7.6 The Nonisolated System—Conservation of Energy
We have seen examples in which an object, modeled as a particle, is acted on by various forces, resulting in a change in its kinetic energy. This very simple situation is the first example of the nonisolated system—a common scenario in physics problems. Physical problems for which this scenario is appropriate involve systems that interact with or are influenced by their environment, causing some kind of change in the system. If a system does not interact with its environment it is an isolated system, which we will study in Chapter 8.
The work–kinetic energy theorem is our first example of an energy equation appropriate for a nonisolated system. In the case of the work–kinetic energy theorem, the interaction is the work done by the external force, and the quantity in the system that changes is the kinetic energy.
We have seen only one way to transfer energy into a system so far—work. We mention
below a few other ways to transfer energy into or out of a system. The details of these processes will be studied in other sections of the book. We illustrate these in Figure 7.17 and summarize them as follows:
Work, as we have learned in this chapter, is a method of transferring energy to a system by applying a force to the system and causing a displacement of the point of application of the force.
Mechanical waves (Chapters 16–18) are a means of transferring energy by allowing a disturbance to propagate through air or another medium.
Heat (Chapter 20) is a mechanism of energy transfer that is driven by a temperature difference between two regions in space.
Matter transfer (Chapter 20) involves situations in which matter physically crosses the boundary of a system, carrying energy with it.
Figure 7.17 Energy transfer mechanisms. (a) Energy is transferred to the block by
work; (b) energy leaves the radio from the speaker by mechanical waves; (c) energy transfers
up the handle of the spoon by heat; (d) energy enters the automobile gas tank by
matter transfer; (e) energy enters the hair dryer by electrical transmission; and (f) energy
leaves the light bulb by electromagnetic radiation.
Electrical Transmission (Chapters 27–28) involves energy transfer by means of electric currents. This is how energy transfers into your hair dryer , stereo system, or any other electrical device.
Electromagnetic radiation (Chapter 34) refers to electromagnetic waves such as light, microwaves, radio waves, etc. Examples of this method of transfer include cooking a baked potato in your microwave oven and light energy traveling from the Sun to the Earth through space.
One of the central features of the energy approach is the notion that we can neither create nor destroy energy—energy is always conserved. Thus, if the total amount of energy in a system changes, it can only be due to the fact that energy has crossed the boundary of the system by a transfer mechanism such as one of the methods listed above. This is a general statement of the principle of conservation of energy. We can describe this idea mathematically as follows:
7.7 Situations Involving Kinetic Friction
A book sliding to the right on a horizontal surface slows down in the presence of a force of kinetic friction acting to the left. The initial velocity of the book is vi , and its final velocity is vf . The normal force and the gravitational force are not included in the diagram because they are perpendicular to the direction of motion and therefore do not influence the book’s speed.
The work–kinetic energy theorem is valid for a particle or an object that can be modeled as a particle. For these kinds of situations, Newton’s second law is stil valid for the system, even though the work–kinetic energy theorem is not.
For particle under constant acceleration, we know that the following relationship
where vi is the speed at t = 0 and vf is the speed at time t .
For a nonisolated system, we can equate the change in the total energy stored in the system to the sum of all the transfers of energy across the system boundary. For an isolated system, the total energy is constant—this is a statement of conservation of energy. If a friction force acts, the kinetic energy of the system is reduced and the appropriateequation to be applied is
The conclusion of this discussion is that the result of a friction force is to transform kinetic energy into internal energy, and the increase in internal energy is equal to the decrease in kinetic energy.
Quick Quis 7.11
You are traveling along a freeway at 65 mi/h. Your car has kinetic energy. You suddenly skid to a stop because of congestion in traffic. Where is the kinetic energy that your car once had?
(a) All of it is in internal energy in the road.
(b) All of it is in internal energy in the tires.
(c) Some of it has transformed to internal energy and some of it transferred away by mechanical waves.
(d) All of it is transferred away from your car by various mechanisms.
Answer : (c). The brakes and the roadway are warmer, so their internal energy has increased. In addition, the sound of the skid represents transfer of energy away by mechanical waves.
Power is the time rate of energy transfer or valid for any means of energy transfer. If an external force is applied to an object (which we assume acts as a particle), and if the work done by this force in the time interval ∆t is W, then the average power during this interval is defined as
In a manner similar to the way we approached the definition of velocity and acceleration,we define the instantaneous power P as the limiting value of the average power as ∆t = 0
The SI unit of power is joules per second ( J/s), also called the watt (W)(after James Watt): 1 W = 1 J/s = 1 kg m2s3 . A unit of power in the U.S. customary system is the horsepower (hp):
1 hp = 746 W
A unit of energy (or work) can now be defined in terms of the unit of power. One kilowatt-hour (kWh) is the energy transferred in 1 h at the constant rate of 1 kW =1 000 J/s. The amount of energy represented by 1 kWh is
1 kWH = (103W) (3600s) = 3.6 x 106 J
Quick quis 7.12
An older model car accelerates from rest to speed v in 10 seconds. A newer, more powerful sports car accelerates from rest to 2v in the same time period. What is the ratio of the power of the newer car to that of the older car?
(a) 0.25 (b) 0.5 (c) 1 (d) 2 (e) 4
Answer : (e). Because the speed is doubled, the kinetic energy is four times as large. This kinetic energy was attained for the newer car in the same time interval as the smaller kinetic energy for the older car, so the power is four times as large.
7.8 Energy and Automobile
Automobiles powered by gasoline engines are very inefficient machines. About 67% of the energy available from the fuel is lost in the engine. Approximately 10% of the available energy is lost to friction in the transmission, drive shaft, wheel and axle bearings, and differential. Friction in other moving parts transforms approximately 6% of the energy to internal energy, and 4% of the energy is used to operate fuel and oil pumps and such accessories as power steering and air conditioning. This leaves a mere 13% of the available energy to propel the automobile.
For large objects, the resistive force fa associated with air friction is proportional to the square of the speedFa = 1/2
where D is the drag coefficient, / is the density of air, and A is the cross-sectional area of the moving object.