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112 Conservation of Energy
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onsider 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
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Solve problems using the law of conservation of energy Analyze collisions to find the change in kinetic energy
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In our everyday world, it may not seem as if energy is conserved A hockey puck eventually loses its kinetic energy and stops moving, even on smooth ice A pendulum stops swinging after some time The money model can again be used to illustrate what is happening in these cases Suppose you have a total of $50 in cash One day, you count your money and discover that you are $3 short Would you assume that the money just disappeared You probably would try to remember whether you spent it, and you might even search for it In other words, rather than giving up on the conservation of money, you would try to think of different places where it might have gone Law of conservation of energy Scientists do the same thing as you would if you could not account for a sum of money Whenever they observe energy leaving a system, they look for new forms into which the energy could have been transferred This is because the total amount of energy in a system remains constant as long as the system is closed and isolated from external forces 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 Conservation of mechanical energy The sum of the kinetic energy and gravitational potential energy of a system is called mechanical energy 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
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Figure 1 1-9 A decrease in potential energy is equal to the increase in kinetic energy 100 N 00 J 200 J
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
Imagine a system consisting of a 100-N ball and Earth, as shown in Figure 11-9 Suppose the ball is released from 200 m above the ground, which you set to be the reference level Because the ball is not yet moving, it has no kinetic energy Its potential energy is represented by the following equation: PE mgh (100 N)(200 m) 200 J
200 m KE PE 100 J 100 J 100 m KE PE 200 J 00 J
The ball s total mechanical energy, therefore, is 200 J As the ball falls, it loses potential energy and gains kinetic energy When the ball is 100 m above Earth s surface: PE mgh (100 N)(100 m) 100 J
000 m Ground KE PE
Section 112 Conservation of Energy
Weight PE 200 J
100 N
What is the ball s kinetic energy when it is at a height of 100 m The system consisting of the ball and Earth is closed and isolated because no external forces are acting upon it Hence, the total energy of the system, E, remains constant at 200 J E KE 200 J PE, so KE E PE
20 m KE 200 J
200 J
100 J = 100 J
Figure 1 1-10 The path that an object follows in reaching the ground does not affect the final kinetic energy of the object
When the ball reaches ground level, its potential energy is zero, and its kinetic energy is 200 J The equation that describes conservation of mechanical energy can be written as follows Conservation of Mechanical Energy KEbefore PEbefore KEafter PEafter
When mechanical energy is conserved, the sum of the kinetic energy and potential energy present in the system before the event is equal to the sum of the kinetic energy and potential energy in the system after the event
Figure 1 1 For the simple 1-1 harmonic motion of a pendulum bob (a), the mechanical energy the sum of the potential and kinetic energies is a constant (b)
What happens if the ball does not fall down, but rolls down a ramp, as shown in Figure 11-10 If there is no friction, there are no external forces acting on the system Thus, the system remains closed and isolated The ball still moves down a vertical distance of 200 m, so its loss of potential energy is 200 J Therefore, it gains 200 J of kinetic energy As long as there is no friction, the path that the ball takes does not matter Roller coasters 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 Skiing 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 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 Pendulums 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 At the instant the bob is released, all the energy is in the form of potential energy, but as the bob swings downward, the energy is converted to kinetic energy Figure 11-11 shows a graph of the changing potential and kinetic energies of a pendulum When the bob is at the lowest point, its gravitational potential energy is zero, and its kinetic energy is equal to the total mechanical
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