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Our plan is to come up with a mathematical representation of a frequency modulated oscillations of carrier waves that incorporate given sound waves.
Sound waves are changes in the air pressure and can be represented by some function of time m(t) (m stands for modulation, as we want this signal to modulate the frequency of sinusoidal carrier signal).
Practically, m(t) is the changes in the current of some circuit that contains a microphone, that converts the air pressure changes into synchronous changes in the electric current.
Let's discuss the concept of angular frequency of the carrier signal in more mathematical terms.
Frequency f, if constant, is the number of periods of oscillations per unit of time (second).
It's often expressed in radians per second ω=2πf and unmodulated oscillations of an electric current in the carrier LC circuit that produces base oscillations would be I(t) = A·cos(ω·t)
This sinusoidal function of time I(t) that represents oscillations of electric current can be viewed as an X-coordinate of a point on a circle of radius A (amplitude of oscillations) with a center at the origin of coordinates, that rotates along a circle counterclockwise with constant angular speedω radians per second, assuming that at t=0 it is located on the X-axis at point (A,0).
So, angular frequency in more mechanical terms is angular speed.
From now on we will consider the above presented rotation of a point as mathematical representation of sinusoidal oscillations.
Another important characteristic of an oscillation is its phase φ(t).
The definition of a phase is an angle a rotating point has rotated to during the time of rotation from its start up to a current moment t, which can be expressed as a time-dependent function φ(t).
From this definition immediately follows the analogy between kinematics terms distance, speed and concepts phase, angular frequency used in radio electronics.
This analogy is complete in a sense that the relationship between a phase φ(t) and angular frequency ω(t) is similar to that between a distance S(t) covered by a moving object and its instantaneous speed V(t).
The instantaneous speed at moment in time t in, generally, a non-uniform movement, as a function of time, is a derivative of a distance covered by a moving object from the start of movement up to a position at time t, as a function of time: V(t) = dS/dt = S'(t)
Similarly, angular frequency (sometimes called instantaneous angular frequency or simply instantaneous frequency) is the first derivative of a phase (angle of rotation) as a function of time: ω(t) = dφ/dt = φ'(t)
Knowing speed V(t) of a moving object at every moment of time from start to t, we can restore the distance S(t) covered by this object as a function of time S(t) = ∫[0,t]V(τ)·dτ
Similarly, we can restore the phase φ(t) (that is, an angle a point has rotated from the start of its rotation to a moment in time t), knowing the instantaneous angular frequency ω(t) at each moment of time. φ(t) = ∫[0,t]ω(τ)·dτ
Consider the main equation of oscillations of an electric current in the LC circuit of a carrier without any sound modulation I(t) = A·cos(ω0·t)
The argument to a function cos() is a product of a constant angular frequency (speed) ω0 by time, which is an angular distance of rotation or, using terminology introduced above, a phase of the rotation at time t φ(t) = ω0·t
Therefore, our representation of carrier signal can be expressed in a more general form, suitable even for non-uniform rotation: I(t) = A·cos(φ(t))
In a non-uniform rotation with variable instantaneous angular frequency ω(t) we can always derive this frequency from the phase: ω(t) = φ'(t)
If we want to combine the carrier signal I(t) with some frequency modulating signal m(t) in such a way that the resulting variable instantaneous frequencyωmod(t) of a modulated signal reflected the modulation, we need to satisfy the following equation: ωmod(t) = ω0 + m(t)
where ω0 is the carrier own unmodulated constant frequency determined by it main LC circuit, m(t) is a modulating add-on to reflect the sound waves to be transmitted.
We can even vary the degree by which the modulating affects the output signal by adding a modulating index λ as a factor to a modulator m(t): ωmod(t) = ω0 + λ·m(t)
Knowing the target instantaneous frequencyωmod(t) and the above expression of a phase in terms of this frequency φ(t) = ∫[0,t]ω(τ)·dτ
we can express the modulated phase as φmod(t) = ∫[0,t]ωmod(τ)·dτ
This modulated phase will be an argument to a modulated signal of a carrier Imod(t) = A·cos[φmod(t)] =
= A·cos[∫[0,t]ωmod(τ)·dτ] =
= A·cos[∫[0,t](ω0+λ·m(τ))·dτ] =
= A·cos[ω0·t+λ·∫[0,t]m(τ)·dτ]
The role of modulation index λ in this formula is to define how significantly base carrier frequency should change with a change in sound waves frequency.
Below is a picture of a rather complicated air pressure oscillations (red) and the resulting modulated signal (purple) that represents the carrier frequency modulation by this sound together with equations that represent all the components of this frequency modulation
(you can click the right mouse button and open this picture in another tab for better view)
Unfortunately, amplitude modulation (AM), with its carrier frequency in the range from 540kHz to 1,600kHz, does not fit to transmit good quality (Hi-Fi) sound, especially on higher pitch notes, as was explained in the lecture "Amplitude Modulation" of this chapter.
The maximum audio frequency that is practical to transmit through AM radio is about 4.4kHz, while the sensitivity of the human ear goes up to 20kHz.
In addition, this type of modulation is too sensitive to radio noise.
The necessity to accommodate high quality transmission of sound waves caused new ideas and designs solutions.
A very useful invention was frequency modulation (FM).
First of all, frequency modulation is implemented within a carrier frequency range from 88mHz to 108mHz, which by itself helps to transmit a higher pitch audio signal of up to 15kHz.
Secondly, the basic principle of FM transmission is to represent sound waves, which are just changes in the air pressure, by a carrier signal's deviations from the base frequency assigned to it. Higher air pressure is represented by higher frequency of the carrier signal, lower air pressure is represented by a lower carrier frequency.
Amplitude of the carrier signal frequency remains always the same, only its frequency is fluctuating in synch with changes of the air pressure that carries a sound, as converted by a microphone into electric current.
All the deviations from the base frequency of the carrier signal must be within certain range assigned for each base frequency, so different radio stations, having different base frequencies of signal transmission, do not step over each other.
In the US the FM transmission uses the frequencies from 88mHz to 108mHz divided into 100 channels of 0.2mHz wide. That means, the frequency deviation from the base frequency of each FM transmitter should not exceed 0.1mHz up or down.
A picture below shows how the frequency of the carrier's signal (blue line) is changing in response to the most simple sound waves that represent a single note of a constant intensity (red line). Mathematically, the red line is a graph of an air pressure at some point in space as a function of time - a sinusoid for a single note of constant intensity.
(you can click the right mouse button and open this picture in another tab for better view)
What remains to discuss is a physical implementation of the frequency modulation.
This implementation involves changing the frequency of the transmitting oscillations of an electric current in the LC circuit connected to an antenna in synch with sound waves input from a microphone.
This can be accomplished by adding a variable capacitor into a circuit that is controlled by a current that represents the sound waves on the output of a microphone.
Changing the capacitance C in the LC circuit will effectively change the angular frequency ω=1/√L·C.
There are many different ways to put together such a device, we will leave this to special courses on radio electronics.
Finally, as an example of a more practical case of complicated sound waves that result in superposition of many different sources of sound, here is a picture of the sound waves (red) and modulated carrier signal (purple) with its frequency in synch with the sound waves.
(you can click the right mouse button and open this picture in another tab for better view)
As you see, higher intensity of sound waves (higher pressure of air) is represented by more frequent oscillations of the carrier signal and lower intensity of sound waves is represented by lower frequency of the carrier signal.
This picture intentionally presents the carrier signal of, generally, lower range of frequencies in order to visualize the individual oscillations.
In practice that range of oscillations of the carrier signal should be significantly higher to better represent each curve of sound waves.
Let's approach amplitude modulation (AM) from a mathematical standpoint.
High frequency oscillations of the electric current in the LC circuit with an inductor of inductance L and a capacitor of capacitance C can be described by a function I(t) = Ac·cos(ωc·t)
where I(t) is the circuit's own oscillation of the current, Ac is the amplitude of these oscillations, ωc = 1/√L·C is an angular frequency of these oscillations.
