Spring, Chapter : A. near the surface of the earth. The forces of gravity and an ideal spring are conservative forces. With only the forces of an ideal spring and gravity acting on a ass, energy F F will be exchanged between the potential kinetic fors of energy. T s The su of the potential energies (spring and ass) and kinetic energy (ass) will reain the sae during transfers of energy k between the potential and kinetic fors. Huan action is not a T conservative force. The work ground done by the huan body, Figure. Potential energy introduced into a spring and a ass. Forces acting downward on the earth are not shown. whether positive (adding energy) or negative (reoving energy), will increase or decrease the total energy available, respectively. The huan body generates forces by releasing cheical energy (stored by the electroagnetic forces in olecules) to contract uscles. These cheical reactions cannot be reversed to store echanical energy; e.g., the kinetic energy of a baseball when caught cannot be stored as potential energy by the huan body for later use. Huan action can, however, increase or decrease the potential energy stored by a spring or stored by a ass (due to gravity), as shown in Figure.. As described in chapter 9, the work done by the huan stretching the spring a distance, s = x, fro its natural length, stores a spring potential energy, PE S = Slowly lifting a ass,, a distance, s, as shown on the right in Figure., an external huan force, F = g, does an aount of work, w = Fs = gs, and the gravitational force stores an equal aount of potential energy in the ass. Defining the gravitational potential energy of the ass to be zero at the height of the tabletop, after being lifted to the height, s = h, above the table, the potential energy of the ass (due to gravity) is, kx. PE G = gh Eq.. 89
Spring, and kinetic energy, are both properties of a ass. The gravitational force can transfer the potential energy of the ass to kinetic energy of the sae ass. For gravity, the potential energy does not reside in an interediate object, like it did for the potential energy stored by a spring being transferred to a ass. B. Transfers of gravitational potential energy, PE, to KE, near the surface of the earth. Once a ass, raised to an initial height h, is released, as shown in Figure., the gravitational force acting on the ass akes it fall. In the process gravity does work on the ass transferring its potential energy to its kinetic energy. As the potential energy decreases the kinetic energy of the ass increases. The conservative gravitational force akes no change in the total energy, E = KE + PE, where, PE = PE G, if gravity is the only force involved (subscript, G, reoved to siplify notation). The total energy will reain constant, but the energy in the two coponents, the potential energy, PE, and the kinetic energy, KE, changes as the ass falls. The ass, when released at t =, has the gravitational potential energy, PE = gh and a kinetic energy, KE = (the ass has the speed v = when released). The total energy at the tie of release, E = KE + PE = + gh will have this value at any other tie during the fall. Just prior to hitting the tabletop, the ass has lost all of the gravitational potential energy (given to it by lifting), so that, PE =, while gaining the kinetic energy, KE = v. The total energy just prior to hitting the table is, therefore, G E = KE + PE = v +. The total energy at release and just before hitting the table ust have the sae value, if only conservative forces (gravity and springs) are involved E = E, and allows a prediction for the speed at the tie the ass hits the table: ground h Figure. A ass just released at a height, h, fro a table. Eq.. 9
Spring, v E = E = gh v = gh, v = gh. The units of g in this expression ust use the equivalent unit of force, N = kg /s (see Chapter 9) to ake the conversion: g = 9.8 N/kg, to g = 9.8 /s. It will be shown later that g has an acceleration interpretation as well as one relating ass and weight, and a siilar expression for the speed of an object oved a distance under a constant force and acceleration will appear in a later chapter. This analysis sees to ignore the possibility that the ass could fall off the table. In contrast to a spring where a natural length spring is the clear choice for the zero of a spring s potential energy, gravity does not have a unique state that is logically defined as the zero of gravitational potential energy. The proper place for the gravitational potential energy to be zero ust be chosen for each situation and soe training is needed to ake a wise choice. C. The zero of potential energy. Near the surface of the earth, the gravitational potential energy of a ass will be a iniu when the separation of the ass fro the center of the planet is the sallest allowable value. If a ass could never fall off a table, the logical place for zero gravitational potential energy is at the surface of the table. If the ass can fall to the ground, a ore logical choice for a state with zero potential energy is at the surface of the earth. If the ass can fall to the botto of the baseent (below ground level), the baseent floor becoes the logical place to define as the location with zero potential energy. PE = The potential energy ust decrease for a ass carried into a baseent. The potential energy of d the ass, taken as zero on the surface of the earth, is negative at any point lower than the surface, PE = gd as shown in Figure.3. Figure.3 A ass is carried into a baseent a distance, d, below the surface, changing PE fro to gd. 9
