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CHAPTER 6 – ENERGY Because of the nature of the questions asked by the class, it will not be necessary to cover this chapter in its entirety. Rather, selected sections will be presented briefly. Energy is one of the four quantities (along with space, time and matter) that make up the “physical world”. Space, time and matter are defined in terms of fundamental properties: length (in meters), time (in seconds) and mass (in kilograms). Energy is a derived quantity, whose units are Joules 1 Joule = 1 kg-m2/s2 = 1 (kg-m/s2)m = 1 N-m The last representation above suggests that energy can be associated with a force which acts over a certain distance or displacement. (Think of pushing a crate across a floor 50 m to the right by applying a force to the right on the crate.) Doing this might make you feel physically tired, i.e. like you had done a lot of “work”! So, in physics, work is defined as the product of force (in the direction of motion) times the distance covered (i.e. the magnitude of the displacement), or Work W = Fd (in N-m). Since the units here are the same as for energy (above), it follows that “work” and “energy” are essentially the same. Specifically, we say that energy is the ability (or the capacity) to do work. Note: Energy and Work are scalars, i.e. they have no direction! In everyday terms, we sometimes talk about many different kinds of energy: mechanical, thermal, chemical, electrical, electromagnetic, nuclear. (See Table 6.1, p.244.) In reality, however, there are only two distinct kinds of energy: 1. Kinetic energy – energy due to motion; and 2. Potential energy – energy due to position or shape Furthermore, the total energy within any system remains constant. (This is called the Law of Conservation of Energy). This means that every physical process can be viewed as a series of exchanges of energy. When there is more than one body in the system, energy lost by one (or more) of the bodies must be gained by the remaining bodies. Even when there only a single body, an increase (or decrease) in that body’s potential energy must be accounted for by a loss (or gain) in its kinetic energy, and vice versa. Kinetic Energy As we saw early in the course, the kinetic energy possessed by a moving body is given the formula 1 2 Kinetic Energy (KE) K 2 mv (kg-m2/s2). (Note that an object at rest will have KE = 0 J.) Because energy is a scalar, the v here represents the magnitude of the velocity vector, i.e. the speed. (Don’t confuse KE with linear momentum p = mv, which also depends on mass and velocity, but which is a vector!) Potential Energy Potential energy (energy due to position or shape) actually comes in a number of types: local gravitational potential energy, universal gravitational potential energy, spring potential energy, electrostatic potential energy, etc. The only one we will discuss in this course is the simplest of these, namely local gravitational potential energy. Unlike kinetic energy, which is zero when an object is at rest, potential energy can be assumed to be zero at any location we choose. What matters is the amount by which potential energy changes as the object moves. Such a change can be either positive or negative, depending on whether the body gains or loses potential energy. Since we’ve already seen that work and energy have the same units, it makes sense that the amount of local gravitational potential energy gained when an object’s height increases is the work done against gravity to lift it. Likewise, the amount of local gravitational potential energy lost when an object descends is the work done by gravity as it drops. Since we know that the local force of gravity is just the weight mg, this work is given by the formula W = Fd = mgd . So, we can define the local gravitational potential energy (relative to any location where we want the energy to be zero) by Local Gravitational Potential Energy (PE) = mgy . Note that, above the zero energy height, y (and hence PE) will be positive, while below the zero energy height, y and PE will be negative. For instance, suppose we choose PE to be zero on the surface of a desk which is 1 m high. A 1 kg book would then have: PE = 1(9.8)0 = 0 J on the desk (where y = 0 m) PE = 1(9.8)(+2) = +19.6 J 2 m above the desk PE = 1(9.8)(–1) = –9.8 J on the floor (1 m below) Example A 3 kg stone is released from rest at the top of a 20 m cliff. As it falls, its velocity and its displacement can be found using our earlier equations v f vi at 0 9.8t [1] 1 2 2 y vi t at 0 4.9t [2] . 2 Find how long it takes for the stone to reach the bottom. Assuming the local gravitational potential energy (PE) is zero at the top of the cliff (i.e. at the moment of release), find the KE, PE and the sum of the two at the top, at t = 1.0 s, at t = 1.5 s and at the bottom. Solution At the bottom, y 20 m, so from [2] 20 2 t 2.02 s 20 4.9t , i.e. 4.9 At the top, t = 0 s and y 0 m, so: 1 2 v f 9.80 0 m/s, i.e. KE = 30 0 J. 2 PE = mgy 39.80 0 J. It follows that KE + PE = 0 J. At 1.0 s: v f 9.81.0 9.8 m/s, i.e. KE = 1 3 9.82 144 J. 2 From [2], y 4.91.0 4.9 m, so PE = mgy 39.8 4.9 144 J. 2 It follows that KE + PE = 0 J. At 1.5 s: 1 2 3 14 . 7 324 J. v f 9.81.5 14.7 m/s, i.e. KE = 2 From [2], y 4.91.5 11.03 m, so PE = mgy 39.8 11.03 324 J. 2 It follows that KE + PE = 0 J. At the bottom, t 2.02 s and y 20 m, so: 1 2 v f 9.82.02 19.8 m/s, i.e. KE = 3 19.8 588 J. 2 PE = mgy 39.8 20 588 J. It follows that KE + PE = 0 J . Location/Time Top (0 s) 1.0 s 1.5 s Bottom (2.02 s) Summary KE PE 0J 0J 144 J –144 J 324 J –324 J 588 J –588 J Total 0J 0J 0J 0J This example clearly illustrates the principle of conservation of energy: as the rock falls, it accelerates and gains kinetic energy, but at the same time it loses exactly the same amount of potential energy, thus keeping the total energy (0 J, in this particular case) constant throughout! Recall that other forms of energy (thermal, chemical, electrical, nuclear, etc), when considered on a microscopic scale, are really explained in terms of kinetic and potential energy. See Table 6.1 on page 244. For example, everyone knows that adding heat to substance (solid, liquid or gas) raises its temperature. In fact, temperature is essentially a measure of the average kinetic energy of the molecules. Think of gas in a closed container. Its molecules move very rapidly, continually colliding with the walls of the container (and sometimes colliding with each other). What happens to the container if you heat the gas? If you cool the gas? What would happen theoretically if the temperature were lowered to “absolute zero” (0 Kelvin)? Now that we have gained a basic understanding of kinetic and potential energy and conservation of total energy, we are in a position to answer a couple of your questions.