Simple air pressure oscillations resulting from a sound can be expressed by a similar function
P(t) = As·cos(ωs·t)
where P(t) is the oscillation of the air pressure around a source of sound, As is the amplitude of these air pressure oscillations, ωs is an angular frequency of these oscillations.
Note that our design requires ωc to be substantially greater than ωs.
Also note that real sound is a combination (superposition) of different overtones with different amplitudes and different angular frequencies and phases, like P(t) = As1·cos(ωs1·(t+φs1)) +
+ As2·cos(ωs2·(t+φs2)) +
+ As3·cos(ωs3·(t+φs3)) + ...
Amplitude modulation alters the amplitude of the carrier's oscillations by changing it in synch with sound oscillations.
The simplest way to achieve it mathematically is to incorporate the sound waves into an amplitude of carrier's waves: Im(t) = [Ac+P(t)]·cos(ωc·t)
Here is an example of this type of modulation of a carrier's signal.
Assume, the carrier's high frequency oscillations have an amplitude Ac=4 and angular frequency ωc=20, which can be described by an equation I(t) = 4·cos(20t)
(you can click the right mouse button and open this picture in another tab for better view)
The sound makes air pressure oscillations that combine two different tones, one with an amplitude As1=2 and angular frequency ωs1=1 and another with an amplitude As2=1 and angular frequency ωs2=2, which can be described by the following equation P(t) = 2·cos(t) + cos(2t)
Then the modulated signal can be described by an equation Im(t) =
= [4+2·cos(t)+cos(2t)]·cos(20t)
that graphically looks like this:
Just as a demonstration of the importance of having a high carrier frequency of a signal to properly represent a sound, here is what the transmitted signal would look like if the LC circuit of a carrier has a frequency comparable to a frequency of sound.
Assume, the carrier's oscillations have an amplitude Ac=4 and angular frequency ωc=3, which can be described by an equation I(t) = 4·cos(3t)
The sound waves are assumed to be as above P(t) = 2·cos(t) + cos(2t)
Then the modulated signal can be described by an equation Im(t) =
= [4+2·cos(t)+cos(2t)]·cos(3t)
The following graph represents both the sound wave (purple) and modulated signal (blue):
As you see, the representation of sound waves by a modulated signal is far from exact. This lower frequency of amplitude modulation cannot be used for transmitting sound.
Now we have some basic understanding of how a radio signals (oscillations of electromagnetic field) are transmitted and received.
But, by themselves, these oscillations do not carry any useful information, which can be voice, image, data or any other type of information.
The oscillations of an electromagnetic field are only carriers of the information and their frequency represents the carrying frequency of the radio communication.
Transmitting and receiving information using the carrying radio waves requires certain additional design that is called modulation.
This lecture is about basic principles of amplitude modulation (AM) to transmit sound waves (oscillations of the air or other medium).
Electromagnetic oscillations are synchronous waves of electric and magnetic field forces perpendicular to each other (see "Field Waves" of this topic "Waves") that can be represented graphically as follows
(you can click the right mouse button and open this picture in another tab for better view)
Since these two kinds of forces act synchronously (in phase), we will represent the intensity of the electromagnetic field as a simple two-dimensional sinusoidal graph
At this point we'd like to state that the frequency of the carrier waves represented above should be relatively high as compared to frequency of the sound waves to accomplish successful sound transmission. The reason for this will be obvious after we explain what amplitude modulation is.
Our first task is to convert sound into oscillations of electric current. This is accomplished by a microphone or any other sound capturing device.
Notice that the oscillations of the electric current must represent the sound in its frequency and amplitude. Moreover, real sound is usually a combination of different frequencies of oscillation of air or other medium with different amplitudes, and the corresponding oscillations of electric current should reflect all this multitude of different parameters.
Considering we have accomplished producing an electric current oscillations in some circuit that reflects the sound. Here is a graphical representation of these oscillations:
The next task is to transmit these oscillations.
Experiments show that these oscillations are of too low frequency and not well propagated using any kind of an antenna. Its the high frequency oscillations that needed to do it with some sort of success.
Here comes the amplitude modulation. This method is using the high frequency of the carrier waves produced by an electronic RLC oscillator (red on a picture below), superimposing on its oscillations the changes in amplitude in synch with oscillations of the sound frequency (green on a picture below).
The result is the carrier oscillations modulated by amplitude by sound waves (blue on a picture below):
It should be obvious now why the carrier frequency of radio waves oscillations must be significantly higher than the sound waves frequency. If this difference is not significant, it would be impossible to distinguish the signal of the carrier from oscillations of sound waves.
Sound waves frequencies are those in the range from 20Hz to 20,000Hz (1Hz is 1 period of oscillations per second, which is, actually, sec−1), while the range of carrier frequencies for amplitude modulation (AM range) is from 540,000Hz to 1,600,000Hz (that is from 540kHz to 1,600kHz). Such a difference in ranges assures that the sound waves reflected in the amplitude of carrier waves are distinguishable from the carrier waves themselves.
For example, WCBS radio station in New York broadcasts at the carrier's frequency f=880kHz=0.88·106sec−1
At the same time, the sound frequency of High A note is 440Hz. That means that for each period of this sound wave the carrier makes 2,000 oscillations.
The higher the sound's pitch (which is, higher frequency of sound waves) - the less difference there is between its frequency and the frequency of the carrier radio waves. With the highest frequency that human ear can hear, which is about 20,000Hz, the WCBS radio waves with frequency of 880kHz are only 440 times more frequent. Generally, it's still sufficient to select the sound waves from the carrier's but some small details of sound waves might be lost. That's why AM radio cannot carry really high fidelity sound, but for regular speech it works fine.
Incidentally, considering the speed of propagation of radio waves is, approximately, c=0.3·109m/sec,
the WCBS radio waves have the length of λ = c/f ≅ 341 m
These relatively long waves allow AM radio signals to go around large objects and even be reflected from the ionosphere. As a result, AM radio can be received at a significantly more distant from a transmitter location than waves of a shorter length.
What remains to discuss is a physical implementation of the amplitude modulation.
This implementation involves reducing the amplitude of the oscillations of an electric current in the LC circuit connected to an antenna.
This can be accomplished by adding a variable resistor into a circuit that is controlled by a current that represents the sound waves on the output of a microphone.
There are many different ways to put together such a device, we will leave this to special courses on radio electronics.
Radio transmission is a process of generating oscillations of electromagnetic field.
In this lecture we'll attempt to put together a simple generator, kind of those created in the beginning of the radio era.
In the previous lecture we discussed a simple LC circuit and connected to it a receiving antenna as a device that can capture the oscillations of electromagnetic field and induce the oscillations of electric current of a particular frequency in the circuit.
This device accepts all electromagnetic waves of all frequencies into an antenna and, because of the resonance of a certain frequency with a circuit's own frequency, selects a particular frequency out of many received ones.
In theory, the same device can work in reverse - if there is an oscillating with certain frequency electric current in the circuit, it will induce the oscillations of an electric current in an antenna, which, in turn, will create electromagnetic waves around this antenna, oscillating with the same frequency.
If these electromagnetic waves are of certain strength, the device will serve as a transmitter.
To make this design work, we have to assure an uninterrupted oscillations of the current in our circuit with its own inherent frequency.
Generally speaking, the oscillations in our circuit are damped because of natural resistance. Let's design a system with some sort of energy support to compensate the damping effect of resistance in a circuit.
Consider a swing at the children's playground. After initial push it will eventually stop because of a friction, unless we give another push. That new push should be synchronized with a movement of a swing and directed along its movement. So, periodic push in some direction at every moment when a swing moves in that direction is a solution.
Let's design an electronic circuit that does similar supporting pushes to electric current.
