find the kinetic energy of the limb when it reaches the roof. b. What is the speed of the limb when it reaches the roof? Conservation of Mechanical Energy

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Sketch the initial and final conditions. Choose a reference level. Section 11.2 Conservation of Energy Conservation of Mechanical Energy

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Conservation of Energy Identify the known and unknown variables. Section 11.2 Unknown: PE i = ? KE f = ? PE f = ? v f = ? Known: m = 22.0 kg g = 9.80 m/s 2 h limb = 13.3 m Δv i = 0.0 m/s 2 h roof = 6.0 m KE i = 0.0 J Conservation of Mechanical Energy

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Conservation of Energy Section 11.2 a. Set the reference level as the height of the roof. Solve for the initial height of the limb relative to the roof. h = h limb – h roof Conservation of Mechanical Energy Substitute h limb = 13.3 m, h roof = 6.0 m = 13.3 m – 6.0 m = 7.3 m

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Conservation of Energy Section 11.2 Identify the initial potential energy of the limb. PE i = mgh Substitute m = 22.0 kg, g = 9.80 m/s 2, h = 7.3 m PE i = ( 22.0 kg) (9.80 m/s 2 ) (7.3 m) = 1.6×10 3 J Conservation of Mechanical Energy

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The tree limb is initially at rest. Conservation of Energy Section 11.2 Solve for the initial kinetic energy of the limb. KE i = 0.0 J Conservation of Mechanical Energy PE f = 0.0 J because h = 0.0 m at the reference level. The kinetic energy of the limb when it reaches the roof is equal to its initial potential energy because energy is conserved. KE f = PE i KE f = 1.6×10 3 J

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Conservation of Energy Section 11.2 b. Solve for the speed of the limb. Conservation of Mechanical Energy Substitute KE f = 1.6×10 3 J, m = 22.0 kg

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Conservation of Energy Are the units correct? Velocity is measured in m/s and energy is measured in kg·m 2 /s 2 = J. Do the signs make sense? KE and the magnitude of velocity are always positive. Section 11.2 Conservation of Mechanical Energy

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Conservation of Energy A collision between two objects, whether the objects are automobiles, hockey players, or subatomic particles, is one of the most common situations analyzed in physics. Because the details of a collision can be very complex during the collision itself, the strategy is to find the motion of the objects just before and just after the collision. Analyzing Collisions Section 11.2

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Conservation of Energy What conservation laws can be used to analyze such a system? If the system is closed and isolated, then momentum and energy are conserved. However, the potential energy or thermal energy in the system may decrease, remain the same, or increase. Therefore, you cannot predict whether or not kinetic energy is conserved. Analyzing Collisions Section 11.2

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Conservation of Energy Consider the collision shown in the figure. “Before” is the same in all three cases. Section 11.2 Analyzing Collisions

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Conservation of Energy In case 1, the momentum of the system before and after the collision is represented by the following: Analyzing Collisions Section 11.2 p i = p Ci +p Di = (1.00 kg)(1.00 m/s)+(1.00 kg)(0.00 m/s) = 1.00 kg · m/s P f = p Cf +p Df = (1.00 kg)(–0.20 m/s)+(1.00 kg)(1.20 m/s) = 1.00 kg·m/s Thus, in case 1, the momentum is conserved.

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Conservation of Energy Analyzing Collisions Section 11.2 Is momentum conserved in case 2 and in case 3? (Yes)

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Conservation of Energy Consider the kinetic energy of the system in each of these cases. The kinetic energy of the system before and after the collision is represented by the following equations: Analyzing Collisions Section 11.2

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Conservation of Energy In case 1, the kinetic energy of the system increased. If energy in the system is conserved, then one or more of the other forms of energy must have decreased. Perhaps when the two carts collided, a compressed spring was released, adding kinetic energy to the system. This kind of collision is sometimes called a superelastic or explosive collision. Analyzing Collisions Section 11.2

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Conservation of Energy After the collision in case 2, the kinetic energy is equal to: Analyzing Collisions Section 11.2 Kinetic energy remained the same after the collision. This type of collision, in which the kinetic energy does not change, is called an elastic collision.

