Chapter 1 – Excerpt 6
This is the last excerpt from Chapter 1 and also the shortest one.
Another quantity dependent on mass and velocity is ½ m|v|2, which is called kinetic energy. Note that it is a nonnegative scalar quantity.
Kinetic energy is just one form of the most familiar and fundamental physical quantity, namely energy. Although the concept of energy is a commonplace one, it is rather difficult to define. Essentially, energy is the capacity to effect motion with respect to a force.11
Don’t despair if you don’t fully grasp the concept of energy from this definition, any more than we despair that we haven’t provided you with a better definition. About a century ago, the famous French mathematician Henri Poincaré (who turned to writing popular science when he got too old to do mathematics, and was so good at it that he was eulogized as “a kind of bard of science”) stated “we cannot give a general definition of energy”. According to some modern textbooks:”there is no completely satisfactory definition of energy”12; "we really don’t know what energy is but we know the many forms it takes"12.1; "a kind of ‘quick-change artist’ with a whole trunkful of disguises"12.2. And our favorite Nobel Laureate, Richard Feynman, also a brilliant explicator, went further: “It is important to realize that in physics today, we have no knowledge of what energy is”13.
The SI unit of energy is the joule, which is named after James P. Joule and abbreviated J. A joule is a newton-meter. Thus, one joule is the amount of energy it takes to lift a small apple (about .1 kg) up one meter from the ground. On the other hand, the food energy your person obtains by eating such an apple is 50 Calories, or about 200 kilojoules, since one Calorie is exactly 4184 joules by definition. Unlike Newton, Joule did not have any cats and did have a wife. But that is just as well, since he was infamous for meeting with Lord Kelvin to conduct one of his famous experiments while he was a newlywed on his honeymoon, so one can only imagine how he would have neglected a pet cat.
Here we will just focus on the forms of energy relevant to classical mechanics, i.e. mechanical energy, which takes two forms – kinetic energy and potential energy. Kinetic energy and potential energy are often obscurely symbolized by T and U, respectively, but we will instead use the more obvious K.E. and P.E.
Kinetic energy is due to motion. From the above definition of K.E. = ½ mv2, it is clear that only moving objects have kinetic energy, and the faster they’re moving, the more kinetic energy they have.
In contrast, potential energy is due to position. It is stored energy that could be released by a change of position. For example, a compressed or stretched spring or elastic string has a potential energy that depends on how far it is from the relaxed position. The elastic potential energy of a spring either compressed or expanded a distance x is P.E. = ½ kx2, where k is the spring constant (i.e. the spring’s force is F = -kx). Gravitational potential energy is possessed by elevated objects with respect to a lower position, for they have the potential to move to that lower position. For an object elevated at height h, the gravitational potential energy is P.E. = weight×height = mgh14. Potential energy is illustrated in Figures 1-12, 1-13 and 1-14.
Kinetic energy and gravitational potential energy are clearly interchangeable. Consider an object at rest at height h. As it falls, its motion is described by Equations (2) with the initial speed v0 = 0 and the initial height z0 = h:
K.E. = ½ mv2 = ½ m(-gt)2 = ½ mg2t2
and its potential energy is
P.E. = mg(h – ½gt2) = mgh – ½ mg2t2
So its total mechanical energy is
M.E. = K.E. + P.E. = ½ mg2t2 + (mgh – ½ mg2t2) = mgh
which is independent of the time t. Thus, as it falls, all its potential energy is gradually transformed into kinetic energy, with the total always being constant, equal to the initial potential energy it had when it had no kinetic energy because it was still.
Energy considerations also give us a very easy way to calculate the speed of the object when it hits the ground, without knowing what the time t is then. At that point, all the energy is kinetic, so we just solve ½ mv2 = mgh to get v = (2gh)½. Note that although we didn’t need to know the landing time to obtain this result, we can now use it to easily determine that time: t= (2h/g)½. Not only did we avoid calculus here, but we didn’t even have to solve a non-trivial quadratic equation, as we did when analyzing motion in the first section of this chapter. This is, in a sense, the opposite approach. Energy considerations tend to lead to simpler formulations of physical laws.
It is not just the two forms of mechanical energy that are interchangeable. More generally, all types of energy can be interchanged. For example, when the falling object just discussed reaches the ground and stops moving, it has neither kinetic energy nor gravitational potential energy. But its energy was transferred to the spot of ground it hit, which heated up.
Energy takes many forms: heat, light, chemical, mechanical, electrical, radiant, atomic, nuclear, etc. As Eugene Hecht said, “we bake apple pies with thermal energy and defend them with nuclear energy“15. The transformations between the different forms of energy account for all the physical phenomena that we observe, such as current, combustion, and color changes, as well as motion.
Although energy can be transformed between its different types, energy cannot be either created or destroyed – the total amount of energy in the universe is a constant. This is the greatest of the conservation laws in physics, and one of the greatest of all generalizations in physics. It is known as the Principle of the Conservation of Energy. At various times throughout history, it has appeared that it was being violated, i.e. that energy was being created or destroyed, but further investigation of the suspect phenomena instead led to the discovery of new forms of energy, and occasionally even to new laws of physics.
Unlike virtually all of the other concepts in this chapter, the idea of energy as a physical concept was not only not due to either Newton or his cat, but it was not even known by either of them.
There are many additional concepts of linear classical mechanics, such as elasticity, impulse and power, as well as work, which have been omitted from this chapter in the interests of saving space and patience. If you wish to pursue them, we suggest you start by pouncing on a mouse connected to the Internet.
11 – The laborious nature of this definition is due to the circumventing of the definition of the physical concept of work – thus we are doing work to avoid “work” here! (Technically, energy is the capacity to do work, where work is defined as force x distance, for the motion through a distance of an object due to a force, or W = ∫Fdx.) Linguistically, the root of “energy” is the Greek ergon, which means work. <return-to-text>
12 – Hecht, Eugene, Physics: Calculus, page 222, Brooks/Cole ©2000. <return-to-text>
12.1 – Kirkpatrick, Larry and Gerald Wheeler, Physics: A World View, third edition, page 150, Saunders College Publishing ©1998. <return-to-text>
12.2 – March, Robert, Physics for Poets, fourth edition, page 56, McGraw-Hill ©1996. The quoted phrase replaces the following more poetic one from the first edition: “some mysterious, chameleonlike entity that appears in a variety of guises”.<return-to-text>
13 – Feynman, Leighton & Sands, The Feynman Lectures on Physics. Vol. I, page 4-2. Addison-Wesley ©1963. <return-to-text>
14 – The elastic potential energy is the amount of work it takes to compress or expand a spring that distance, and the gravitational potential energy is the amount of work it takes to lift the object to that height. <return-to-text>
15 – Hecht, Eugene, Physics: Calculus, page 223, Brooks/Cole ©2000. <return-to-text>
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