Energy Lost From Vibrations

Friction has the effect of pinching the x-t graph of a vibrating object.

Until now, we have been making the relatively unrealistic assumption that a vibration would never die out. For a realistic mass on a spring, there will be friction, and the kinetic and potential energy of the vibrations will therefore be gradually converted into heat. Similarly, a guitar string will slowly convert its kinetic and potential energy into sound. In all cases, the effect is to "pinch" the sinusoidal x-t graph more and more with passing time. Friction is not necessarily bad in this context - a musical instrument that never got rid of any of its energy would be completely silent! The dissipation of the energy in a vibration is known as damping.

Self-Check	Most people who try to draw graphs like those shown above will tend to shrink their wiggles horizontally as well as vertically. Why is this wrong?
Answer	The horizontal axis is a time axis, and the period of the vibrations is independent of amplitude. Shrinking the amplitude does not make the cycles any faster.

In the graphs on the left, I have not shown any point at which the damped vibration finally stops completely. Is this realistic? Yes and no. If energy is being lost due to friction between two solid surfaces, then we expect the force of friction to be nearly independent of velocity. This constant friction force puts an upper limit on the total distance that the vibrating object can ever travel without replenishing its energy, since work equals force times distance, and the object must stop doing work when its energy is all converted into heat. (The friction force does reverse directions when the object turns around, but reversing the direction of the motion at the same time that we reverse the direction of the force makes it certain that the object is always doing positive work, not negative work.)

Damping due to a constant friction force is not the only possibility however, or even the most common one. A pendulum may be damped mainly by air friction, which is approximately proportional to v², while other systems may exhibit friction forces that are proportional to v. It turns out that friction proportional to v is the simplest case to analyze mathematically, and anyhow all the important physical insights can be gained by studying this case.

If the friction force is proportional to v, then as the vibrations die down, the frictional forces get weaker due to the lower speeds. The less energy is left in the system, the more miserly the system becomes with giving away any more energy. Under these conditions, the vibrations theoretically never die out completely, and mathematically, the loss of energy from the system is exponential: the system loses a fixed percentage of its energy per cycle. This is referred to as exponential decay.

A nonrigorous proof is as follows. The force of friction is proportional to v, and v is proportional to how far the objects travels in one cycle, so the frictional force is proportional to amplitude. The amount of work done by friction is proportional to the force and to the distance traveled, so the work done in one cycle is proportional to the square of the amplitude. Since both the work and the energy are proportional to A², the amount of energy taken away by friction in one cycle is a fixed percentage of the amount of energy the system has.

Self-Check	The figure shows an x-t graph for a strongly damped vibration, which loses half of its amplitude with every cycle. What fraction of the energy is lost in each cycle?
Answer	Energy is proportional to the square of amplitude, so its energy is four times smaller after every cycle. It loses three quarters of its energy with each cycle.

It is customary to describe the amount of damping with a quantity called the quality factor, Q, defined as the number of cycles required for the energy to fall off by a factor of 535. (The origin of this obscure numerical factor is e^2π, where e=2.71828... is the base of natural logarithms.) The terminology arises from the fact that friction is often considered a bad thing, so a mechanical device that can vibrate for many oscillations before it loses a significant fraction of its energy would be considered a high-quality device.

Exponential decay in a trumpet.

The decay of a musical sound is part of what gives it its character, and a good musical instrument should have the right Q, but the Q that is considered desirable is different for different instruments. A guitar is meant to keep on sounding for a long time after a string has been plucked, and might have a Q of 1000 or 10000. One of the reasons why a cheap synthesizer sounds so bad is that the sound suddenly cuts off after a key is released.

Q of a stereo speaker.

We will see later in the chapter that there are other reasons why a speaker should not have a high Q.