One of many kinds of circuits that delivers this solution is below.
It utilizes a triode as an ON/OFF switch to turn ON or OFF the current from a battery that feeds a circuit exactly at the moments it's needed to support loss of energy due to circuit resistance.
Assume, there are some oscillations in a circuit that contains inductor L1 and capacitor C.
These oscillations cause the variable electric current in the inductor L1, which, in turn, induces the electromotive force in the inductor L2.
Considering every circuit has certain resistance, the oscillations of electric current in the L1C circuit, if not connected to the inductor L2, would be damped and their amplitude would gradually diminish towards zero.
Note, however, that the oscillations of the electric current in the L1C circuit create a variable magnetic field with a variable magnetic flux going through the inductor L1 and, since it's paired with inductor L2, through L2 as well.
Since this magnetic flux is variable, it induces an electromotive force in inductor L2 proportional to a speed of change of the magnetic flux.
This variable electromotive force, in turn, causes a grid of a triode to be periodically charged positively or negatively.
When the grid is positive, electrons from the hot cathode go through a grid to the anode and feed the capacitor, thus compensating losses of energy due to resistance of a circuit.
If, however, the grid is negative, there is no current going from a cathode to an anode and capacitor is not receiving any charge from a battery.
To properly calculate the resulting oscillations of the current in the circuit we have to include this extra dosage of electricity supplied by a battery when the triode is open. This significantly complicates the calculations, and we will not go through this.
All we can state is that, with properly evaluated parameters of all components of this circuit, the feed from a battery will be exactly at the time the capacitor needs it, and the oscillations of the current in the circuit will be sustained as long as the battery has energy in it.
The above is just an idea. To practically implement it, we need to go through some calculations, which are beyond the scope of this course.
The above example is based on feeding a capacitor with additional charge. Alternatively, we can design a circuit, when a battery would feed an inductor with extra electromotive power.
Many other ideas and designs for transmitters were developed throughout the time, all have certain pluses and minuses, but the main principle of synchronously turn on the flow of new energy into an oscillator to compensate the losses due to resistance is still at the foundation of them all.
This lecture is about radio signals and receiving of these signals by a device we call "radio".
We will not talk in this lecture about how to produce and transmit radio signals, only about receiving them.
In this and subsequent lectures we will use the concept of forced oscillation presented in lectures "Forced Oscillation 1", "Forced Oscillation 2" and "Problems 4" of the "Mechanical Oscillations" topic of the "Waves" part of this course.
We strongly recommend to familiarize yourselves with the material presented there prior to study the material of this lecture.
The space around us is filled with radio signals, which are, in essence, oscillations of electromagnetic field.
We don't see them because our eyes are sensitive to certain range of frequencies of oscillations of electromagnetic field called visible spectrum.
Radio frequencies used to carry information to our radio, television, wireless phones and other devices are outside of this visible spectrum and, therefore, are invisible, but they do exist and we live in space filled with oscillations of electromagnetic field.
These oscillations are not synchronous because they are produced by many unrelated to each other sources, including our Sun, planet Earth, numerous sources of electromagnetic field around our own devices, including transmitters that intentionally produce oscillating electromagnetic fields and those produced unintentionally because any electric device is a source of electromagnetic field around it.
So, the space around us is filled with chaotic waves of electromagnetic field, and our purpose is to "catch" specific waves that have specific frequency of oscillations, used to transmit specific information.
The circuit that contains an inductor and a capacitor can help, and here is how.
Any oscillations of electromagnetic field will cause the oscillations of electric current in a conductor (like a piece of wire) in this field. This is the Faraday's Law of Induction.
While oscillations of electromagnetic field cannot be felt or observed by us directly, the induced electric current in a conductor is something tangible, we can use it in some devices to measure, observe, feel and use.
The problem is, chaotic oscillations of electromagnetic field with many different frequencies superposed on each other cause exactly similar chaotic oscillations of the electric current induced in the conductor.
Using a circuit with an inductor and a capacitor, we can select a particular oscillations of a particular frequency in this conductor, while suppressing the others.
Out of numerous oscillations with different frequencies superposed on each other we will select a particular frequency and use it to receive specific information.
The principle of selection of a particular frequency of oscillation out of many oscillations of different frequencies superposed on each other is based on a relationship between an oscillating external electromotive force and the own frequency of oscillation in a circuit that includes an inductor and a capacitor (LC circuit).
It is exactly analogous to forced mechanical oscillations and a principle of resonance presented in the lectures mentioned above.
Consider the following circuit that includes antenna, inductor and a variable capacitor (the one with moving plates, so its capacitance can be varied.)
Antenna is a conductor, where a chaotically oscillating electromagnetic field around it produces chaotically oscillating induced electric current.
The oscillations of an electric current in the antenna are transferred to forced oscillations of the current in the inductor of a circuit through some transformer device.
Using this setup we have introduced an external electromotive force into an LC circuit that has its own parameters (inductance of an inductor and capacitance of a capacitor), including its own natural frequency of oscillation ω0 = 1/√L·C
(see "Electronic Oscillator" topic of this course's "Waves" part.)
External electromotive force induced by an antenna in the inductor of a circuit is a combination of many oscillations of electric current with many different wave frequencies.
Each one will interact with the circuit's own frequency of oscillations ω0.
At this point we would like to mention once more that the equations for an electric current in the LC circuit are exactly the same as those for mechanical oscillations, including forced oscillations with periodic external force.
So, the concept of resonance in an LC circuit is exactly the same as that described for mechanical oscillations in lecture "Forces Oscillation 2" of the "Mechanical Oscillations" topic of this part of the course.
It means that, out of many different frequencies of oscillations of the external electromotive force, the oscillations with a frequency equal to the circuit's own natural frequency will cause an increase in value of electric current oscillating with this frequency, while all other external waves with frequencies not equal to the circuit's own frequency will remain as low intensity noise of oscillations of electric current in the LC circuit.
Notice that the capacitor in the circuit presented above is a variable one, that is it's capable to change its capacitance, usually achieved by changing the position of its plates relative to each other, thus changing the common area of opposing plates and, consequently, changing the capacitanceC.
Change in capacitance results in a change of a circuit's own frequency of oscillation. That, in turn, allows to select a different external frequency of oscillations out of all waves chaotically received by an antenna, thus performing a tuning to any specific frequency within a range of values specific for a particular technical characteristics of a circuit.
The explanation above is just a principle of selection of electromagnetic oscillations of a specific frequency using an LC circuit. The real devices in radio, TV, phones etc. are much more complex and involve significant enhancements to the process, but it's important to be familiar with this principle, as it lies in the foundation of most electronic devices we use.
The previous lecture considered a simple circuit that contained a battery, a capacitorC and an inductorL, while ignoring an internal resistance of a circuit R.
This lecture continues the exploring the oscillating characteristics of this circuit, but, in addition, we will address more practical case of the existence of an electrical resistanceR in the circuit.
These circuits are called RLC-circuits.
As in the previous lecture, we put an AB-switch in the A position to charge the capacitor from a battery.
Then we switch to the B position, battery is no longer a factor, and capacitor is charged.
Initially, the voltage between the plates of a capacitor is at maximum, so the electric current is at maximum as well.
As the charge goes from one plate to another through an inductor and a resistor, the voltage and the electric current are diminishing.
Assume, the electric current in the circuit is changing with time t as I(t), the voltage drop between the plates of the capacitor is VC(t), the voltage drop on the inductor is VL(t) and the voltage drop on the resistor is VR(t).
By definition of the capacitanceC of a capacitor, C = Q(t)/VC(t) or C·VC(t) = Q(t)
where Q(t) is the charge on its plates at time t.
