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# Oscillations

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If you want to find the hidden secrets of the universe, you must think in terms of energy, frequency, and vibration. This famous quote from Tesla cannot be closer to the truth. An oscillation is a periodic motion that can be repeated in a cycle, such as a wave. Because of de Broglie's hypothesis, we learned that all matter has properties of particles and waves. Oscillations are one of the most important phenomena in physics, as they are to describe the nature of particles in quantum mechanics. They are also important for understanding how society works in the 21st century. All electronic devices, the internet, TV signals, communication systems, and medical imaging are all using electromagnetic waves. Now that we know how important oscillations are, let's learn more about them and their properties.

## Defining Oscillations

Oscillatory motion is a movement that repeats itself. So, an oscillation is a back-and-forth motion about an equilibrium position. An equilibrium position is a location where the net force acting on the system is zero. A vibrating string of a guitar is an example of an oscillation.

A guitar string oscillates, JAR (CC BY 2.0)

## Period and Frequency of Oscillations

The frequency is defined as the inverse of the period. For example, a large period implies a small frequency.

$$f=\frac1T$$

Where $$f$$ is the frequency in hertz, $$\mathrm{Hz}$$, and $$T$$ is the period in seconds, $$\mathrm{s}$$.

The period is the time required to complete one oscillation cycle. The period of an oscillation cycle is related to the angular frequency of the object's motion. The expression for the angular frequency will depend on the type of object that is oscillating. The equation that relates the angular frequency denoted by $$\omega$$ to the frequency denoted by $$f$$ is

$$\omega=2\pi f.$$

Substituting $$\dfrac{1}{f}$$ for $$T$$ and rearranging for $$T$$ we obtain

$$T=\frac{2\pi}\omega.$$

Where $$\omega$$ is the angular frequency in radians per second, $$\frac{\mathrm{rad}}{\mathrm s}$$. If we think about it, this expression makes sense, as an object with a large angular frequency will take a lot less to make one complete oscillation cycle.

## Harmonic Oscillators

A harmonic oscillation is a type of oscillation in which the net force acting on the system is a restoring force. A restoring force is a force acting against the displacement in order to try and bring the system back to equilibrium. An example of this is Hooke's Law given by

$$F_s=ma_x=-k\Delta x,$$

where $$m$$ is the mass of the object at the end of the spring in kilograms, $$\mathrm{kg}$$, $$a_x$$ is the acceleration of the object on the $$\text{x-axis}$$ in meters per second squared, $$\frac{\mathrm m}{\mathrm s^2}$$, $$k$$ is the spring constant that measures the stiffness of the spring in newtons per meter, $$\frac{\mathrm{N}}{\mathrm m}$$, and $$\Delta x$$ is the displacement in meters, $$\mathrm{m}$$.

If this is the only force acting on the system, the system is called a simple harmonic oscillator. This is one of the most simple cases, as the name suggests.

Most oscillations occur in the air or other mediums, where there is some type of force proportional to the system's velocity, such as air resistance or friction forces. These may act as damping forces. The equation for the damping force is

$$F_{damping}=-cv,$$

where $$c$$ is a damping constant in kilograms per second, $$\frac{\mathrm{kg}}{\mathrm s}$$, and $$v$$ is the velocity in meters per second, $$\frac{\mathrm{m}}{\mathrm s}$$.

As a consequence, part of the system's energy is dissipated in overcoming this damping force, so the amplitude of the oscillation will start to decrease as it reaches zero. These types of harmonic oscillators are called damped oscillators. We can write Newton's Second Law for the case where there is a restoring force and a damping force acting on the system,

$$ma=-cv-kx.$$

Writing the above expression as a differential equation, we obtain

$$m\frac{\operatorname d^2x}{\operatorname dt^2}+c\frac{\operatorname dx}{\operatorname dt}+kx=0.$$

The solution to the above equation is an exponential function. The damping term will exponentially dissipate the oscillations until the system decays to rest.

$x=A_0e^{-\gamma t}\cos\left(wt+\phi\right),$ where $$\gamma=\frac c{2m}$$

$$x=A_0e^{-\frac c{2m}t}\cos\left(wt+\phi\right)$$

We can prove this is a solution by differentiating it and substituting it into the differential equation:

$$\begin{array}{rcl}\frac{\operatorname dx}{\operatorname dt}&=&-A_0\omega e^{-\frac c{2m}t}\sin(\omega t+\phi)\;-A_0\frac c{2m}e^{-\frac c{2m}t}\cos(\omega t+\phi)\\\frac{\mathrm d^2x}{\mathrm dt^2}&=&\begin{array}{c}-A_0\omega^2e^{-\frac c{2m}t}\cos(\omega t+\phi)\;+A_0\omega\frac cme^{-\frac c{2m}t}\sin(\omega t+\phi)\;+A_0\frac{c^2}{4m^2}e^{-\frac c{2m}t}\cos(\omega t+\phi)\end{array}\end{array}.$$

