Conservation of Energy

Transcription

1 Conservation of Energy In the past few lectures, we ve explored systems where mechanical energy was conserved We had to apply some pretty specific restrictions to the systems before we could assume mechanical energy was conserved However, we also know that conservation of energy is a fundamental concept of physics, which must be true if the laws of nature don t change over time In the general case, there must be other forms of energy, not just kinetic and potential energies associated with the motion and position of the center of mass or a system

2 Example: Consider a fireman sliding down a pole: As he falls, he holds his velocity constant Yet clearly his potential energy is decreasing So mechanical energy is not conserved Where does the missing energy go?

3 Example 2: Climbing a rope in gym class You pull yourself up at (more or less) constant velocity So you are gaining potential energy, but not losing kinetic energy Where is the extra energy coming from? In both cases, the answer lies in the fact that neither you, the fireman, the pole, nor the rope are simple particles, but rather complex systems of particles Such systems can absorb energy (for example, by increasing the motion of parts of the system with respect to the center of mass) They can also emit energy (if they contain an engine, battery, food, etc.) So we need to account for the ernal energy of the system

4 A more general expression of conservation of energy is needed We start by combining two ideas we ve already covered 1. The work done on an system is equal to the change in kinetic energy 2. The work done by conservative ernal forces is equal to the opposite of the change in potential energy Thus we can say: KE = W = W + W = W U KE + U = W tot ext ext ext

5 But we ve already seen that systems of particles can store energy in other forms than kinetic or potential energy, so we need to add this ernal energy as well This leads to the general equation for energy conservation (good for all systems!): KE + U + E = W ext In applying this equation, it will be crucial to be clear on what is ernal to the system and what is external

6 For example, consider a skydiver falling at constant velocity If we take the system to be the skydiver only, we have: K = U = 0 0 W = W + W ext grav air E = W = W + W ext grav air There s little more we can say, since the work done by the air can only be determined through a detailed model of the eraction between the air and skydiver This is true for all types of frictional work There s no potential energy associated with the skydiver alone

7 Let s now add the air to the system In this case, we still have no change in potential or kinetic energy But the only external work is done by gravity, so we have: E = Wext = Wgrav = mg y So we see that as the skydiver falls ( y is negative) the ernal energy of the air+skydiver increases by a welldefined amount In other words, the work done by gravity goes o random motion of molecules in the skydiver and air, and not o center-of-mass kinetic energy

8 We can also define the system to include the entire earth In this case, there is no external work done, and the change in kinetic energy is still 0 However, we do know have a change in potential energy associated with the ernal gravitational force between the skydiver and the earth: U + E = This is the same result we got before Implies ernal energy of earth isn t changing, which makes sense Selecting which objects are in the system is a matter of convenience 0 mg y + E = E = mg y 0

9 Example You wake up one morning on an empty stomach, and notice that there s an infinite upwards staircase in front of you. You eat a PowerBar (235 calories) and start climbing. How far up do you get before collapsing from exhaustion? A calorie is equal to 4.186J, and a food calorie is 1000 real calories So you have an initial ernal energy supply of 984kJ Assuming the system is you+the stairs+the earth, conservation of energy tells us: K + U + E = 0

10 We can assume that your kinetic energy is not changing (I m not asking you to accelerate up the stairs!), and also assume that the only change in ernal energy is your use of the energy from the PowerBar So we have: U + E = Where I ve assumed your mass is 60kg 0 mgh 984kJ = 0 984kJ 984kJ h = = = 1670m 2 mg 60kg 9.8m/s Of course, you probably can t really go this high, since some of the energy in the PowerBar goes o maaining your body temperature, running your heart, heat in the air, etc.

11 Frictional Work We might be tempted to assume the the work done on an object by friction is given by W = fs, where f is the frictional force and s is the distance the object moves To see why this isn t so, consider a block being pulled at constant velocity across a table top We ll define the system to be the block only f T

12 Applying conservation of energy gives: KE + U + E = Wext 0 0 = W W Since the net force on the block is zero, we know that T = f E = fs W W = fs E f And we know that the ernal energy of the block isn t constant when friction acts i.e., the block gets warmer ernal energy increases Therefore, work done by friction is always less than fs T = Ts W f f f