Energy Dissipation

Conservation of Energy

To understand energy dissipation, we will first need to understand the law of conservation of energy.

Okay, so if energy cannot be created or destroyed, how can it dissipate? We will answer that question in more detail a little further down the road, but for now, remember that although energy cannot be created or destroyed, it can be converted into various forms. It is during the conversion of energy from one form to another that energy can become dissipated.

Physical Interactions

Energy dissipation helps us to understand more about physical interactions. By applying the concept of energy dissipation, we can better predict how systems will move and act. But, to fully comprehend this, we will first need to have some background on energy and work.

A single-object system can only have kinetic energy; this makes perfect sense because energy is usually the result of interactions between objects. For example, potential energy can result from the interaction between an object and the earth's gravitational force. In addition, work done on a system is often the result of the interaction between the system and some outside force. Kinetic energy, however, only relies on the mass and velocity of an object or system; it does not require interaction between two or more objects. Therefore, a single-object system will always only have kinetic energy.

A system involving the interaction between conservative forces can have both kinetic and potential energy. As referred to in the example above, potential energy can result from the interaction between an object and the earth's gravitational force. The force of gravity is conservative; therefore, it can be the catalyst for allowing potential energy to enter a system.

Mechanical Energy

Mechanical energy is kinetic energy plus potential energy, leading us to its definition.

Seeing as how mechanical energy is the sum of an object's kinetic and potential energy, its formula would look something like this:

$$E_\text{mec} = KE + U\mathrm{.}$$

Work

Work is energy transferred into or out of a system due to an outside force. Conservation of energy requires that any change to a type of energy within a system must be balanced by an equivalent change of other types of energies within the system or by a transfer of energy between the system and its surroundings.

Fig. 2 - When the athlete picks up and swings the hammer, work is done on the hammer-earth system. Once the hammer is released, all that work is gone. The kinetic energy must balance out the potential energy until the hammer hits the ground.

For example, take the hammer toss. For now, we will only focus on the hammer's motion in the vertical direction and ignore air resistance. While the hammer sits on the ground, it has no energy. However, if I perform work on the hammer-earth system and pick it up, I give it potential energy that it did not have before. This change to the system's energy has to be balanced out. While holding it, the potential energy balances the work I did on it when I picked it up. Once I swing and then throw the hammer, however, all the work I was doing disappears.

This is a problem. The work that I was doing on the hammer is no longer balancing out the hammer's potential energy. As it falls, the vertical component of the hammer's velocity increases in magnitude; this causes it to have kinetic energy, with a corresponding decrease in potential energy as it approaches zero. Now, everything is okay because the kinetic energy caused an equivalent change for the potential energy. Then, once the hammer hits the ground, everything returns to how it was initially, as there is no further energy change in the hammer-earth system.

If we had included the motion of the hammer in the horizontal direction, as well as air resistance, we would need to make the distinction that the horizontal component of the hammer's velocity would decrease as the hammer flies because the frictional force of air resistance would slow the hammer down. Air resistance acts as a net external force on the system, so mechanical energy is not conserved, and some energy is dissipated. This energy dissipation is directly due to the decrease in the horizontal component of the hammer's velocity, which causes a change in the hammer's kinetic energy. This kinetic energy change directly results from air resistance acting on the system and dissipating energy from it.

Definition of Dissipated Energy

We have been talking about the conservation of energy for a long time now. Okay, I admit there was a lot of setup, but now it's time to address what this article is all about: energy dissipation.

A typical example of energy dissipation is energy lost to frictional forces.

For instance, let's say Sally is about to go down a slide. At first, all her energy is potential. Then, as she goes down the slide, her energy is transferred from potential to kinetic energy. However, the slide is not frictionless, which means that some of her potential energy turns into thermal energy due to friction. Sally will never get this thermal energy back. Therefore, we call that energy dissipated.

We can calculate this "lost" energy by subtracting Sally's final kinetic energy from her initial potential energy:

$$\text{Energy Dissipated}=PE-KE.$$

The result of that difference will give us how much energy was converted to heat due to the non-conservative frictional force acting on Sally.

Energy Dissipators Types

As we saw above, the resultant dissipated energy was due directly to a non-conservative force acting on Sally.