Now we can go back to the differential equation and prove that we found a solution for it.

$$m\frac{\operatorname d^2x}{\operatorname dt^2}+c\frac{\operatorname dx}{\operatorname dt}+kx=0$$

$$\begin{array}{rcl}\frac{A_0c^2e^{\displaystyle\frac{-bt}{2m}}\cos\left(\omega t+\phi\right)}{4m}+\cancel{A_0c\omega e^\frac{-bt}{2m}\sin\left(\omega t+\phi\right)}\;-A_0\omega^2me^\frac{-bt}{2m}\cos\left(\omega t+\phi\right)\;-\frac{A_0c^2e^{\displaystyle\frac{-bt}{2m}}\cos\left(\omega t+\phi\right)}{2m}&-\cancel{A_0c\omega e^\frac{-bt}{2m}\sin\left(\omega t+\phi\right)}+A_0ke^\frac{-bt}{2m}\cos\left(\omega t+\phi\right)=&0\end{array}$$

$$\begin{array}{rcl}-\frac{\cancel{A_0}c^2\cancel{e^{\displaystyle\frac{-bt}{2m}}\cos\left(\omega t+\phi\right)}}{4m}-\cancel{A_0}\omega^2m\cancel{e^\frac{-bt}{2m}\cos\left(\omega t+\phi\right)}\;+\;\cancel{A_0}k\cancel{e^\frac{-bt}{2m}\cos\left(\omega t+\phi\right)}&=&0\end{array}$$

$$-\frac{c^2}{4m^2}-\omega^2+\frac km=0$$

$$\omega=\sqrt{\frac km-\frac{c^2}{4m^2}}.$$

The damped oscillators with oscillations and an amplitude that decreases with time are called underdamped oscillators. While the ones that do not oscillate and immediately decay to equilibrium position are called overdamped oscillators. The boundary limit between under-damping and over-damping is called critical damping. To confirm the damped oscillator is undergoing critical damping we verify that the damping coefficient $$\gamma$$ is equal to the system's natural angular frequency $$\omega_0$$. The damping coefficient $$\gamma$$ can be determined with the following equation:

$$\gamma=\frac c{2m},$$

where $$c$$ is a damping constant measured in units of kilograms per second, $$\frac{\mathrm{kg}}{\mathrm s}$$, and $$m$$ is the system's mass in kilograms, $$\mathrm{m}$$.

The angular frequency for the damped oscillator can be defined in terms of the damping coefficient and the natural angular frequency.

$$\begin{array}{rcl}\omega&=&\sqrt{\frac km-\frac{c^2}{2m}}\\\omega&=&\sqrt{\omega_0-\gamma}\end{array}$$

These 3 cases can be summarized as follows:

• Underdamping: $$\omega_0>\gamma$$
• Critical damping: $$\omega_0=\gamma$$
• Overdamping: $$\omega_0<\gamma$$

There is also another type of oscillator called forced oscillators. In these, the oscillations are caused by an external force that is a periodic force. If the frequency of this force is equal to the system's natural frequency, this causes a peak in the amplitude of oscillation. The natural frequency is the frequency at which an object will oscillate when it is displaced out of equilibrium.

## Oscillations in a Spring-Mass System

We will consider the simplest case of Simple Harmonic Motion to understand oscillations in a spring-mass system. For a spring, we already know the equation for Newton's second law:

$$F_s=ma_x=-k\Delta x.$$

Rearranging for the acceleration we obtain

$$a_x=-\frac km\Delta x.$$

So, comparing the equation for a spring with the general equation for harmonic motion $$a=-\omega_0^2x$$, we can derive the angular frequency $$\omega$$ for a spring, which is given by the equation

$$\omega_0^2=\frac km,$$

expressed more explicitly as

$$\omega_0=\sqrt{\frac km}.$$

Where $$m$$ is the mass of the object at the end of the spring in kilograms, $$\mathrm{kg}$$, and $$k$$ is the spring constant that measures the stiffness of the spring in newtons per meter, $$\frac{\mathrm N}{\mathrm m}$$.

The formula for the time period of an oscillating spring-mass system is

$$T_s=2\pi\sqrt{\frac mk}.$$

What is the period of oscillation for a spring-mass system with a mass of $$4\;\mathrm{kg}$$ and a spring constant of $$1\;{\textstyle\frac{\mathrm N}{\mathrm m}}$$?

$$T_s=2\pi\sqrt{\frac{4\;\mathrm{kg}}{1\;{\displaystyle\frac{\mathrm N}{\mathrm m}}}}$$

$$T_s=2\pi\sqrt{\frac{4\;\mathrm{kg}}{1\;{\displaystyle\frac{\frac{m\;kg}{s^2}}m}}}$$

$$T_s=4\pi\;\mathrm s$$

## Graphing oscillations

If we plot the displacement as a function of time for an object undergoing simple harmonic motion, we would identify the period as the time between two consecutive peaks or any two analogous points on two waves with the same phase. To locate the amplitude, we look at the highest peak in distance.

