A. La Rosa Lecture Notes PH 21 ENERGY CONSERVATION The Fisrt Law of Thermodynamics and the Work/Kinetic-Energy Theorem ENERGY [This section taken from The Feynman Lectures Vol-1 Ch-4] 1. What is energy? 2. Ideal machines and reversible processes 2.A Reversible process 2.B The principle of non-perpetual motion 2.C Comparing reversible and non-reversible machines. 2.D Universal behavior of reversible machines. Figuring out the potential energy TRANSFER of ENERGY Heat-transfer Q Macroscopic external Work W done on a system ENERGY CONSERVATION LAW The work/kinetic-energy theorem Case: inelastic collision Generalization of the work/kinetic-energy theorem Fundamental Energy Conservation Law Inelastic collision (revisited) Case: Pure Thermodynamics The First Law of Thermodynamics 1
ENERGY [This section taken from The Feynman Lectures Vol-1 Ch-4] 1. What is energy? There is a fact or law governing all natural phenomena that are known to date, called the conservation of energy. It states that: There is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes It is not a description of a mechanism It is a mathematical principle It says that it is a numerical quantity which does not change when something happens A strange fact that we can calculate some numbers and when we finish watching nature go through her tricks and calculate the number again, it is the same Conservation of blocks Little John plays with 28 indestructible blocks At the beginning of the day, his mother puts him, with his blocks, into a room. At the end of every day, she counts the blocks very carefully, and discovers a law: No matter what little John does with the blocks, there are always 28 remaining. One day she finds only 26. There is consternation. But when she looks through the window, she find that 2 blocks lies on the garden. Another day she finds! Later she finds out that a friend came over to visit John and left 2 block. She starts enjoying the counting blocks game. But from now on she will close the windows and will not allow extra blocks entering the room. Everything goes all right Each block weights 2 grams number of (weight of the box) - 5 grams blocks seen 2 grams constant Original height of the buthtub is 4 cm. Each block raises the water by 2 cm 2
m number of blocks seen (weight of the box) - 5 grams 2 grams height of the water - 4 cm constant 2 cm In the complexity of her world she finds a whole series of terms representing ways of calculating how many blocks are in places where she is not allowed to look. As a result she finds, A complex formula, a quantitity that has to be computed, which always stays the same. What is the analogy of this to the conservation of energy? The most remarkable aspect that must be abstracted from this picture is that there are no block The analogy has the following points: When we are calculating the energy, sometimes it leaves the system and goes away, or sometimes some comes in. To verify the conservation, we have to track that we have not put any in nor taking out. The energy has a large number of different forms, and there is a formula for each: gravitational, electrical, radiant energy, Quoting Feynman, In physics, today, we have no knowledge of what energy is. 2. Ideal machines and reversible processes 2.A Reversible process Y Consider a frictionless lever (an ideal machine indeed.) 1 g Figure 1. A massless, friction free, weight lifting machine. We wish to lift a block of mass m.
Y For that purpose, the block of mass m is placed at the bottom on the right side of the lever, and a mass m is placed at the left of the lever at a height h= 1 m. Here m is slightly smaller than m. [ m = m - with ] m 1 m g U internal = 2 1 m i Figure 2. Lever in its initial position. Since m < m the system in Fig. 2 will remain stationary. The lever will be able to lift the block of mass m if some additional energy K were added to the mass m. Y K m 1 m g Figure. A little bit of kinetic energy is given to m so it can start lifting the m block. Let s choose the magnitude of K such that, afterwards, m lowers down to the ground level height=, and the lever becomes stationary. Thus the lever machine has lifted the block m to a height : 4
K m 1 m g Figure 4. The amount of kinetic energy given to m is just to make m to go down to the bottom level (while lifting the m block.) At this stage, we clamp the lever to, momentarily, avoid further motion. K The motion could continue if we just unclamp the lever (this occur because m is slightly less than m, so a small torque acts on the lever). When m reaches a height of, it would have a kinetic energy K. We can place a spring at the proper position such that, when the mass m reaches the height 1 m, the energy K gets stored into the spring. Right after this occurs, we clamp the lever again. Y m Spring compressed a bit 1 m g Figure 5. Lever and masses m and m are back to their initial positions (as in Figure 2). If we wanted to lift the m block again, all we need to do is to unclamp the lever. What we have, then, is a reversible process. The work (energy) that the lever invested in lifting the m block by a distance (Fig. 4) was at the expense of lowering the mass m by. By operating the lever in reverse, mass m is lifted at the expense of lowering the m block (Fig. 5). 5