We prefer to deal with the electric current I(t), which is defined as amount of moving charge per unit of time, which is mathematically expressed as a derivative of a charge by time, I(t) = dQ(t)/dt
Differentiating the above equation between voltage and charge on a capacitor, we obtain the dependency between voltage and current on a capacitor: C·dVC(t)/dt = I(t) or V'C(t) = I(t)/C
Any current that goes along a wire creates a magnetic field around this wire. Since the electric current in our inductor is variable, the magnetic field that is generated by this current is variable.
According to the Faraday's Law, the variable magnetic field going through a wire loop generates electromotive force (EMF) equal to a rate of change of the magnetic field flux.
Therefore, VL(t) = dΦ(t)/dt or VL(t) = Φ'(t)
In turn, magnetic flux Φ(t) going through inductor, as a function of time t, is proportional to an electric current I(t) going through its wire Φ(t) = L·I(t)
where L is a coefficient of proportionality that depends on physical properties of the inductor called inductance of the inductor.
From this we derive the following equation that describes the relationship between VL(t) and I(t): VL(t) = L·I'(t)
According to the Ohm's Law, the voltage drop on the resistor is VR(t) = R·I(t)
Finally, since the circuit contains no sources of electrical energy, sum of all voltage drops should be equal to zero: VC(t) + VL(t) + VR(t) = 0
Now we have the following system of four equations with four unknown functions: V'C(t) = I(t)/C VL(t) = L·I'(t) VR(t) = R·I(t) VC(t) + VL(t) + VR(t) = 0
Differentiating the last equation by t and substituting voltages through their expressions in terms of electric current, we obtain L·I"(t) + R·I'(t) + I(t)/C = 0
The last equation resembles the one that describes damped mechanical oscillations of an object on an ideal spring m·x"(t) + c·x'(t) + k·x(t) = 0
where x(t) is a displacement of an object on a spring from the neutral position, m is an object's mass, c is a damping coefficient, k is a spring's elasticity
(see lecture UNIZOR.COM - "Physics 4 Teens" - "Waves" - "Mechanical Oscillations" - "Viscosity Damping 1")
As we described in the lecture "Viscosity Damping 1", the general solution to a homogeneous linear differential equation of the second order, like the one above, can be found by finding two functions that satisfy this equation x1(t) and x2(t), called partial solutions, and describing the general solution as a linear combination of these two functions x(t) = A·x1(t) + B·x2(t)
The coefficients A and B depend on two initial conditions x(0) and x'(0).
In this case of electronic oscillation mathematics are exactly the same, which is quite remarkable.
Let's proceed with a solution along the same path as presented in the lecture on damped mechanical oscillations.
We will seek partial solutions based on function I(t) = eγ·t.
where γ - some (generally, complex) number.
The reason for this is that, substituting this function with its first and second derivatives into the differential equation, we get I'(t) = γ·eγ·t I"(t) = γ²·eγ·t
and the differential equation is γ²·L·eγ·t + γ·R·eγ·t + eγ·t/C = 0
Canceling eγ·t, which never equal to zero, we get a quadratic equation for γ γ²·L + γ·R + (1/C) = 0
with any of its solutions (generally, complex), used in function eγ·t, delivering the partial solution of the original differential equation.
Since a quadratic equation has, generally speaking, two solutions among complex numbers γ1 and γ2, we have two solutions to our original differential equations: I1(t)=eγ1·t and I2(t)=eγ2·t.
Moreover, since our equation is a homogeneous linear differential equation of the second order, all linear combinations of the above two solutions deliver a general solution that looks like I(t) = A·eγ1·t + B·eγ2·t
where constants A and B are defined by initial conditions on a circuit.
Switching from theory to concrete results for our differential equation presented above, our first task is to solve the quadratic equation to find γ1 and γ2
The quadratic equation for γ is γ²·L + γ·R + (1/C) = 0
This is exactly analogous to damped mechanical oscillations (see three "Viscosity Damping" lectures in the "Mechanical Oscillations" topics of this course), where the corresponding equation looks like γ²·m + γ·c + k = 0
(m - mass, c - damping coefficient, k - elasticity of a spring).
Solutions to this equation that describes the (generally, complex) exponent of damped mechanical oscillations are γ1,2 = −c/(2m)±√(c/(2m))²−k/m
Substituting inductanceL of an electronic oscillator for massm of mechanical one, resistanceR of an electronic oscillator for damping coefficientc of mechanical one and inverse of capacitance1/C of an electronic oscillator for elasticityk of mechanical one, we can use all the calculations of damped mechanical oscillations to obtain corresponding results for damped oscillations of electronic oscillator.
Solutions to our quadratic equation for γ, therefore, are: γ1,2 = −R/(2L) ±
± √(R/(2L))²−1/(L·C)
Following the same logic as in "Viscosity Damping" lectures, we consider three cases: Δ = (R/(2L))²−1/(L·C) > 0 Δ = (R/(2L))²−1/(L·C) = 0 Δ = (R/(2L))²−1/(L·C) < 0
Case 1:Δ > 0
Let ω² = Δ = (R/(2L))²−1/(L·C), where ω is some positive number that is less than R/(2L)
Then both solutions to our quadratic equations are negative γ1,2 = −R/(2L) ± ω < 0
General solution to our differential equation is I(t) = A·e(−R/(2L)+ω)·t +
+ B·e(−R/(2L)−ω)·t
Both components of this general solution for I(t) with any coefficients A and B contain an exponential function with negative multiplier at time variable t.
That means, after some initial fluctuation, the electric current will be monotonously diminishing by absolute value, and there will be no oscillations.
This is an overdamped circuit, analogous to overdamping of mechanical oscillations within a viscous environment with high viscosity.
High viscosity of the environment for mechanical oscillations is equivalent to high resistance of the electrical circuit.
The graph of electrical current I(t) as a function of time t would look similar to this one:
Case 2:Δ = 0
In this case γ1,2 = −R/(2L) ± 0 = −R/(2L)
Unfortunately, in this case our quadratic equation has only one solution, we will call it γ1.
But, to come up with a general solution to a homogeneous linear differential equation of the second order we need two linearly independent partial solution, like in Case 1 above.
To look for a second solution, we will expand our search and, following the same logic as in "Viscosity Damping 2" lecture of "Mechanical Oscillations" topic, we will try to find a solution to our differential equation in the form I(t) = t·eγ·t
There is one particular value of variable γ that makes this function a solution to our differential equation. Just substituting this function together with its first and second derivatives in the differential equation renders the value γ2 = −R/(2L)
as it was derived in the lecture "Viscosity Damping 2".
Therefore, we have now two linearly independent partial solutions to our differential equation: I1(t) = eγ1·t = e−(R/(2L))·t I2(t) = t·eγ2·t = t·e−(R/(2L))·t
General solution to our differential equation is I(t) = A·e−(R/(2L))·t +
+ B·t·e−(R/(2L))·t
Since all exponents are negative number multiplied by time, the asymptotic behavior of this function is similar to the one presented in Case 1 - after some initial fluctuation, the electric current will be monotonously diminishing by absolute value, and there will be no oscillations.
The graph of electrical current I(t) as a function of time t would look similar to the one of Case 1 above.
Case 3:Δ < 0
Let ω² = −Δ = −(R/(2L))²+1/(L·C), where ω is some positive number.
Then both solutions to our quadratic equations are complex numbers γ1,2 = −R/(2L) ± ω·i
One particular solution to our differential equation is a function with complex values I1(t) = eγ1·t = e(−R/(2L)+ω·i)·t
and another is I2(t) = eγ2·t = e(−R/(2L)−ω·i)·t
If a function with complex values, like Z(t)=X(t)+i·Y(t), where X(t) and Y(t) are real values functions and i²=−1, satisfies a linear differential equation, its real and imaginary components must satisfy this equation too, because, if a+b·i=0 then a=0 and b=0 too.