Displacement vs Time for a system in simple harmonic motion. From this graph, we can identify the amplitude and period of oscillation, Yapparina, Wikimedia Commons (CC0 1.0).

We can also graph the displacement as a function of time for damped oscillators, to visually understand and compare their characteristics. Critical damping provides the quickest way for the amplitude to reach zero. Overdamping takes you faster to the zero position, but decaying oscillations still occur. Underdamped oscillations take more time to reach an amplitude of zero.

## Oscillations - Key takeaways

• An oscillation is a back-and-forth motion about an equilibrium position. An equilibrium position is a location where the net force acting on the system is zero.
• A harmonic oscillation is a type of oscillation where the net force acting on the system is a restoring force. A restoring force is a force acting against the displacement in order to try and bring the system back to equilibrium.
• The period is the time required to complete one oscillation cycle. The frequency is defined as the reciprocal of period, $$f=\frac1T$$.
• If the restoring force is the only force acting on the system, the system is called a simple harmonic oscillator. A damping force may also act on an oscillating system. It is some type of force proportional to the system's velocity, such as air resistance or friction forces, $$F_{damping}=-cv$$.
• For damped oscillators, part of the system's energy is dissipated in overcoming the damping force, so the amplitude of the oscillation will start to decrease as it reaches zero. The damped oscillators with oscillations and an amplitude that decreases with time are called underdamped oscillators. The overdamped oscillators are the ones that do not oscillate and immediately decay to the equilibrium position.

• The boundary limit between an underdamped and overdamped oscillator is called critical damping. To confirm the damped oscillator is undergoing critical damping we verify that the damping coefficient $$\gamma=\frac c{2m}$$ is equal to the system's angular frequency $$\omega=2\pi f$$. These threecases can be summarized as follows:

• Underdamping: $$\omega_0>\gamma$$
• Critical damping: $$\omega_0=\gamma$$
• Overdamping: $$\omega_0<\gamma$$
• In Forced oscillators, the oscillations are caused by an external force that is a periodic force. If the frequency of this force is equal to the system's natural frequency this causes a peak in the amplitude of oscillation.

## Frequently Asked Questions about Oscillations

The period is the time taken for one oscillation cycle. The period for Simple Harmonic Motion is related to the angular frequency of the object's motion. The expression for the angular frequency will depend on the type of object that is undergoing the Simple Harmonic Motion.

An oscillator is an object that moves back and forth about an equilibrium position.

An oscillation is back and forth movement about an equilibrium position.

To find the frequency we first need to get the period of the cycle. To do so we find the time it takes to complete one oscillation cycle. This can be done by looking at the time between two consecutive peaks or any two analogous points. After we find the period, we take its inverse to determine the frequency.

To find amplitude we look for the peak values of distance.

## Final Oscillations Quiz

Question

If the only force acting on an oscillating system is a restoring force that varies linearly with displacement from the equilibrium position, we have:

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Answer

A damped oscillator.

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Question

A damping force is proportional to a system's:

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Answer

Acceleration.

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In damped oscillators:

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Answer

The amplitude starts to increase with time but then suddenly goes to zero.

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The oscillators that do not oscillate and immediately decay to equilibrium position are called:

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Answer

Underdamped oscillators.

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Question

The damped oscillators with oscillations and an amplitude that decreases with time slowly are called:

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Answer

Underdamped oscillators.

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Question

To confirm the damped oscillator is undergoing critical damping we verify that the damping coefficient $$\gamma$$:

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Answer

Is equal to the system's angular frequency.

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Question

If the damping coefficient $$\gamma$$ is greater than the system's angular frequency $$\omega$$:

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Answer

We have an underdamped oscillator.

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If the damping coefficient $$\gamma$$ is smaller than the system's angular frequency $$\omega_0$$:

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We have a critically damped oscillator.

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Oscillators in which the oscillations are caused by an external force that is a periodic force are called:

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Forced oscillators.

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The greater the mass of a system:

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The greater the period of oscillation.

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Question

... takes more time for the amplitude to reach zero.

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Overdamping.

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... provides the quickest way for the amplitude to reach zero

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Answer

Critical damping.

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Question

Which of the following are harmonic oscillators?

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Simple harmonic oscillator.

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Which of the following is an example of a restoring force?

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Gravitational Law.

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Question

A spring with a large spring coefficient will yield:

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Answer

A small period of oscillation.

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