Notice that for operating this reversible process, we (the external agent) just need to clamp and (! ) unclamp the lever during the process; i.e. we do not input to, nor extract from, the machine any energy. 2.B The principle of non-perpetual motion. Reversible and non-reversible machines The machine described above is called a reversible machine. It is an ideal machine (where friction is absent), which is in fact unattainable no matter how careful our design. Its concept is however useful, for comparing it with other non-reversible machines. A non-reversible machine includes all real machines. They are subjected to friction and other adverse factors that detriment their motion. Non-perpetual motion Let s now consider the following hypothesis: There is not such a thing like perpetual motion. (This is basically a general statement of the conservation of energy.) For the case of lifting-weight machines: If, when we have lifted and lowered a lot of weights and restored the machine to the original condition, we find that the net result is to have lifted a weight, then we would have a perpetual motion machine, (because we can use that lifted weight to run something else, which can be repeated again and again.) Accordingly, for the case of weight-lifting machines, in the absence of perpetual motion, after bringing the machine to its initial state, the net lifted weight should be zero. 2.C Comparing reversible and non-reversible machines. Consider the reversible machine shown in Figures 1 and 2. Let s call it machine-a. This machine lowers the mass m by and lifts a m weight to a maximum height ; and then it is run in reverse (as described above, Figures 1 to 5.) Consider another machine-b which is not necessarily reversible. This machine lifts a m weight to a maximum distance Y, while lowering a mass m by one meter (m m). to its initial position (similar to the process as 6
K described above, Figures 1 to 5.) The mechanism of how machine-b operates is unknown. [Maybe it just drops the mass m from the height, and at the bottom it trampolines the mass m, the latter the shooting to a height Y.] We now prove that Y cannot be greater than More general, we state that it is impossible to build a weight-lifting machine that, by lowering a mass m by one meter, it will lift a weight any higher than it will be lifted by a reversible machine. Proof: Suppose Y >. That is, the special unknown design of machine-b allows lifting the m block to a height Y while bringing the mass m one meter down. The design is assumed to be so good that Y>. Y m Machine-B Unknown arbitrary design 1 m Y g K Figure 6. Once the m block is at Y, we can let it free-fall to a height and thus obtain free energy. We then use the machine-a (with the lever in figure 1 with the left side located down. The mass m (being already at the lower position) is placed at the left side of the lever; and the block of mass m (being already at the height ) is placed at the right side of the lever (see figure 7 below.) Machine-A m 1 m g Figure 7. We now run the the machine-a backwards (raising the mass m ). At the end of this process we would have brought all the masses (m and m to their initial height (shown in Figure 6). In addition we would have had an additional energy (when the m block was thrown from Y to ). But this would constitute perpetual motion, which we postulated is not possible. Therefore Y cannot be greater than. 7
m Y Accordingly, among all the machines that can be designed, the reversible machines are the best. They lift the m block to the highest height. 2.D Universal behavior of reversible machines All reversible machines must lift the m block to exactly the same height. Notice the proof of this statement is similar. If one reversible machine-c were to lift the m weight to a height Z>, we could free fall that block to a height and then operate machine-a in reverse. This would constitute perpetual generation of free energy, which is not possible. Therefore Z cannot be higher than. But cannot be smaller than Z either. (The same argument used above with Z and interchanged). Therefore, all reversible machines lift the m block up to the same height. This is a remarkable observation because it permits us to analyze the height at which different machines are going to lift something without looking at the interior mechanism. If somebody makes an enormous elaborated series of lever that lift a m block a certain distance by lowering a mass m by one unit distance, and we compare it with a simple lever which does the same thing and is fundamentally reversible, his machine will lift it no higher, but perhaps less height. If his machine is reversible, we also know exactly how high it will lift. In summary, we have a universal law: Every reversible machine, no matter how it operates, which drops one kilogram by one meter and lift a -Kg weight always lifts it the same distance. The question is, what is the value of?. Figuring out the potential energy K Y m 8
m K m m K m By design, the height of the boxes is. We claim that has to be equal to. So = 1/ meter. Notice two blocks practically were not lifted. The net effect on the right is to lift one ball by, while on the right one block was down. (equivalent to lowering one ball by ) Macroscopic and microscopic contributions to the energy The total energy of a system has two distinct contributions: R CM U internal = V CM 2 m u i 1 2 i i i, j 1 2 P ij Disordered energy (gas molecules inside the container.) u i = velocities P ij = potential energies E macro = (1/2) Mv CM 2 + + (1/2) I 2 + Mgz Ordered energy (of the can cylinder) 9