To represent functions I1,2(t) above in a canonical form for complex numbers, we will use the Euler's formula ei·x = cos(x) + i·sin(x)
Since real and imaginary parts of a solution to a linear differential equation must be solutions in their own right, function I1(t) gives two linearly independent real valued partial solutions I1a(t) = e(−R/(2L)·t·cos(ω·t)
and I1b(t) = e(−R/(2L)·t·sin(ω·t)
Similarly, function I2(t) gives two linearly independent real valued partial solutions I2a(t) = e(−R/(2L)·t·cos(ω·t)
and I2b(t) = −e(−R/(2L)·t·sin(ω·t)
Functions I2a(t) and I2b(t) are linearly dependent on I1a(t) and I1b(t) and are not needed to construct a general solution to our differential equation.
From partial solutions I1a(t) and I1b(t) we can construct a general solution to our differential equation I(t) = e(−R/(2L)·t·
·[A·cos(ω·t) + B·sin(ω·t)]
The electric current of this kind gives true oscillations, albeit damped ones with an amplitude exponentially diminishing because of the presence of a resistor in a circuit.
Its graph looks like this
From the "Electromagnetism" part of this course we know that variable (sinusoidal) oscillations of electric current in the wire frame can be induced by rotating this frame in the magnetic field.
Using the terminology of the mechanical oscillations, the oscillations of the electric current in such a case should be called forced.
There are, however, circuits where oscillations of electric current are free from any external forces.
Let's consider the following simple circuit that contains a battery, a capacitorC and an inductorL, ignoring for now an electrical resistance of a circuit.
The above is the LC-circuit because it consists of an inductor and a capacitor.
When an AB-switch is in the A position, the capacitor will be charged from a battery.
Now we switch to the B position, battery is no longer a factor, and capacitor is charged.
Initially, the voltage between the plates of a capacitor is at maximum, so the electric current is at maximum as well.
As the charge goes from one plate to another through an inductor, the voltage and the electric current are diminishing sinusoidally from some maximum at time t=0 to zero at t=T/4, where T is a period.
Assume (and the supporting calculations of the validity of this assumption follow), the electric current is diminishing with time t as I(t) = I0·cos(ωt)
where I0 is the maximum value of a current ω is angular speed T = 2π/ω is a period
The speed of change of the current is the first derivative of this function dI/dt = −I0·ω·sin(ωt)
This speed of change also changes with time sinusoidally, but its maximum at t=T/4 coincides with minimum of the current itself.
As we know, the change in the current in the inductor causes the change of the magnetic field around it, which, in turn, causes the appearance of a self-inductedelectromotive force. You can refer to "Self-induction" chapter of the "Electromagnetism" part of this course for details.
The electromotive force is stronger, when the speed of change of the magnetic field is stronger, and it forces the electric current to continue its existence even after one plate of the capacitor lost its charge to another plate.
The maintained by the magnetic field current will make the capacitor to be charged again, but in opposite direction (the plate, that used to positive, now is negative and vice versa).
As the charge accumulates on the capacitor, it slows down and stops the flow of the current induced by a changing magnetic field. When the current is stopped, the situation resembles the one at time t=0, but with reversed charges on the capacitor's plates. So, the current starts flowing again, but in opposite direction.
The internal resistance R of a circuit will result in some lost energy and, therefore, the maximum charge on the plates of a capacitor will diminish with time. This exactly resembles damped mechanical oscillations.
Let's get into some theory of this LC-circuit without taking into consideration the internal resistance of a circuit.
Assume that the current in a circuit is I(t), the voltage drop between the plates of the capacitor is VC(t) and the voltage drop on the inductor is VL(t).
By definition of the capacitanceC of a capacitor, C = Q(t)/VC(t) or C·VC(t) = Q(t)
where Q(t) is the charge on its plates at time t.
We prefer to deal with the electric current I(t), which is defined as amount of moving charge per unit of time, which is mathematically expressed as a derivative of charge by time, I(t) = dQ(t)/dt
Differentiating the above equation between voltage and charge on a capacitor, we obtain the dependency between voltage and current on a capacitor: C·dVC(t)/dt = I(t) or V'C(t) = I(t)/C
Any current that goes along a wire creates a magnetic field around this wire. Since the electric current in our inductor is variable, the magnetic field that is generated by this current is variable.
According to the Faraday's Law, the variable magnetic field going through a wire loop generates EMF equal to a rate of change of the magnetic field flux.
Therefore, VL(t) = dΦ(t)/dt or VL(t) = Φ'(t)
In turn, magnetic flux Φ(t) going through inductor, as a function of time t, is proportional to an electric current I(t) going through its wire Φ(t) = L·I(t)
where L is a coefficient of proportionality that depends on physical properties of the inductor called inductance of the inductor.
From this we derive the following equation for VL(t): VL(t) = L·I'(t)
Finally, since the circuit contains no sources of electrical energy, sum of all voltage drops should be equal to zero: VC(t) + VL(t) = 0
Now we have the following system of three differential equations with three functions: V'C(t) = I(t)/C VL(t) = L·I'(t) VC(t) + VL(t) = 0
Differentiating the third equation by t and substituting voltages through their expressions in terms of electric current, we obtain I(t)/C + L·I"(t) = 0 or I"(t) + I(t)/(L·C) = 0
The last equation resembles the one that describes harmonic mechanical oscillations of an object on an ideal spring x"(t) + (k/m)·x(t) = 0
where x(t) is a displacement of an object on a spring from the neutral position, k is a spring's elasticity and m is an object's mass.
The general solution to the above equation, as we have shown, when considered the periodic mechanical oscillations, is x(t) = C1·cos(ω·t) + C2·sin(ω·t)
where ω² = k/m C1 is initial displacement of the object from the neutral position of equilibrium C2 is defined by the initial velocity of an object and is zero, if there is no initial velocity.
Analogously, the general solution of our differential equation for the electric current in an LC-circuit is I(t) = C1·cos(ω·t) + C2·sin(ω·t)
where ω² = 1/(L·C) C1 and C2 are defined by initial conditions.
The period of these oscillations of electric current in the LC-circuit is T = 2π/ω = 2π√L·C
This is known as the Thomson's formula.
As presented in the lecture "Forced Oscillation 1", the general solution to a differential equation that describes forced oscillation looks like x(t)=F0·cos(ω·t)/[m(ω0²−ω²)] +
+ D·cos(ω0·t−φ)
where F0 is an amplitude of a periodic external force; ω - angular speed of a periodic external force; m - mass of an object on a spring; k - elasticity of a spring; ω0=√k/m is an inherent natural angular speed of oscillation of this particular object on this particular spring; t is time; D and φ parameters are determined by initial conditions for x(0) and x'(0).
Consider forced oscillations used as an example in that lecture.
A picture below represents forced oscillations of a system with its own natural angular speed ω0=2(rad/sec) with periodic force applied to it that has its angular speed ω=5(rad/sec).
Initial displacement and speed are at zero.
Are the forced oscillations periodic?
If yes, what is the period of oscillations?
Solution
The result of a periodic force acting on periodic movement of the system is a periodic movement of this system but with a longer period.
If a period of a force is T1 and the own natural period of a system T2, then smallest number T that is a multiple of both T1 and T2 (if it exists!) will be a period of system with periodic external force acting on it.
Such a period will exist, if T1/T2 is a rational number.
Indeed, if T1/T2=m/n, then T=T1·n=T2·m will be divisible by T1, getting integer n as a result, and by T2, getting integer m as a result.
The period of an external force with angular speed ω=5(rad/sec) is T1 = 2π/5(sec)
The natural period of a system with its own natural angular speed ωh=2(rad/sec) is T2 = 2π/2 = π(sec)
In this case T=2π(sec) would be a smallest time interval that contains integer number of both periods T1 (T/T1=2) and T2 (T/T2=5).
Answer
The system will be in a periodic motion with a period T=2π(sec).
Problem B
Let's solve a problem that combines free oscillations, viscosity and external periodic force acting on an oscillating object.
Making all the reasonable assumptions, derive the equation of motion of such oscillations.