All energy dissipators work by utilizing non-conservative forces to do work on the system. Friction is a perfect example of a non-conservative force and an energy dissipator. The friction from the slide did work on Sally which caused some of her mechanical energy (Sally's potential and kinetic energy) to transfer to thermal energy; this meant that the mechanical energy was not perfectly conserved. Therefore, to increase the dissipated energy of a system, we can increase the work done by a non-conservative force on that system.

Other typical examples of energy dissipators include:

• Fluid friction such as air resistance and water resistance.
• Damping forces in simple harmonic oscillators.
• Circuit elements (we will talk in more detail about damping forces and circuit elements later on) such as wires, conductors, capacitors, and resistors.

Energy Dissipated by Damping Force

Now, we'll talk expand upon another kind of energy dissipator: damping.

Similar to friction's effect on a system, a damping force applied to an oscillating object can cause energy to dissipate. For example, damped springs in the suspension of a car allow it to absorb the shock of the car bouncing as it drives. Normally, the energy due to simple harmonic oscillators will look something like Fig. 4 below, and with no outside force such as friction, this pattern would continue forever.

Fig. 3 - The total energy in a spring oscillates between storing all of it in kinetic energy and all of it in potential energy.

However, when there is damping in the spring, the above pattern will not go on forever because with each new rise and fall, some of the spring's energy will be dissipated due to the damping force. As time goes on the total energy of the system will decrease, and eventually, all of the energy will be dissipated from the system. The motion of a spring affected by damping would therefore look like this.

Examples of damping include:

• Viscous drag, such as air drag on a spring or the drag due to a liquid one places the spring into.
• Resistance in electronic oscillators.
• Suspension, such as in a bike or a car.

Damping should not be confused with friction. While friction can be a cause of damping, damping applies solely to an influence's effect to slow or prevent the oscillations of a simple harmonic oscillator. For example, a spring with its lateral side to the ground would experience a frictional force as it oscillates back and forth. Fig. 5 shows a spring moving to the left. As the spring slides along the ground, it feels the force of friction opposing its movement, directed right. In this case, the force \(F_\text{f}\) is both a frictional and damping force.

Fig. 4 - In some cases, friction can act as a damping force on a spring.

Therefore, it is possible to have simultaneous friction and damping forces, but that does not always imply their equivalence. The force of damping only applies when a force exerts to oppose the oscillatory motion of a simple harmonic oscillator. If the spring itself was old, and its components hardened, this would cause the reduction of its oscillatory motion and those old components could be considered causes of damping, but not friction.

Energy Dissipated in Capacitor

There is no one general formula for energy dissipation because energy can be dissipated differently according to the system's situation.

In the realm of electricity and magnetism and circuits, energy is stored and dissipated in capacitors. Capacitors act as energy stores in a circuit. Once they charge up completely, they act as resistors because they don't want to accept any more charges. The formula for energy dissipation in a capacitor is:

$$Q=I^2X_\text{c} = \frac{V^2}{X_\text{c}},\\$$

where \(Q\) is the charge, \(I\) is the current, \(X_\text{c}\) is the reactance, and \(V\) is the voltage.

We can understand how energy dissipates inside a capacitor through careful analysis of the above equation. Capacitors are not meant to dissipate energy; their purpose is to store it. However, capacitors and other components of a circuit in our non-ideal universe are not perfect. For example, the above equation shows that lost charge \(Q\) equals the voltage in the capacitor squared \(V^2\) divided by the reactance \(X_\text{c}\). Thus, the reactance, or a circuit's tendency to oppose a change in the current, causes some of the voltage to drain from the circuit, resulting in energy dissipated, usually as heat.

You can think of the reactance as the resistance of a circuit. Note that replacing the reactance term for resistance yields the equation

$$\text{Energy Dissipated} = \frac{V^2}{R}.$$

This is equivalent to the formula for power

$$P=\frac{V^2}{R}.$$

The above connection is enlightening because power equals the rate at which energy changes with respect to time. Thus, the energy dissipated in a capacitor is due to the energy change in the capacitor over a certain time interval.

Energy Dissipation Example

Let's do a calculation about energy dissipation with Sally on the slide as an example.