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Energy Redux

Potential Energy -> Kinetic Energy
The image above shows the planet Jupiter as it looks in the infrared. The bright white spot near the bottom left of the planet is due to the impact of a piece of comet Shoemaker-Levy in July of 1994. Jupiter's powerful gravitational field provided enormous potential energy for the comet piece. As it hit the atmosphere, this potential energy was converted to kinetic energy equivalent to thousands of conventional nuclear explosions. The kinetic energy appears as a hot plume of gas rising thousands of kilometers above the atmosphere. Click on the image to see a time-lapse movie of the plume. Click here to get more information, pictures, and animations on the the Great Comet Crash.

Now that momentum has had its share of the spotlight, we need to backtrack a little to look at a property of the universe which is every bit as fundamental as momentum conservation, although more difficult to observe directly. The new universal property we will study is Conservation of Energy. We have already seen that Newton observed (as did others before him) that the quantity mv² is conserved in collisions between hard objects. We can see directly how this kinetic energy term comes in by turning back to our old familiar friends, the 4 equations for motion. For example, suppose you throw a ball into the air. If we ignore air friction, then the equations for motion tell us that the ball will rise for a distance, stop, then fall straight down until it hits our hand again. To determine how far the ball goes up, we turn to an equation for motion, like v² = v² _0 - 2g(y - y_0) where y is the maximum height, y_0 is the point at which the ball left our hand, v_0 is the initial velocity of the ball as it leaves our hand, and we want v, the final velocity, to be zero. Thus, we find

If we want to know what the velocity of the ball is when it returns to our hand after reaching its maximum height, we start from the maximum, where the ball velocity is initially zero, and get

Now, we note that -g(y_0 - y) = g(y - y_0) which is just one-half times the original velocity as the ball left our hand. Therefore, the ball returns with the same velocity. This makes us think that there is something conserved about the motion since apparently the ball carried with it something that allowed it to recover all the motion it lost going up against gravity. The ``something'' in this case is called energy. The energy the ball starts off with is kinetic, i.e. associated with its motion. The energy when its at the top of its path and ready to start falling back we term potential energy. This potential energy is associated with the ball's position in the earth's gravitational field.

Balls (and everything else) always fall from high to low positions, so its natural to think that a ball's height determines how much potential energy it has. The simplest relationship with height we can think of is mg times height. The mg represents the force due to gravity. We include it since the amount of energy stored should depend on the force accelerating the object and this is determined by mg. Hence, a relationship for kinetic energy and potential energy which is consistent with the equations for motion is:

where all we have done is to take our earlier expression and multiply both sides by the mass, m. This formula includes both kinds of energy, kinetic and potential. When the ball is in flight, we think of the kinetic energy as being turned into potential energy (the ball rises to higher and higher values of y as it slows down) and after it reaches its maximum height, the ball falls, converting potential energy to kinetic as it does so (the value of y decreases, so potential is decreasing as kinetic is increasing). At any point along the ball's flight some of its energy appears as kinetic and some as potential. At the end points, the energy is all potential (top of its flight) or all kinetic (beginning of the throw upward and end of the flight when the ball returns to our hand). At any given time, the total energy of the ball is termed

where K, as we have already seen, is the short-hand for kinetic energy, U is the short-hand for potential energy, and E stands for the total of the two. Since K and U trade off during the ball's flight, E is constant throughout. In other words, total energy is conserved!

Forces, like gravity, which maintain a constant balance between kinetic and potential energy are termed conservative forces. All of the fundamental forces in nature are conservative. Complicated forces like friction and air resistance are NOT conservative. These complicated forces transfer some of the total energy into heat. The heat escapes from the system and does not appear as motion or change in position.

The plot you produced in exercise 4-1 is called an energy diagram. It shows how the energy in the system is maintained between kinetic and potential as a function of time or, more usually, position. Textbooks will often present energy diagrams as a way of understanding the dynamics of a physical system. These diagrams are particularly powerful now that you know about slopes and their connection to derivatives because, it turns out, there is a direct connection between the potential energy of a particle subjected to a conservative force and the force itself. We will not prove it, but we can take it as a definition of potential energy that, if the potential energy as a function of position is called U(x) and the force as a function of position is called F(x), then

hence the energy diagram tells us not only potential energy, but kinetic energy (assuming we know the total energy and can use energy conservation) and the force as a function of position (from the slope of the potential energy curve). That's a lot of informational value for one plot!

Springs produce conservative forces. You'll remember that the formula for the force due to a spring is F = -kx where k is the spring constant. The potential energy associated with the spring is U_{spring} = ½ kx², (confirm this in Maple by finding the derivative of U with respect to x and don't forget the minus sign!) provided the mass at the end of the spring is moving in the horizontal plane only. The spring stores its potential energy in the molecular bonds of its structure whenever the spring is compressed or stretched by distance x from its equilibrium position.

Do exercises 4-2 and 4-3 to learn more about the character of the spring's potential energy and the use of energy diagrams.

There are further deep connections between the motion of objects on springs and physical concepts like conservation of energy, but to see them, we need first to broaden our perspective on motion.