Solution
Let's assume that oscillations and all the forces are acting along the X-axis, so we have to derive a position of an object relative to the equilibrium point, which is at the origin of X-coordinate, as a function of time t.
If the position of an object relative to an equilibrium point is x(t), we assume the following:
(a) a spring acts on it, according to the Hooke's Law, with a force Fs(t)=−κ·x(t), where κ is a spring elasticity
(b) viscosity force is proportional to the speed x'(t) of an object and equals to Fv(t)=−c·x'(t), where c is viscosity damping coefficient
(wherever necessary, we will assume that viscosity c is sufficiently small in order to observe some oscillations)
(c) an external force is periodic and equals to Fx(t)=F0·cos(ω·t), where F0 is a constant amplitude and ω represents an angular speed of oscillations of an external force.
Now, according to the Newton's Second Law, the combination of all these forces should be equal to mass m of an oscillating object times its acceleration x"(t): m·x"(t) = −κ·x(t) − c·x'(t) +
+ F0·cos(ω·t)
It can be transformed into more traditional format x"(t)+(c/m)·x'(t)+(κ/m)·x(t) =
= (F0/m)·cos(ω·t)
This differential equation is not homogeneous because the right side contains some function that is not linearly dependent on x(t).
If two solutions of the above non-homogeneous differential equation are known, their difference is, obviously, a solution to a corresponding homogeneous equation xh"(t) + (c/m)·xh'(t) +
+ (κ/m)·xh(t) = 0
Therefore, to find all solutions to a non-homogeneous differential equation, it is sufficient to find one particular solution xp(t) and add to it a general solution to a corresponding homogeneous equation xh(t).
Let's start by finding a general solution to the corresponding homogeneous differential equation xh"(t) + (c/m)·xh'(t) +
+ (κ/m)·xh(t) = 0
It's a linear differential equation of the second order, so its general solution is any linear combination of any two linearly independent particular solutions.
Let's look for solutions to this differential equation among functions that look like this: xh(t) = eγt
to find a pair of γ's (may be complex) that result in eγt to be a solution of our homogeneous differential equation.
Using this approach, we obtain xh'(t) = γ·eγt xh"(t) = γ²·eγt
Substituting this into our homogeneous equation: γ²·eγt+(c/m)·γ·eγt+(κ/m)·eγt=0
Since eγt is not zero, we have a quadratic equation for γ γ² + (c/m)·γ + (κ/m) = 0
Its discriminant is Δ = (c/m)²−4κ/m
For sufficiently small (by assumption) viscosity c this discriminant will be negative.
So, our quadratic equation for γ will have two complex solutions: γ1,2 = −c/(2m)±√(c/(2m))²−κ/m
Let ωh² = −(c/(2m))² + k/m
Then γ1,2 = −c/(2m)±√−ωh² =
= −c/(2m) ± ωh·i
where i is an imaginary unit, i²=−1
From this, the solution to our homogeneous differential equation, that does not take into account external periodic force, is xh1,2(t) = eγ1,2·t =
= e[−c/(2m)±ωh·i]·t =
= e−c·t/(2m)·e±ωh·i·t
The first member of this product e−c·t/(2m) is an exponentially diminishing amplitude of oscillation (because of viscosity).
The second one, according to Euler's formula, is e±ωh·i·t = cos(ωh·t) ± i·sin(ωh·t)
If function z(t)=p(t)+i·q(t) with complex values satisfies some linear differential equation with real coefficients, its real and imaginary components must satisfy this equation independently.
Therefore, xh1(t) = e−c·t/(2m)·cos(ωh·t)
and xh2(t) = e−c·t/(2m)·sin(ωh·t)
are two independent solutions and the general solution to our linear homogeneous equation of the second order is any linear combination of these two solutions: xh(t) = e−c·t/(2m)·
·[A·cos(ωh·t)+B·sin(ωh·t)]
where A and B are any real numbers.
Using simple trigonometric identity cos(α−β) =
= cos(α)·cos(β) + sin(α)·sin(β)
the above expression can be transformed into xh(t) = R·e−c·t/(2m)·cos(ωh·t−φ)
where R = √A²+B² and cos(φ)=A/R, sin(φ)=B/R.
Parameters R and φ are defined by initial conditions - initial displacement of an object from the equilibrium xh(0) and initial speed xh'(0).
Let's check that the above function is indeed a solution to our homogeneous differential equation xh"(t) + (c/m)·xh'(t) +
+ (κ/m)·xh(t) = 0
Calculate all the components: xh(t) = R·e−c·t/(2m)·cos(ωh·t−φ)
That proves that the general solution to our homogeneous differential equation of the second order is the function xh(t) = R·e−c·t/(2m)·cos(ωh·t−φ)
where R is an amplitude and φ is a phase shift of oscillations, both determined by initial conditions xh(0) and xh'(0).
Now, to find all solutions to our original non-homogeneous differential equation x"(t)+(c/m)·x'(t)+(κ/m)·x(t) =
= (F0/m)·cos(ω·t)
it is sufficient to find one particular solution xp(t) and add to it a general solution to a corresponding homogeneous equation xh(t) that we have found above.
Considering the trigonometric character of the right side of our original equation, let's try to find a particular solution in a form xp(t) = A·cos(ω·t) + B·sin(ω·t)
where A and B are some constants.
Then x'p(t) =
= −A·ω·sin(ω·t)+B·ω·cos(ω·t)
x"p(t) =
= −A·ω²·cos(ω·t)−B·ω²·sin(ω·t)
Substitute these functions into our original differential equation, getting
−A·ω²·cos(ω·t)−B·ω²·sin(ω·t)−
−(c/m)·A·ω·sin(ω·t)+
+(c/m)·B·ω·cos(ω·t)+
+(κ/m)·A·cos(ω·t)+
+(κ/m)·B·sin(ω·t) =
= (F0/m)·cos(ω·t)
In order for this to be true for any time moment t, the corresponding coefficients at cos(.) and sin(.) functions on the left and right side of an equation must be equal.
The coefficient at cos(.) on the left side is supposed to be equal to (F0/m): −A·ω²+(c/m)·B·ω+(κ/m)·A =
= (F0/m)
There is no sin(.) on the right side, so the coefficient at sin(.) on the left side is supposed to be equal to zero: −B·ω²−(c/m)·A·ω+(κ/m)·B = 0
So, we have two linear equations with two variables A and B, which can easily be solved −A·ω²+(c/m)·B·ω+(κ/m)·A =
= (F0/m) −B·ω²−(c/m)·A·ω+(κ/m)·B =
= 0
Transform into canonical form ((k/m)−ω²)·A+(c/m)·ω·B =
= (F0/m) −(c/m)·ω·A+((κ/m)−ω²)·B =
= 0
Solutions are: A = (F0/m)·(k/m−ω²)/det B = (F0/m)·(c/m)/det
where determinant det = ((k/m)−ω²)² + (c/m)²·ω²
Now the general solution x(t) to our original non-homogeneous differential equation is a combination of the above partial solution xp(t)=A·cos(ω·t)+B·sin(ω·t)
(where constants A and B are defined above)
and the general solution to a corresponding homogeneous solution xh(t) = R·e−c·t/(2m)·cos(ωh·t−φ)
That is, x(t) = xp(t) + xh(t) =
= A·cos(ω·t) + B·sin(ω·t) +
+ R·e−c·t/(2m)·cos(ωh·t−φ)
What's interesting about this analytical expression of such a complex oscillations is the fact that a component related to damped oscillations with the system's natural angular speed ωh is gradually decreasing in amplitude as time passes, because viscosity exponentially diminishes its amplitude R·e−c·t/(2m) to zero.
That leaves the oscillations with major angular speed that depends only on the driving angular speed ω of an external force.
The following graph renders the gradual transformation of damped oscillations under an external periodic force into oscillations mostly dependent on the angular speed of that external force and less and less dependent on the natural properties of a spring and environment.