Spring, Only changes in potential energy, PE, are constrained by the conservation of energy: KE + PE =, (see previous chapter). Changes in the potential energy of a ass are insensitive to the height PE = gd at which the potential energy is defined to be zero. Brought up fro the baseent the potential d energy starts with a negative value and is zero at the surface resulting in a potential energy change that is PE = positive. Figure.4 Defining PE to be zero in the baseent. Defining the zero of potential energy the baseent, as shown in Figure.4, the potential energy of the ass, raised fro the baseent to the earth's surface, starts at zero and becoes higher at the surface, resulting in a change in potential energy that is again positive. D. Energy conservation in the action of springs and gravity There are any phenoena that are the result of the action of just the two conservative forces, gravity and an ideal spring. A ass fired upward by a copressed spring, as shown in Figure.5, is an exaple of a process involving only conservative forces. The process can be thought of as taking the potential energy stored in a spring, transferring it to the kinetic energy of the ass. When the ass reaches its highest point, where the speed, v, is zero, the energy has been copletely transferred to the gravitational potential energy of the ass. A detailed analysis shows how the speed and height of the ass are related to the initial energy stored in the spring. The ass,, is placed on the spring copressed a distance, x, and PE S 9 kx k x h=h gh h=h h=h h= PE G gh gh Figure.5 A copressed spring firing a ass upward.
Spring, then released. The initial (the tie of release) values of the kinetic energy and the two potential energies are: KE = (ass starts with v = ), PE = kx (spring PE), and, S PE = gh (gravitational PE). G The total energy at the initial tie is the su of these: E = ( KE + PE + PE ) S G =+ kx gh + The only forces acting on the ass are the spring force and the gravitational force, both conservative forces, which will conserve energy through the flight of the ass. When the ass reaches the height, h = h = h + x, it will loose contact with the spring, have a speed, v, and the spring will have transferred all of its potential energy, PE S =, to the kinetic and potential energy of the ass. The value of the total energy at this tie is then: E = ( KE + PE + PE ) = v + gh S G and ust have the sae value as the initial value, E = E. Equating the two expressions of the total energy yields a prediction for the speed, v : E v = E v gh kx gh + = + k = x g ( h h ) ( h h = x ) k v = x gx. When the ass reaches the highest point, h= h, the speed and kinetic energy of the ass will be zero (it would ove higher if the speed were not zero). The gravitational potential energy, PE gh, will be a axiu, and the total energy will be G = E = ( KE+ PES + PEG) = + + gh. A prediction for the highest point reached by the ass is obtained by equating the initial and final values of the total energy: E = E gh = kx + gh h kx = + h. g 93
Spring, The conservation of the total energy has lead to predictions of the speed and height of the ass at critical points in its flight in ters of the initial conditions. E. The effects of work by non-conservative forces on a ass In this section the energy conservation concept is odified to include the effects of nonconservative forces. Instead of a spring firing a ass, as shown in Figure.5, the ass is now lifted slowly with your hand (so that it never leaves your hand) fro the initial height, h, up to the height, h F H, increasing the gravitational potential energy of the ass by, h h PEG = gh gh = g( h h). This otion involves the introduction of energy by the non-conservative force of a huan being. The force of the hand is larger than the weight only for a short tie at the start of the upward otion and, ust be slightly less for a short tie at the end of the otion. The agnitude of the force that your hand applies to the ass is equal to its weight as the ass oves upward with a constant speed, as shown in Figure.6. The total energy of the ass at the final height (the speed is zero) is entirely in potential energy. The aount of work done by the huan ( w