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KE A B Horizontal position C
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11 Energy and Its Conservation
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energy in the system Note that the total mechanical energy of the system is constant if we assume that there is no friction You will learn more about pendulums in 14 Loss of mechanical energy The oscillations of a pendulum eventually come to a stop, a bouncing ball comes to rest, and the heights of rollercoaster 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 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 Usually, this new energy causes the temperature of objects to rise slightly You will learn more about this form of energy, called thermal energy, in 12 The following strategies will be helpful to you when solving problems related to conservation of energy
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When solving problems related to the conservation of energy, use the following strategies
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1 Carefully identify the system Make sure it is closed In a closed system, no objects enter or leave the system 2 Identify the forms of energy in the system 3 Identify the initial and final states of the system 4 Is the system isolated a If there are no external forces acting on the system, then the system is isolated and the total energy of the system is constant Ebefore Eafter b If there are external forces, then the following is true Ebefore W Eafter 5 If mechanical energy is conserved, decide on the reference level for potential energy Draw bar graphs showing initial and final energy like the bar graphs shown to the right
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PEf KEf
Section 112 Conservation of Energy
Conservation of Mechanical Energy During a hurricane, a large tree limb, with a mass of 220 kg and a height of 133 m above the ground, falls on a roof that is 60 m above the ground 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
Before (initial) vi hi 00 m/s 133 m
Analyze and Sketch the Problem
Sketch the initial and final conditions Choose a reference level Draw a bar graph Known:
m hlimb hroof PEi PEf 220 kg 133 m 60 m g vi KEi KEf vf 980 m/s2 00 m/s 00 J
PEi After (final)
Ground reference
vf hf 60 m
Unknown:
Solve for the Unknown
a Set the reference level as the height of the roof Solve for the initial height of the limb relative to the roof h hlimb hroof 133 m 60 m Substitute hlimb 133 m, hroof 73 m Solve for the initial potential energy of the limb PEi mgh (220 kg)(980 m/s2)(73 m) Substitute m 16 103 J
Bar Graph
60 m
220 kg, g
980 m/s2, h
73 m
Identify the initial kinetic energy of the limb KEi 00 J The tree limb is initially at rest The kinetic energy of the limb when it reaches the roof is equal to its initial potential energy because energy is conserved KEf PEi PEf 00 J because h 00 m at the reference level 16 103 J b Solve for the speed of the limb KEf vf
1 mv 2 2 f 2KEf m 2(16 1 03 J) 220 kg
Math Handbook
Square and Cube Roots pages 839 840 Substitute KEf 16 103 J, m 220 kg
12 m/s
Evaluate the Answer
Are the units correct Velocity is measured in m/s and energy is measured in kg m2/s2 J Do the signs make sense KE and the magnitude of velocity are always positive
11 Energy and Its Conservation
15 A bike rider approaches a hill at a speed of 85 m/s The combined mass of the bike and the rider is 850 kg Choose a suitable system Find the initial kinetic energy of the system The rider coasts up the hill Assuming there is no friction, at what height will the bike come to rest 16 Suppose that the bike rider in problem 15 pedaled up the hill and never came to a stop In what system is energy conserved From what form of energy did the bike gain mechanical energy 17 A skier starts from rest at the top of a 450-m-high hill, skis down a 30 incline into a valley, and continues up a 400-m-high hill The heights of both hills are measured from the valley floor Assume that you can neglect friction and the effect of the ski poles How fast is the skier moving at the bottom of the valley What is the skier s speed at the top of the next hill Do the angles of the hills affect your answers 18 In a belly-flop diving contest, the winner is the diver who makes the biggest splash upon hitting the water The size of the splash depends not only on the diver s style, but also on the amount of kinetic energy that the diver has Consider a contest in which each diver jumps from a 300-m platform One diver has a mass of 136 kg and simply steps off the platform Another diver has a mass of 102 kg and leaps upward from the platform How high would the second diver have to leap to make a competitive splash
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