The parameters chosen for this graph are: A = 0.2
B = 0.3
ω =8
R = 4
c = 1
m = 1
ωh=5
φ = π/4
An object, oscillating in viscous environment, moves according to the following equation of motion x(t) = 4·e-t/2·cos(5t−π/4)
where t is time and x(t) is deviation from the equilibrium.
Determine the moments of time {ti} = {t0, t1, t2, ...}, when the object returns to the equilibrium.
Solution
Since we are talking about damping oscillations with diminishing displacements from the equilibrium, the motion is not exactly periodic, but we use the term period in a sense that an object returns to the equilibrium point again and again, albeit deviating from it less and less.
In this context period means the interval of time between consecutive crests or consecutive troughs of the trajectory, or between moments in time, when an object returns to the equilibrium position after reaching its local maximum deviation in both directions from the equilibrium.
Calculations below show that this interval does not change with time and remains constant, while the amplitude of oscillations around the equilibrium gets smaller with time.
To find the times when the object returns to the equilibrium, we have to find points, where x(t)=0.
Since coefficient in front of cos(...) never equals to zero, the solutions of an equation x(t)=0 are those moments in time, when cos(5t−π/4)=0.
Solving the equation: cos(5t−π/4) = 0 5t−π/4 = π/2+N·π 5t = 3π/4+N·π t = 3π/20+N·π/5 t = (3+4N)·π/20
where N ≥ 0 - an integer.
Interval between any two consecutive time moments of return to the equilibrium equals to π/5 (constant).
This makes a period of oscillation to be 2π/5 to allow an object to move from the equilibrium to one direction, return to the equilibrium, move to another direction and return again.
Substituting N=0,1,2..., we will get values {t0=3π/20, t1=7π/20, t2=11π/20, ...}
The first few members of this sequence are t0 ≅ 0.471 t1 ≅ 1.100 t2 ≅ 1.728
Answer
{ti} = (3+4N)·π/20
where N ≥ 0 - an integer.
Problem B
Consider the same object and environment as in Problem A.
Determine the moments of time {τi} = {τ0, τ1, τ2, ...}, when the absolute value of object's deviation from the equilibrium reaches its local maximum.
Solution
Local extremes (maximum and minimum) of function x(t) are characterized by the fact that the derivative x'(t) at these points equals to zero. x'(t) = −2·e-t/2·cos(5t−π/4) −
− 20·e-t/2·sin(5t−π/4) = 0
From this follows cos(5t−π/4)+10·sin(5t−π/4)=0 tan(5t−π/4) = −1/10 5t−π/4 = arctan(−1/10) + N·π 5t = arctan(−1/10) +
+ (4N+1)·π/4 t = 0.2·arctan(−1/10) +
+ (4N+1)·π/20
where N ≥ 0 - an integer.
Since arctan(−1/10) ≅ −0.0996686525,
the first few members (N=0,1,2) of this sequence are τ0 ≅ 0.137 (local max) τ1 ≅ 0.765 (local min) τ2 ≅ 1.394 (local max)
Answer
{τi} = 0.2·arctan(−1/10) +
+ (4N+1)·π/20
where N ≥ 0 - an integer.
Problem C
Determine the values of object's deviations from the equilibrium {A0, A1, A2, ...}, when they are at local maximums by absolute value.
Solution
Ai = x(τi)
Therefore, the first few members of this sequence are A0 ≅ 3.716 (local max) A1 ≅ −2.714 (local min) A2 ≅ 1.983 (local max)
Answer
{Ai} = {4·eτi·cos(5τi−π/4)}
where values {τi} were determined in Problem B.
An object of mass m=0.3kg sits on a table and is attached to a wall by a horizontal spring with elasticity κ=2N/m.
This object can slide left and right on a surface of a table with the static and dynamic friction coefficient μ=0.1.
Initially, the spring is in equilibrium position, not stretched, not squeezed.
What is the maximum distance λ we can stretch a spring such that the object would remain in that position on a stretched spring and not moved back towards equilibrium?
Assume that gravity acceleration is g=9.8 m/sec².
Solution
The necessary condition for an object to remain in place with a stretched spring is that the force of a spring that pulls the object back to equilibrium Fs is not sufficient to overcome the static friction force Ff. Fs = κ·d = 2·d (N) Ff = μ·m·g = 0.1·0.3·9.8 (N)
The necessary condition for an object to remain in place on a stretched spring is Fs ≤ Ff
That is, 2(N/m)·λ(m) ≤ 0.1·0.3·9.8 (N)
from which follows λ ≤ 0.147 (m) Answer
The maximum distance to stretch a spring with an object not being pulled back (critical distance) is λ = 0.147m = 147mm
Problem B
Consider the same set-up as in the Problem A.
Our task now is to stretch a spring beyond the critical distanceλ, so that the object will be pulled back by a spring, but it should reach the equilibrium point and stop there with no further movements.
What is the distance d to stretch a spring to achieve this result?
Solution
Let's approach this problem from the energy standpoint.
As we know, the potential energy of a spring stretched by a distance d is U(d) = κ·d²/2
(see "Energy of Oscillation" lecture in this chapter of a course)
Using the data from Problem A, U(d) = 2·d²/2
This energy should be fully used to overcome the constant friction force Ff on a distance d. Ff = μ·m·g = 0.294 (N)
On a distance d the force Ff performs work W(d) = Ff·d = 0.294·d
which should be equal to initial potential energy U(d).
Therefore, W(d) = U(d) 0.294·d = 2·d²/2 d = 0.294 (m)
Answer
In order to initiate a movement of an object back to the equilibrium point such that it will stop at that point, we have to stretch the spring by 0.294m = 294mm
Incidentally, this is a double critical distance, which could've been predicted since the initial point and a point of stop must be symmetrical relative to a critical point, as was proved in the "Energy of Oscillation" lecture.
So far, our understanding of waves was very much related to a concept of some material object to oscillate, that is some material particles going back and forth, up and down, left and right or, in short, making some repetitive movements.
But Physics deals not only with material in a usual sense objects. A very important part of what physicists deal with is fields.
Let's spend a few minutes to talk about a nature of the concept field.
In Mathematics it's customary to analyze the properties of some mathematical objects (like N-dimensional vector space) without really having any real life object that corresponds exactly to this mathematical concept.
Mathematicians care only about properties, regardless of the objects that possess these properties. Moreover, there might be completely different mathematical objects that possess exactly the same properties, and the beauty of Mathematics allow us not to think about objects that possess these properties, but to concentrate only on analysis of the properties themselves.
With properly analyzed properties, mathematicians can apply the results to any mathematical object that possess these properties.
In short, properties are of primary interest, not objects that possess these properties.
With fields physicists do the same. They analyze the properties of the fields using some experiments, building a model that corresponds to these experiments and analyzing the properties of the model.
If at certain time they come up with a new experiment that contradicts the model, they change (enhance, update) the model and continue analyzing the model.
We don't really know what fields really are, but we can observe, experiment with, measure their properties, and, as long as our experiments do not contradict the results of our theoretical analysis, we accept the results of the theory.
So, for our purposes we consider a field as some area of space that has certain properties. In particular, we can experiment with some material objects and see how they behave inside these space areas we call fields, and that's how we observe the fields' properties.
The primary subject of analysis in this chapter is an electromagnetic field.
Our analysis will be based on material presented in the previous part of this course called "Electromagnetism".
The purpose of this lecture is to explain that electromagnetic field can oscillate and propagate even without any "material" (in a usual sense) carrier, allowing the light from stars to reach us through Space vacuum.
Let's start from the familiar fact that around a line of moving electrons (electric current) there exists a magnetic field.
If, for example, we take a straight wire and connect it to a direct current (DC) battery, the permanent flow of electrons through this wire will create constant magnetic field around it with concentric magnetic field lines.