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Conservation of Energy Collisions between hard, elastic objects, such as those made of steel, glass, or hard plastic, often are called nearly elastic collisions. After the collision in case 3, the kinetic energy is equal to: Analyzing Collisions Section 11.2 Kinetic energy decreased and some of it was converted to thermal energy. This kind of collision, in which kinetic energy decreases, is called an inelastic collision. Objects made of soft, sticky material, such as clay, act in this way.

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Conservation of Energy In collisions, you can see how momentum and energy are really very different. Momentum is almost always conserved in a collision. Energy is conserved only in elastic collisions. Momentum is what makes objects stop. Analyzing Collisions Section 11.2

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Conservation of Energy It is also possible to have a collision in which nothing collides. If two lab carts sit motionless on a table, connected by a compressed spring, their total momentum is zero. If the spring is released, the carts will be forced to move away from each other. The potential energy of the spring will be transformed into the kinetic energy of the carts. The carts will still move away from each other so that their total momentum is zero. Analyzing Collisions Section 11.2

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Conservation of Energy The understanding of the forms of energy and how energy flows from one form to another is one of the most useful concepts in science. The term energy conservation appears in everything from scientific papers to electric appliance commercials. Analyzing Collisions Section 11.2 Practice Problems p. 297: 15,17; p.301: 27

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Section Check Write the law of conservation of energy and state the formula for mechanical energy of the system. Question 1 Section 11.2

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Section Check The law of conservation of energy states that in a closed, isolated system, energy can neither be created nor destroyed; rather, energy is conserved. Under these conditions, energy changes from one form to another, while the total energy of the system remains constant. Since energy is conserved, if no other forms of energy are present, kinetic energy and potential energy are inter-convertible. Mechanical energy of a system is the sum of the kinetic energy and potential energy if no other forms of energy are present. That is, mechanical energy of a system E = KE + PE. Answer 1 Section 11.2

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Section Check Two brothers, Jason and Jeff, of equal masses jump from a house 3-m high. If Jason jumps on the ground and Jeff jumps on a platform 2-m high, what can you say about their kinetic energy? Question 2 Section 11.2 A. The kinetic energy of Jason when he reaches the ground is greater than the kinetic energy of Jeff when he lands on the platform. B. The kinetic energy of Jason when he reaches the ground is less than the kinetic energy of Jeff when he lands on the platform. C. The kinetic energy of Jason when he reaches the ground is equal to the kinetic energy of Jeff when he lands on the platform. D. Neither Jason nor Jeff possesses kinetic energy.

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Answer: A Answer 2 Reason: The kinetic energy at the ground level is equal to the potential energy at the top. Since potential energy is proportional to height, and since the brothers have equal masses, Jason will have greater kinetic energy since he falls from a greater height. Section 11.2 Section Check

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A car of mass m 1 moving with velocity v 1 collides with another car of mass m 2 moving with velocity v 2. After collision, the cars lock together and move with velocity v 3. Which of the following equations about their momentums is true? Question 3 Section 11.2 A. m 1 v 1 + m 2 v 2 = (m 1 + m 2 )v 3 B. m 1 v 2 + m 2 v 1 = (m 1 + m 2 )v 3 C. m 1 v 1 + m 2 v 2 = m 1 v 2 + m 2 v 1 D. m 1 v m 2 v 2 2 = (m 1 + m 2 )v 3 2

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Answer: A Answer 3 Reason: By the law of conservation of momentum, we know that total momentum before collision is equal to total momentum after collision. Section 11.2 Section Check Before collision, the car of mass m 1 was moving with velocity v 1 and the car of mass m 2 was moving with velocity v 2. Hence, the total momentum before collision was ‘m 1 v 1 + m 2 v 2 ’. After collision, the two cars lock together. Hence, the mass of the two cars is (m 1 + m 2 ) and the velocity with which the two cars are moving is v 3. Hence, the total momentum after collision is ‘(m 1 + m 2 )v 3.’ Therefore, we can write, m 1 v 1 + m 2 v 2 = (m 1 + m 2 )v 3.

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Chapter 10 & 11 Test Information The test is worth 50 points total. Matching: 10 questions, 1 point each. Multiple Choice: 14 questions, 1 point each. Problems: 7 questions for a total of 28 points. Know how to use the formulas on the green formula sheet. When Mechanical Energy is Conserved: E i = E f KE before +PE before = KE after + PE after When Mechanical Energy is Lost to Heat due to Friction: E i > E f KE before +PE before = KE after + PE after + heat

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