Combining these two equations gives us this important result:
Or, alternatively,
As the kinetic energy of a system increases, its potential energy decreases by the same amount, and vice versa. As a result, the sum of the kinetic energy and the potential energy in a system is constant. We define this constant as E, the mechanical energy of the system:
This law, the conservation of mechanical energy, is one form of the more general law of conservation of energy, and it’s a handy tool for solving problems regarding projectiles, pulleys, springs, and inclined planes. However, mechanical energy is not conserved in problems involving frictional forces. When friction is involved, a good deal of the energy in the system is dissipated as heat and sound. The conservation of mechanical energy only applies to closed systems. Example 1
A student drops an object of mass 10 kg from a height of 5 m. What is the velocity of the object when it hits the ground? Assume, for the purpose of this question, that g = –10 m/s2.
Before the object is released, it has a certain amount of gravitational potential energy, but no kinetic energy. When it hits the ground, it has no gravitational potential energy, since h = 0, but it has a certain amount of kinetic energy. The mechanical energy, E, of the object remains constant, however. That means that the potential energy of the object before it is released is equal to the kinetic energy of the object when it hits the ground. When the object is dropped, it has a gravitational potential energy of:
By the time it hits the ground, all this potential energy will have been converted to kinetic energy. Now we just need to solve for v:
Example 2
Consider the above diagram of the trajectory of a thrown tomato:
1. At what point is the potential energy 美国GREatest?
2. At what point is the kinetic energy the least?
3. At what point is the kinetic energy 美国GREatest?
4. At what point is the kinetic energy decreasing and the potential energy increasing?
5. At what point are the kinetic energy and the potential energy equal to the values at position A?
The answer to question 1 is point B. At the top of the tomato’s trajectory, the tomato is the 美国GREatest distance above the ground and hence has the 美国GREatest potential energy. The answer to question 2 is point B. At the top of the tomato’s trajectory, the tomato has the smallest velocity, since the y-component of the velocity is zero, and hence the least kinetic energy. Additionally, since mechanical energy is conserved in projectile motion, we know that the point where the potential energy is the 美国GREatest corresponds to the point where the kinetic energy is smallest. The answer to question 3 is point E. At the bottom of its trajectory, the tomato has the 美国GREatest velocity and thus the 美国GREatest kinetic energy. The answer to question 4 is point A. At this point, the velocity is decreasing in magnitude and the tomato is getting higher in the air. Thus, the kinetic energy is decreasing and the potential energy is increasing. The answer to question 5 is point C. From our study of kinematics, we know that the speed of a projectile is equal at the same height in the projectile’s ascent and descent. Therefore, the tomato has the same kinetic energy at points A and C. Additionally, since the tomato has the same height at these points, its potential energy is the same at points A and C. Keep this example in mind when you take SAT II Physics, because it is likely that a similar question will appear on the test. Thermal Energy There are many cases where the energy in a system seems simply to have disappeared. Usually, this is because that energy has been turned into sound and heat. For instance, a coin sliding across a table slows down and comes to a halt, but in doing so, it produces the sound energy of the coin scraping along the table and the heat energy of friction. Rub your hands together briskly and you will feel that friction causes heat. We will discuss thermal energy, or heat, in 美国GREater detail in Chapter 9, but it’s worth noting here that it is the most common form of energy produced in energy transformations. It’s hard to think of an energy transformation where no heat is produced. Take these examples:Friction acts everywhere, and friction produces heat. Electric energy produces heat: a light bulb produces far more heat than it does light. When people talk about burning calories, they mean it quite literally: exercise is a way of converting food energy into heat. Sounds fade to silence because the sound energy is gradually converted into the heat of the vibrating air molecules. In other words, if you shout very loudly, you make the air around you warmer!英语作文
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