The fact that electrons are moving signifies the existence of the electric field of some intensity, and the appearance of a magnetic field we attribute to properties of electric field.
The next logical step is that around a line of moving electrons, when the movement is not constant, but variable, there exists a variable magnetic field.
For example, if we have a wire loop and connect it to alternate current (AC) source of electricity that always changes the direction certain number of times per second, the magnetic field going through this wire loop also will change the direction of its magnetic field lines with each cycle of an electric current.
The variability of the flow of electrons is caused by a variable electric field, and the appearance of a variable magnetic field we attribute to variability of an electric field.
The lecture "Straight Line Current" of this course (follow the menu items UNIZOR.COM - "Physics 4 Teens" - "Electromagnetism" - "Magnetism of Electric Current") explains this process in details for DC.
The lecture "Induction and AC" of this course (follow the menu items UNIZOR.COM - "Physics 4 Teens" - "Electromagnetism" - "Alternating Current Induction") explains this process in details for AC.
Our next experimental fact is the electromagnetic induction - a variable magnetic field around an electric wire causes appearance of a variable electric current in the wire that can only be caused by a variable electric field.
So, we can conclude that a variable electric field causes the existence of a variable magnetic field, while a variable magnetic field causes the existence of a variable electric field.
Combining everything we mentioned above, we see that a variable electromagnetic field (a field where variable electric and magnetic forces exist) that consists of logically and physically chained together variable electric field and variable magnetic field, mutually causing each other, is self-sustained and capable of propagating through vacuum without existence of any wires as a medium.
This chain can schematically be represented as follows.
Here a variable (oscillating) electric field causes the existence of a variable (oscillating) magnetic field in its neighborhood. That magnetic field causes the existence of another variable electric field further in space. This process repeats itself, thus facilitating the propagation of electromagnetic oscillations of an electromagnetic field.
Any field has certain forces acting on certain objects (or other fields).
The electromagnetic field at any point can be characterized by two forces - intensity vector of electric component of a field (E) and intensity vector of its magnetic component (B).
Here are a few facts about electromagnetic fields that require deeper immersion into their structure and are beyond the scope of this course.
1. Propagation of electromagnetic waves (called radiation in many cases) is perpendicular to both intensity vectors, electric and magnetic.
That makes these waves transverse.
2. Electric and magnetic intensity vectors are sinusoidal, perpendicular to each other and are in phase. Combining that property with their perpendicularity to a direction of propagation prompts the following graphical representation of electromagnetic waves.
(you can click the right mouse button and open this picture in another tab for better view)
3. Speed of propagation depends only on the properties of environment where the radiation is taking place.
In particular, the ability of electric field to permeate an environment (ε) and the ability of magnetic field to permeate an environment (μ) determine this speed.
The electric permittivity of vacuum is ε0 ≅ 8.8542·10−12
(F/m - Farad per meter).
The magnetic permeability of vacuum is μ0 ≅ 1.2566·10−6
(H/m - Henry per meter).
Speed of electromagnetic waves c can be expressed in terms of electric permittivity and magnetic permeability as c² = 1/(ε0·μ0)
According to the formula above and values of electric and magnetic constants, in vacuum the speed of electromagnetic waves is c = 299,792,458 (m/sec)
which in many cases is rounded to 300,000 km/sec.
Analyze the oscillations of a pendulum in a uniform gravitational field.
Solution
The picture above defines a pendulum with a thread length L and a point-mass m hanging in a gravitational field that produces a constant gravity acceleration g.
Let angle φ be a deviation of the thread from a vertical. This angle is a function of time, so we will use notation φ(t) and will attempt to include it in some equation in order to find this function.
There are two forces acting on a point-mass m: gravity, directed vertically down and equal to m·g, and tension of a thread T.
The force of gravity can be represented as a sum of two forces: directed along a thread and equal to m·g·cos(φ(t)) and perpendicular to a thread equal to m·g·sin(φ(t)).
The tension force T balances the force m·g·cos(φ(t)), as both directed along a thread in opposite directions, thus canceling each other.
The force m·g·sin(φ(t)) is the source of pendulum moving back an force, as it is always directed towards a vertical position of equilibrium.
For the purposes of this analysis we assume that initially we have moved a pendulum from its initial position, so its thread makes an angle φ(0)=φ0 with a vertical and let it go without any push, so φ'(0)=0.
According to the Second Newton's Law, the force is equal to a product of the mass and its linear acceleration.
Therefore, the force F=m·g·sin(φ(t)) that always acts along a tangential to a trajectory of the point-mass towards its equilibrium point should be equal by absolute value to a product of mass m and linear acceleration along a circular trajectory on a thread of the length L, which is equal to L·φ"(t).
Also, this force F always acts towards the vertical. Choose a positive direction of the deviation φ from a vertical as counterclockwise. For positive angles φ, as on the picture above, the direction of the force F would be negative. For negative angles φ the direction of the force F would be positive.
Therefore, the signs of the force F and deviation from the vertical are opposite to each other. For small angles φ the sign of sin(φ) would be the same as the sign of φ.
It results in the following differential equation −m·g·sin(φ(t)) = m·L·φ"(t)
or φ"(t) + (g/L)·sin(φ(t)) = 0
Since mass m can be canceled out, our first very important conclusion is that pendulum oscillation does not depend on the mass of an object hanging at its end.
The differential equation above is quite complex and we will not attempt to solve it exactly. However, as many physicists do, we can approximate it under certain conditions.
The condition usually accepted is that the angle of deviation φ(t) is very small and, consequently, function sin (φ(t)) is almost identical to function φ(t).
That brings us to an equation φ"(t) + (g/L)·φ(t) = 0
which is easy to solve, as it is a regular equation for harmonic oscillations.
The general solution to it, as explained in the previous lecture, is φ(t) = C1·cos(ωt) + C2·sin(ωt)
where ω=√g/L.
Constants C1 and C2 are determined by initial conditions.
In case of initial conditions φ(0)=φ0 and φ'(0)=0 the values of these constants are: C1 = φ0 C2 = 0
The solution of pendulum oscillations in this case is φ(t) = φ0·cos(√g/L·t)
The angular frequency of these oscillations is ω=√g/L.
The period of oscillations is T = 2π/ω = 2π√L/g.
The amplitude is A = φ0.
Just as a reminder, this analysis is applicable only to small oscillations of a pendulum around its equilibrium point, when an angle (in radians) and its sine are approximately equal to each other. Obviously, for any degree of precision the term "small" has its own meaning, the greater the precision - the smaller angle our analysis is applicable to.
Problem B
Analyze the oscillations of a ball of mass m rolling inside a semicircle of radius R in the uniform gravitational field with gravity acceleration g.
Solution
The diagram above depicts the forces acting on a ball rolling inside a semicircle.
The weight P=m·g can be represented as a sum of tangential force F=P·sin(φ(t)) and normal to a circle force that is canceled by a reaction of a circle surface.
We should take into account that the direction of force F is always opposite in sign to a sign of angle φ(t). So, the correct expressions for force F is F = −m·g·sin(φ(t))
According to Newton's Second Law, the force F can be equated to the product of mass and linear acceleration R·φ"(t) of the rolling ball. F = m·R·φ"(t)
This can be expressed as a differential equation for an angle of position φ(t). −m·g·sin(φ(t)) = m·R·φ"(t)
or, canceling mass m and dividing by radius R, φ"(t) + (g/R)·sin(φ(t)) = 0
As you see, the result is equivalent to the one we have obtained for a pendulum in the previous problem.
This differential equation is not easily solvable, so we will do exactly as in the previous problem - assume that an angle φ is so small that sin(φ(t)) can be approximated with φ(t), which leads us to a familiar differential equation for harmonic oscillations φ"(t) + (g/R)·φ(t) = 0
General and specific solutions of this equation are exactly as in the case of a pendulum in Problem A above.
