5.2 Conservation of Energy
Joey Wu and amaltese
5.2 Conservation of Energy
Learning Objectives
- Describe the law of conservation of energy and how it is applied to natural phenomena
- Construct, use, and present arguments to support the claim that when the kinetic energy of an object changes, energy is transferred to or from the object. (MS-PS3-5)
Law of Conservation of Energy
Energy, as we have noted, is conserved, making it one of the most important physical quantities in nature. The law of conservation of energy states that within a closed system, energy can change form, but the total amount of energy is constant. Another way of expressing the law of conservation of energy is to say that energy can neither be created nor destroyed. An important part of using the conservation of energy is selecting the system, meaning that energy is conserved only if the system is closed. In a closed system, objects may not enter or leave, and it is isolated from external forces so that no work can be done on the system.
In the analysis of the behavior of an object, you must make sure you have included everything in the system that is involved in the motion. For example, if you are considering a ball that is acted on by gravity, you must include the earth in your system. If considered by itself, one can tell that the kinetic energy of the ball is increasing as it falls, but only by including the earth in the system can you see that the increasing kinetic energy is balanced by an equivalent loss of potential energy. The sum of the kinetic energy and the potential energy of an object is often called the mechanical energy.
Mechanical Energy
Mechanical energy is kinetic energy and any potential energy related to the potential for an object to move. We deal with all other forms of energy by lumping them into a single group called other energy (OE). Then we can state the conservation of energy in equation form as
KEi + PEi + OEi = KEf + PEf + OEf
where subscript i stands for initial and subscript f stands for final.
Examples
Suppose a cannon is sitting on top of a 50.0 m high hill and a 5.00 kg cannon ball is fired with a velocity of 30.0 m/s at some unknown angle. What is the velocity of the cannon ball when it strikes the earth?
Solution:
Since the angle at which the cannon ball is fired is unknown, we cannot use the usual equations from projectile motion. However, at the moment the cannon ball is fired, it has a certain KE due to the mass of the ball and its speed and it has a certain PE due to its mass and it height above the earth. Those two quantities of energy can be calculated. When the ball returns to the earth, its PE will be zero. Therefore, its KE at that point must account for the total of its original KE+PE.
Etotal = KE + PE = 1/2 mv2 + mgh
= (1/2) (5.00kg) (30m/s)2 + (5.00kg)(9.8m/s2)(50m)
= 2250J + 2450J = 4700J
1/2mvf2 = 4700J, plug in m = 5kg, we can get
vf = 43.4 m/s
All types of energy and work can be included in this very general statement of conservation of energy. Kinetic energy is KE, mechanical potential energy is represented by PE, and all other energies are included as OE. This equation applies to all previous examples; in those situations OE was constant, and so it subtracted out and was not directly considered.
Explore the simulation below to see how a skate park applies the conservation of energy and its various properties.
MAKING CONNECTIONS: USEFULNESS OF THE ENERGY CONSERVATION PRINCIPLE
The fact that energy is conserved and has many forms makes it very important. You will find that energy is discussed in many contexts, because it is involved in all processes. It will also become apparent that many situations are best understood in terms of energy and that problems are often most easily conceptualized and solved by considering energy.
When does OE play a role? One example occurs when a person eats. Food is oxidized with the release of carbon dioxide, water, and energy. Some of this chemical energy is converted to kinetic energy when the person moves, to potential energy when the person changes altitude, and to thermal energy OE.
Some of the Many Forms of Energy
What are some other forms of energy? You can probably name a number of forms of energy not yet discussed. Electrical energy is a common form that is converted to many other forms and does work in a wide range of practical situations. Fuels, such as gasoline and food, carry chemical energy that can be transferred to a system through oxidation. Chemical fuel can also produce electrical energy, such as in batteries. Batteries can in turn produce light, which is a very pure form of energy. Most energy sources on Earth are in fact stored energy from the energy we receive from the Sun. We sometimes refer to this as radiant energy, or electromagnetic radiation, which includes visible light, infrared, and ultraviolet radiation. Nuclear energy comes from processes that convert measurable amounts of mass into energy. Nuclear energy is transformed into the energy of sunlight, into electrical energy in power plants, and into the energy of the heat transfer and blast in weapons. Atoms and molecules inside all objects are in random motion. This internal mechanical energy from the random motions is called thermal energy, because it is related to the temperature of the object. These and all other forms of energy can be converted into one another and can do work.