Fig. 1 A. MACROSCOPIC COMPONENT ( Ordered energy. ) The total mechanical energy of the system, associated with the macroscopic position and motion of the system as a whole. This mechanical energy comprises: i) Translational kinetic energy of the center of mass (CM) + + the rotational kinetic energy calculated with respect to the CM. ii) Potential energies associated to the position of the center of mass (gravitational potential energy, electrical potential energy, potential energy associated to the spring force, etc.) B. MICROSCOPIC COMPONENT ( Disordered energy. ) The other contribution to the energy is a vast collection of microscopic energies, known collectively as the internal energy U of the system. U comprises: The sum of individual kinetic and potential energies associated with the motion of, and interactions between, the individual particles (atoms and/or molecules) that constitute the system. These interactions involve complicated potential energy functions on a microscopic distance scale. In principle, after an appropriate choice of the zeros of the potential energy functions, one can talk about a definite value of U of the system (when the latter is in a state of thermodynamic equilibrium.) But such calculation of U can be a complicated endeavor. It is relatively simpler to calculate the changes of U. U: When a system changes its state of thermodynamic equilibrium, it is only the changes in the internal energy U that are physically significantly. 1
TRANSFER of ENERGY Different systems can transfer energy among themselves by two processes: (1) Via heat-transfer, driven by temperature differences (2) Via work, driven by external macroscopic forces We will see that, Heat-transfer to a system is fundamentally a microscopic mechanism for transferring energy to a system. Work done on a system is a macroscopic mechanism for transferring energy to a system Heat-transfer Q It can occur via conduction, convection, and radiation The mechanism is fundamentally microscopic (at the atomic and molecular level.) heat transfer is accomplished by random molecular collisions and other molecular interactions. The direction is from the higher to lower temperature (an aspect better explained in the context of the second law of thermodynamics.) Warning: Do not confuse heat-transfer Q with the internal energy U. Heat transfer is not a property of the state of a system (a system in thermal equilibrium does not have an amount of heat or heat-transfer.) On the other hand, a system in thermal equilibrium does have (in principle) a specific internal energy. That is, Q is not a state variable U is a state variable 11
Macroscopic external Work W done on a system The macroscopic external work W done on a system can cause a change in either the internal energy U of the system, or the total mechanical energy E of the system Example where the external work causes a change of purely internal energy Insulation (no heat transfer F Q=) External non conservative force) Fig. 2 Gas enclosed in an insulating container. The insulated walls ensure an absence of heat transfer from or toward the system (the gas.) Movable piston allows an external agent to compress the gas (by pushing the piston), thus doing work on the system. The work on the gas by the external agent results in an increase of the gas temperature (indicative of an increase in the internal energy U.) On the other hand, simply lifting the gas container (described above) would be an example of increasing the mechanical energy of the system, without changing the internal energy. 12
ENERGY CONSERVATION LAW The work/kinetic-energy theorem We are already familiar with the work/kinetic-energy theorem, which establishes the source (work) that causes a change in the kinetic energy of a system. We illustrated this theorem for the case of an individual particle, as well as for a system of particles constituting a rigid body. The later allowed us to solve, in a very straightforward manner, problems involving bodies rolling down an inclined plane, for example.[ But cases involving work done by internal forces in non-rigid bodies were not considered. We will encounter such cases in this section.] Case: Inelastic collision In what follows, we illustrate the need for generalizing the work/kinetic-energy theorem, in order to include cases in which disordered (microscopic) energy is involved. To that effect, let s start consider an inelastic collision. Before the collision Both particles initially at the same temperature and in thermal equilibrium v At rest m M Kinetic energy: K before = ½ m v 2 frictionless i After the collision V ( m + M ) frictionless i 1
Kinetic energy K after = ½ (m +M) V 2 Since the linear momentum is conserved mv = ( m + M ) V K after = ½ (m +M) [ ( m / (m + M) v ] 2 K after = ½ [ ( m 2 / (m + M) v 2 ] = ½ m v 2 [ ( m / (m + M) ] The change in kinetic energy is given by, K = K after - K before = [( m / (m + M) -1 ] ½ m v 2 = - [( M / (m + M) ] ½ m v 2 that is, the kinetic energy is less after the collision than before. According to the work/kinetic-energy theorem this change should have resulted from the work done by the forces acting on the system. But notice, all the external forces acting on the system (normal forces and weight) are perpendicular to the displacement of the particles, hence, their work on the system is zero (W N =, W W =.) ( m + M ) N m N M frictionless i W 1 W 2 Apparently, then, the work/kinetic-energy theorem W total = K appears not to be valid here. 14
The explanation lies in the fact that we are not including the work done by the internal friction forces. Such forces act during the inelastic collision. We say then, W internal-friction = K Thus, in this particular example, we identify the decrease in the kinetic energy in the negative work done by the microscopic internal forces. We would like to highlight that the change in kinetic energy K may include not only the macroscopic kinetic energy (of the center of mass) but also (presumably) an increase also of the microscopic kinetic energy; that is, W internal-friction = K macroscopic + K microscopic (case for the inelastic collision depicted in the figure above) The work/kinetic-energy theorem While we can in principle understand what is going on in the particular example of inelastic collision (where the system under study does not receive external heat-transfer), we would like to explore reformulating the work/kinetic-energy such that include also cases where thermal interaction (heat transfer) from the surrounding environment is allowed. As a firs step, let s express the work/kinetic energy theorem as follows, W internal + W external-non-conservative + W external-conservative = = K CM + K microscopic (generalization of the work/kinetic-energy theorem) Here W external-non-conservative refers to the work done by forces like the one pushing the piston in Fig. 2 above. W external-conservative could be the work done, for example, the gravitational force. 15
