Concept Enhancer - Conservation of Mechanical Energy
CONCEPT ENHANCER - CONSERVATION OF MECHANICAL ENERGY
Conservation of energy is an expression that is commonly used, especially as it relates to stewardship of natural resources. The term conservation is concerned with conserving, protecting or keeping something the same. Conservation of energy implies keeping something the same during the course of an event. The event could be a student sliding down a water slide as illustrated in Figure 3.4 below. For example, does the student have as much energy near the bottom of the water slide as at the top? Energy is measured at the top, the event of water sliding takes place, and energy is measured near the bottom. The two energies are compared to determine if the initial energy equals the final energy, or has energy been lost or gained.
To consider conservation of mechanical energy, the different forms of energy need to be recognized. When you studied conservation of linear momentum in the last Unit, there was only one kind of momentum, which made the application of this principle comparatively easy. However, in conservation of energy there are different forms to consider. You have become acquainted with the forms of mechanical energy in this unit. These forms included gravitational potential energy, elastic potential energy, and kinetic energy. Energy is transferred when work is done on an object. Recall that work done on an object is defined as the product of a force applied through a displacement Fxd. In order for work done to be done, the directions of the applied force and the resulting displacement must be parallel to one another. In case the force varies, then the average force is used in the calculation. Friction, a force that acts on an object through a distance, typically does negative work on an object since it opposes the motion of the object. Thus, the negative work done by friction on an object needs to be considered in the conservation principle.
Summing Up Energies
Let’s go back to the student on the water slide as illustrated in Figure 3.4. Assuming she was sitting at rest at the top of the water slide, then all her energy is in the form of gravitational potential energy - energy due to her position. This gravitational potential energy may be calculated by the expression EG = mgh, where m represents the mass (in kg), g represents the acceleration of gravity (in m/s2), and h is the elevation (in m) relative to some reference point, which in this case is the bottom of the slide. The resulting unit is Joules (J), which happens to be the same unit for work done. The units being the same is not a coincidence. Recall that work being done on an object changes its energy.
Assume that the student has a mass of 50 kg (about 120 pounds) and is initially a height of 3.0 m above the lowest position of the slide. Her gravitational potential energy can be calculated by using the formula
EGinitial = mgh (50 kg)(9.8 m/s2)(3.0 m) = 1470 J
When the student has descended to a height of 1/3h, then she will have 2/3 the gravitational potential energy she had at the top of the slide as indicated in the calculation below.
EGfinal = mgh (50 kg)(9.8 m/s2)(2/3)(3.0 m) = 980 J
Where did the other 1/3 of the initial gravitational potential energy or 490 J go? Well, we know initially the student was not moving. But we know at her new position, she is now moving. We also know from experience that she is probably picking up speed as she goes down the slide. If she is moving and picking up speed as she is going down, she is gaining kinetic energy.
Kinetic energy, energy due to motion, has the formula EK = ½mv2, where m represents the mass (in kg) and v represents the speed (in m/s). The resulting unit for kinetic energy, like gravitational potential energy or any other form of energy, is Joules (J). You do not need to know the mass and the speed of the student in order to calculate her kinetic energy at any position. You could calculate her kinetic energy at any position by using her calculated gravitational potential energy at that position and by using the principle of conservation of mechanical energy. If mechanical energy is conserved and none of the energy is lost to the surroundings (as a result of the work done by friction or air resistance), the total mechanical energy of the student at any position on the slide must remain the same. The total mechanical energy (ET ) of the student is the sum of her gravitational potential energy and kinetic energy as indicated by
ET = EG + EK
Since she was not initially moving at the top of the water slide, her kinetic energy is zero and her total energy at the top of the slide is equal to her initial gravitational potential energy.
ET = EG + EK = 1470 J + 0 J = 1470 J
If her total mechanical energy remains the same at any position on the water slide, the student’s total energy at her new position at 2/3h remains 1470 J. Her kinetic energy at any position on the slide can be found by subtracting her gravitational potential energy from her total mechanical energy at that point. Thus, at 2/3h
EK = ET – EG = 1470 J – 980 J = 490 J
So if she descended to a height of 2/3h, her gravitational potential energy decreased by 1/3rd. In order for her total mechanical energy to remain constant, her kinetic energy must increase by this same amount.
By now knowing her kinetic energy at 2/3h, you could determine her speed by manipulating the formula, EK = ½ mv2. Her speed at this new height would be
v = (2EK/m)½ = [(2)(490 J)/(50 kg)] ½ = 4.4 m/s
What would the student’s EK be at the bottom of the slide? How fast would she be going at the lowest position on the slide? Since she is at the lowest position of the slide, her height would be zero. As a result, her gravitational potential energy at the bottom of the slide would also be zero.
