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Resonance

Written by tutor Jonathan B.

Resonance is the tendency of something to oscillate at a dramatically greater amplitude than other frequencies. This occurs because the oscillating thing is able to easily change energy from two or more forms. In other words, objects tend to oscillate at certain frequencies determined by the ease of energy transfer. Feedback, the screeching sound one can get by holding a microphone close to a speaker, can have a similar effect, but is not resonance in and of itself.

Why Do We Care?

Why should one care about resonance? Growing up, we've never had to take care to create or avoid resonance. Most people don't talk about resonance much, so it doesn't seem to be a concern of most people. Clearly, we can live quite well knowing anything about it, just as people have done for thousands of years. Despite this, knowledge of resonance can be the one thing that stands between people having a normal day and people experiencing a catastrophic disaster resulting in tens (or even hundreds or thousands) of casualties and hundreds of thousands (or even millions if not billions) of dollars lost in a lawsuit. There's no need to be afraid, though. This kind of disaster only happens if an engineer designing a crucial structure neglects to take resonance into account or a certain change is made in a design without checking the resonance (or if something is used in a way or placed in an environment it was not designed for in such a manner that resonance can destroy it, such as shattering a wine glass with one's voice).

Common Examples of Resonance

  • A child on a swing
  • A spring oscillating back and forth (parallel or perpendicular to the length of the spring)
  • A stringed instrument (such as a guitar or violin)

Resonance of a Pendulum

The resonant frequency of a simple pendulum is simply the frequency at which it will naturally swing. A simple pendulum is essentially a mass that can be approximated as a particle attached to a lightweight string. This frequency is given by the equation,

frequency equation for pendulums

where f is frequency, g is acceleration due to gravity (9.8 m/s2), and L is the length of the pendulum. Notice that mass is not in this equation. Mass has no effect on the frequency of a simple pendulum. Mass will have an effect on energy loss through wind resistance and friction, but we can ignore that in order to get a very good estimate without using much more complex math (this applies to other things as well). Notice also that there is nothing in the equation about a starting point for the pendulum. A pendulum will swing at the same frequency regardless of where you initially release the pendulum.

Example:
A simple pendulum has a length of 3 meters with a mass of 100 grams on the end.

Solution:
The only number we need to get from the problem here is the length of the pendulum (the mass is irrelevant when using a simple pendulum). So after plugging the numbers in, we would get,

Solving pendulum resonance equation

which is 0.288 Hz.

Resonance of a Spring

The resonance frequency of a spring with a mass attached to one end, unlike a simple pendulum, DOES depend on the mass. While gravity accelerated the pendulum at an equal rate regardless of the mass, the only driving force in an oscillating spring is the spring itself (in a horizontal position, but it's only part of the force in a vertical position). Due to Newton's Second Law, the acceleration of a mass is dependent upon the force and mass of the mass. The equation for the angular frequency of a mass on a spring is,

Equation for angular frequency of mass on a spring

where ω is the angular frequency (in rad/s), k is the spring constant, and m is the mass. To convert the angular frequency to linear frequency (in Hertz), we can simply divide by 2π, thus giving us,

Solved Equation for Frequency of Mass on a Spring

as the frequency of a spring with a mass attached to one end.

Example:
A spring oscillates at 3 Hz with a spring constant of 50 N/m. What is the mass of the mass on the end of the spring?

Solution:
Before we can use the formula, we need to solve for m. Solving for m gives us,

Putting the numbers in gives us,

which is 0.141 kilograms, or 141 grams.

Resonance of a Pipe

The resonant frequencies of a pipe are fairly straightforward to compute. However, there are different conditions and multiple frequencies that make it slightly more difficult. For a pipe open at both ends, the resonant frequencies are given by the formula,

Resonance Frequencies for a Pipe

where fn is the nth resonant frequency, n is, well, n (basically any positive integer), v is the speed of sound in air (about 343 m/s), and L is the length of the pipe. For a pipe open at only one end and closed at the other, the resonant frequencies are given by the formula,

with all of the same variables as the previous formula with the exception of n being only positive odd integers. The only differences between the pipe with both ends open and the pipe with only one end open are the coefficient in front of the L and the possible values of n.

Another way of saying the nth resonant frequency of a pipe is to say the nth harmonic. For example, the 1st resonant frequency is the 1st harmonic, and the 4th resonant frequency is the 4th harmonic. This applies to both a pipe with both ends open and a pipe with only one end open, with the only difference being that there are no even harmonics for the pipe with only one end open.

Example:
A pipe with both ends open has a length of 50 meters. What is the 5th harmonic of the pipe?

Solution:
Using the formula for a pipe with both ends open and putting the numbers in, we get,

which is 17.15 Hz.

Resonance Practice Quiz

A simple pendulum will swing at a different frequency depending on the starting point from which it is released.

A. True
B. False
The correct answer here would be B.

Which factor(s) affect the frequency of an oscillating spring?

A. only the mass of an attached mass
B. only the length of the spring
C. the spring constant and mass of an attached mass
D. the spring constant and the length of the spring
The correct answer here would be C.

Which pipe will only produce harmonics of an odd integer (eg 1st harmonic, 3rd harmonic, 5th harmonic, and so on)?

A. a pipe with both ends open
B. a pipe with one end open, and one end closed
C. a pipe with both ends closed
The correct answer here would be B.
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