PROBLEM-SOLVING STRATEGIES FOR ENERGY
You will find the following problem-solving strategies useful whenever you deal with energy. The strategies help in organizing and reinforcing energy concepts. In fact, they are used in the examples presented in this chapter. The familiar general problem-solving strategies presented earlier—involving identifying physical principles, knowns, unknowns, checking units, and so on—continue to be relevant here.
Step 1. Determine the system of interest, identify what information is given and what quantity is to be calculated. A sketch will help.
Step 2. Examine all the forces involved and determine whether you know or are given the potential energy from the work done by the forces. Then use step 3 or step 4.
Step 3. If you know that you can apply conservation of mechanical energy simply in terms of potential and kinetic energy, the equation expressing conservation of energy is KEi+PEi=KEf+PEf.
Step 4. You have already identified the types of energy involved (in step 2). Before solving for the unknown, eliminate terms wherever possible to simplify the algebra. For example, choose h=0 at either the initial or the final point, so the gravitational potential energy (PEg) is zero there. Then solve for the unknown in the customary manner.
Step 5. Check the answer to see if it is reasonable. Once you have solved a problem, reexamine the forms of work and energy to see if you have set up the conservation of energy equation correctly. For example, work done against friction should be negative, potential energy at the bottom of a hill should be less than that at the top, and so on. Also check to see that the numerical value obtained is reasonable. For example, the final speed of a skateboarder who coasts down a 3-m-high ramp could reasonably be 20 km/h, but not 80 km/h.
Transformation of Energy
The transformation of energy from one form into others is happening all the time. The chemical energy in food is converted into thermal energy through metabolism; light energy is converted into chemical energy through photosynthesis. In a larger example, the chemical energy contained in coal is converted into thermal energy as it burns to turn water into steam in a boiler. This thermal energy in the steam in turn is converted to mechanical energy as it spins a turbine, which is connected to a generator to produce electrical energy. (In all of these examples, not all of the initial energy is converted into the forms mentioned.)
Figure 4.2.1 A solar powered aircraft flying in the air. Solar cells are on the upper surface of the wings, where they are exposed to sunlight.
Another example of energy conversion occurs in a solar cell. Sunlight impinging on a solar cell (see Figure 4.2.1) produces electricity, which in turn can be used to run an electric motor. Energy is converted from the primary source of solar energy into electrical energy and then into mechanical energy.
EXAMPLE APPLYING THE LAW OF CONSERVATION OF ENERGY
Exercise
Strategy
Choose the Equation.
𝐾𝐸1+𝑃𝐸1=𝐾𝐸2+𝑃𝐸2
𝐾𝐸 = 1/2𝑚𝐯2; 𝑃𝐸 = 𝑚𝐠h
1/2𝑚𝐯12+𝑚𝐠ℎ1=1/2𝑚𝐯22+𝑚𝐠ℎ2
List the known variables.
M = 10kg, v1 = 0, g = 9.80m/s2
h1 = 20m, h2 = 10m
Identify the unknown variables: KE2 and PE2
Plug in the numbers.
½*10*0+10*9.8*20=½*10*𝐯22+10*9.8*10
v2 = 14m/s
Tips for Success
Note that we can solve many problems involving conversion between KE and PE without knowing the mass of the object in question. This is because kinetic and potential energy are both proportional to the mass of the object. In a situation where ΔKE = ΔPE, we know that mg*Δh = (1/2)mv12 – (1/2)mv02.
Dividing both sides by m and rearranging, we have the relationship: 2g*Δh = v12– v02
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Information gathered and edited from:
https://openstax.org/books/physics/pages/9-introduction
https://openstax.org/books/physics/pages/9-1-work-power-and-the-work-energy-theorem
https://www.ck12.org/book/cbse_physics_book_class_xi/section/5.14/
Media Attributions
- SOLAR AIRCRAFT