That is, we are explicitly separating out the macroscopic work (done by external macroscopic forces, conservative and nonconservative) from the work done by microscopic forces. Similarly, we assume also that the kinetic energy changes in both macroscopic (the CM kinetic energy) and microscopic forms i) For the conservative forces component, the work can be derived from a potential energy function E p, W external-conservative = - E p which gives, W internal + W external-non-conservative + (- E p ) = = K CM + K microscopic W internal + W external-non-conservative = K CM + E p ) + K microscopic Calling K CM + E p E macro the macroscopic mechanical energy, the work/kinetic-theorem can be written as, W internal + W external-non-conservative = E macro + K microscopic ii) We can envision that, ultimately, W internal causes a change in microscopic potential energies of the interacting microscopic particles that constitute the system. That is, W internal = - P ij. i j Hence, - P ij + W external-non-conservative = E macro + K microscopic i j 16
W external-non-conservative = E macro + K microscopic + i j P ij Change in macroscopic mechanical energy Change in the internal energy U The last two terms in the right side of the expression above constitute what we called, at the beginning of this section, the disordered Internal Energy U of the system. Through the derivation process followed above, we notice that the work energy is deposited (transformed) into the system as either, macroscopic mechanical energy, or internal energy. The work W external-conservative done by conservative macroscopic external forces has been assimilated into the mechanical energy, while the work W internal done by microscopic forces ended up being grouped into the internal energy term. The expression above also shows that the work energy W external-non-conservative done by external non-conservative forces could end up either as macroscopic mechanical energy or internal energy (that the latter case can occur was illustrated in the example above where a gas was compressed by a piston; the force acting on the piston was the nonconservative force.) Generalization of the work/kinetic-energy theorem As illustrative as the expression above might be, it also reveals its limitations for dealing with cases in which the system is in thermal contact with a body at different temperature. Indeed, in such a case, the system can also receive energy via heat-transfer, a 17
process driven by temperature differences.) Accordingly the expression above needs to be modified or generalized. Q + W external-non-conservative = E macro + K microscopic + i j P ij Heat-transfer into the system In a simplified form Change in macroscopic mechanical energy Change in the internal energy U Q + W external-non-conservative = E macro + U Heat-transfer into the system caused by temperature difference Work done on the system by a nonconservative macroscopic external force Change in macroscopic mechanical energy of the system Change in the internal energy U of the system which constitutes our Fundamental Energy Conservation Law. Inelastic collision (revisited) Both particles initially at the same temperature and in thermal equilibrium v m At rest M frictionless i Kinetic energy K before = ½ m v 2 After the collision V ( m + M ) frictionless i 18
Here Q is the flow of energy by heat transfer, caused by temperature differences. In our case is zero.) W is the work done by external forces. In our case it is zero. U change in the internal energy E change in the mechanical energy In our case E = - [( M / (m + M) ] ½ m v 2 Accordingly, + = - [( M / (m + M) ] ½ m v 2 ) + U which gives, U = [( M / (m + M) ] ½ m v 2 ) That is, the missing (ordered) kinetic energy appears as an increase in the internal (disordered)s energy U of the system. (The increase in the internal energy of the system typically manifest itself in an increase in the temperature of the system. As the temperature of the system increases above the ambient environment because of the increase in the internal energy, heattransfer subsequently occurs from the system to the environment until the system-ambient reach a common temperature. Case: Pure Thermodynamics In pure thermodynamics, one typically considers only systems whose total mechanical energy does not change, E macro =. The general statement of the energy conservation becomes, Q + W external-non-conservative = U 19
sulation o heat transfer Q=) Notice Before F After F Work done by the external force F > W external-non-conservative > sulation o heat transfer Q=) Before F After F Work done by the external force F < W external-non-conservative < It is typical to consider the work done by the system (no the work done on the system by the external non-conservative forces.) Since, according to the Newton s third law, the force exerted by the system is equal in magnitude but opposite in direction, then W external non-conservative = - W done-by-the-system Thus, for pure thermodynamic systems Q Heat-transfer-into-the-system - W done-by-the-system = U When all the terminology is understood, the subscripts are omitted and one simply writes Q - W = U The First Law of Thermodynamics 2
n transfer ) Notice Before F After F Work done by the gas < W < n transfer ) Before F After F Work done by the gas > W > Question: wedge Before After W =? 21