That is one example of using the change in total energy ΔE with the changes in two particular kinds of energy (kinetic and potential). But why stop there? Let's generalize this formula for energy conservation to keep track of all different kinds of energy that we might encounter, such as those listed at the beginning of the chapter: |
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The different terms in the equation correspond to changes in kinetic, potential, radiant, nuclear, electrical, chemical, thermal, and pressure energy, respectively. When solving physics problems using the form of the law of conservation of energy in equation (3.7), many of the kinds of energy listed are unchanged—so you can set them to zero and drop them out of the equation.
As an example, if we want to talk about the conservation of energy for an electric circuit containing a battery and a lamp (that both lights up and radiates heat), we can use only those terms for changes in energy that are important for our closed system (electric, radiant, and thermal) and eliminate all the other changes of energy terms (because they are equal to zero): |
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A pendulum is an example of how kinetic energy and potential energy repeatedly increase and decrease, keeping total energy conserved at all times. At the top of the swing, the pendulum has its maximum gravitational potential energy—but no kinetic energy, because it briefly stops moving as it turns around its motion. At the bottom of the swing, the pendulum is moving at maximum speed and has maximum kinetic energy—but no potential energy, because it is at the lowest point in its swing. Potential and kinetic energy change in tandem during the motion of the pendulum, all the while keeping constant total energy.
In both mathematics and physics we express changes in a quantity by placing the Greek letter Δ (“Delta”) in front of the quantity, such as the change in kinetic energy ΔEk. If ΔEk has a positive value, such as ΔEk= +10 J, then this means that the value of Ek has increased. If ΔEk is negative, such as ΔEk= −3 J, then the value of Ek has decreased. When ΔEk=0 we know that Ek is unchanged or constant.
Many large dams have been built in the last century in order to harness the gravitational potential energy of the water in the reservoir to generate electrical energy. The basic idea of the hydroelectric dam is that the water in the reservoir turns a propeller-like machine called a turbine, converting the water's potential energy into mechanical energy. The turbine then turns a metal shaft that is connected to an electric generator, converting mechanical energy into electrical energy.
The modern piston engine was invented by German Nikolaus Otto in 1867. It is often called a “four-stroke” engine because of the four-step the physical process inside the machine. First is a downward intake stroke—where the piston moves to expand the internal cavity—where air and a small amount of gas enters the chamber. Second is an upward compression stroke, where the piston compresses the air-gas mixture. The third stroke starts with a spark plug igniting the gas, where the explosion causes expansion of the gas to push the piston downwards in a power stroke. And fourth is an upward exhaust stroke, in which the piston expels the spent fuel and air mixture out of the engine (eventually coming out the car's tailpipe). The piston engine has a complicated series of conversions of energy: starting with stored chemical energy of the gasoline; then pressure energy and heat when the gas ignites; and finally mechanical energy when the expanding gas pushes the piston. 