Assuming that her total mechanical energy is conserved, her kinetic energy would be
EK = ET – EG = 1470 J – 0 J = 1470 J
And her speed at the bottom would be
v = (2EK /m)½ = [(2)(1470 J)/(50 kg)] ½ = 7.7 m/s
which is the student’s maximum speed as she slides down the slide.
The distribution of the student’s mechanical energy can be represented in a bar graph. Figure 3.5 illustrates the student’s ET, EG, and EK for the various positions discussed above.
The energy distributions illustrated in Figure 3.5 are for ideal situation in which no energy is lost to the surroundings. Notice that at all three positions, the distribution of the two forms of mechanical energy varies but the total mechanical energy remains the same. As the student slides down, she loses gravitational potential energy but gains the same amount of kinetic energy. As a result, the total mechanical energy is constant.
Identify the System
The total mechanical energy of a system, in this case water slide and a person, cannot change of itself. There is nothing the person can do to gain or lose her total amount of energy. It is possible for a force outside the system to change the amount of energy. Another person standing along side the water slide, could stick out his arm, exert a force, and hinder the motion of the person sliding down the slide. Therefore, an outside agent has done work on the water slider, taking energy from her and slowing her down. Looking just at the water slider, her energy as illustrated in Figure 3.5 appears to be conserved. But when we include the influence of the outside agent doing work on her against her motion and removing some of her energy, then it may appear that the total work and energy are not conserved. For example, if friction does work on an object sliding on a horizontal surface, the result is a loss of kinetic energy since eventually the object will come to stop. Where did the kinetic energy go? It did not change to gravitational potential energy since the object’s relative height did not change. As a result of the work done by the surface and the surrounding air on the object, eventually all the kinetic energy of the block was transferred to other forms of energy which may include thermal and sound energies.
In a more realistic situation, the student will lose some her initial mechanical energy in the form of thermal and sound energies as she slides down the water slide as a result of friction doing work on her. Since friction opposes her motion, the net result is that she will move down the slide and will not be able to obtain the maximum speed of 7.7 m/s calculated above. Recall that when work is done, energy is changed. Work, a force acting through a distance, can also be done in the same direction of the person sliding, and therefore energy is gained and she speeds up. Or the work can be done against the person, slowing her down. But the law of conservation of energy requires that nothing in the system itself can change the energy of that system. The change has to be done by an external agent.
Extending the Concept
Another factor to keep in mind is that energy exists in many forms. By mechanical energy we are limiting our discussion now to kinetic energy, gravitational potential energy, and elastic potential energy. But consider for a moment the operation of a light bulb. Electrical energy enters the lamp and it is changed to light energy. But does all the energy change to light energy? No, some of it changes to heat energy, which is usually not what we desire from the lamp. But in the bookkeeping of energies, the electrical energy will always be equal to the sum of the light energy and the heat energy. So we need to consider the entire system, and be aware of what forms of energy are present.
In the application activity, HOW HOT ARE YOUR HOT WHEELS, you will have the opportunity to apply the Principle of the Conservation of Mechanical Energy to a toy car going around a loop-the-loop. You will need to determine the gravitational potential energy required at the beginning of the event to have the car complete the loop. In addition, you will have to consider friction, which is part of the real world.
Period ______Name ______
CONCEPTUAL PRACTICE
1. Consider the skier in Figure 3.6 with a mass of 60.0 kg. The elevations at A, B and D are: hA = 120.0 m, hB = 80.0 m and hD = 40.0 m. The skier is at rest at position A and has a gravitational potential energy (EG) of 72,000 J. Use 10.0 m/s2 as the approximate value of g.
a.If the skier looses negligible energy to friction from A to B, what is the kinetic energy of the skier at B?
b.If the skier has expended 20,000.0 J to heat by friction by the time he reaches point D, what is his kinetic energy at point D?
c.On another run the skier expends only 12,000 J into heat by friction by the time he reaches C, the bottom of the run. How fast would the skier be traveling at C?
d.Suppose the skier starts from point A, goes across D and then up the hill toward F. By the time the skier has stopped he has transferred 28,000 J of energy to the snow in the form of heat and plowing snow. What elevation do you predict he will reach?
2.How does doubling the height that a barbell is lifted affect its gravitational potential energy?
3.How does doubling the velocity of a car affect its kinetic energy? How does tripling the velocity of a car affect its kinetic energy?
4.How does doubling the velocity of a car affect its stopping distance?
5.Describe the energy changes that take place in a pendulum when it is swinging from side to side, specifically the energies at the ends of the swing and the middle of the swing. How does the total energy change at each point along the path? How does work enter the energy picture?
6.Describe the energy changes on a mass hanging from a spring when the mass oscillates up and down. List the types of mechanical energy present, and show where each is a maximum and where each is zero.
7.When friction causes a moving object to slow down, is work being done on the object? Explain your reasoning.
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8.Consider the hunter, his bow and arrow to all be part of the same system. Describe the energy transfers and the conservation of energy within the system from the time the hunter draws the bow to the time the arrow is in flight.
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