H ) in lifting the ass fro h to the height, h, is wh = g ( h h ). Throwing a ass upward to a axiu height, h, and lifting the ass slowly to the sae height, both involve a transfer of the sae aount of energy fro the huan being to the ass. In throwing, energy is introduced rapidly into kinetic energy of the ass, while in lifting, energy is introduced slowly, however, when reaching the axiu height, the potential energy of the ass is the sae for both otions In free flight, only gravity (a conservative force) affects the otion of a ass, and therefore the total energy of the ass, the su of its kinetic and potential energies, is conserved. When a huan slowly raises a ass, the potential energy of the ass is increased with negligible changes in its kinetic energy. Work done by huans, w H, and F F F H G H = g =+ g Figure.6 Forces on a ass being raised by hand. 94
Spring, other non-conservative forces, such as sliding friction, require special treatent to incorporate their effects on total energy of a ass. Energy conservation, incorporating non-conservative forces can be written as, KE + PE + wnc = KE + PE, (.3) where w NC represents the work done by huans, and also includes the effects of the forces of friction or an explosion on the available energy. Rearranging Equation.3 to show changes in potential and kinetic energies of the ass, yields, wnc KE PE The ter labeled PE, includes changes in potential energies of springs, PE (.4) kx, S = and of a ass, PE G = gh, due to gravity. If the changes in kinetic and potential energies of a ass do not su to zero, energy ust have been added or reoved by a nonconservative force. When a huan being does a positive aount of work on a ass, as when lifting it, Equation.4 predicts an increase in the energy of the ass. No energy is lost or created in the process. Energy is transferred fro the huan to the ass. The ter w NC in Equation.4 corresponds to changes in the energy of the ass on which the non-conservative force acts. For friction, the energy change is nearly always negative and w NC corresponds to the energy that appears as heating the objects in contact. For explosions, the energy change is positive and w NC corresponds to the energy of the explosion added to the otion of the asses it has affected. For cases in which both huans and friction are involved, there are two w NC ters, one for each force. Consider a ass pushed at a constant speed by a huan on a friction-generating table. As the ass oves, no changes occur to the kinetic energy (or potential energy) so that what ever energy the huan adds by doing work, the frictional force reoves by heating the surfaces: w w ( ) = ( ). NC huan NC friction The huan body is very inefficient in converting stored food energy into work on asses, and uch of the energy transfer in this process instead heats the body. Any change in the body heat is accopanied by an equal reduction in the food energy stored by the huan. When a huan slowly lowers a ass (work is negative), the potential energy of the ass decreases, while the body heat increases. Again, inefficiencies in the huan body require additional food energy to be used that appears as heat in the body. The interchange of heat and work is the subject of an entire course in therodynaics. Here, it will suffice to say that once energy is transferred to heat, soe additional heat ust be generated in any attept to utilize that energy to do work. This feature of nature, known as the second law of therodynaics, prevents the construction of achines in perpetual otion with no outside energy source. 95
Spring, Chapter Suary The gravitational force,, is a conservative force and any work it does on a ass changes the gravitational potential energy, PE G, of the ass. For a ass,, near the surface of the earth at a height h above the ground, the ass has a gravitational potential energy, PE gh, in units, N, or equivalently, in joules (J). G = The height at which the gravitational potential energy, PE G, is zero can be adjusted at the beginning of a proble, by defining the location of the height, h =, to fit the situation being addressed (usually the lowest point of the otion). If only conservative forces act on a ass, then the total energy of the ass will be conserved: KE + PE = KE + PE, or equivalently, KE + PE =. Near the surface of the earth, the work, w H, done by a huan being (or by other nonconservative forces), on a ass will result in a corresponding change in the su of the kinetic and potential energies of the ass, wh = KE + PE. When a ass is lifted (or lowered) slowly ( KE = ) by a huan, the work done on the ass is equal to the change in the gravitational potential energy of the ass, wh = PEG. When a ass is lowered slowly the change in the gravitational potential energy of the ass (and the work done by the huan) will be negative, and will appear as an increase in the heat energy of the huan body. 96