Parth G | The Mathematical Bridge That Links Two Completely Different Neighborhoods of Physics @ParthGChannel | Uploaded 2 years ago | Updated 2 hours ago
What links a mass-spring system to an RLC electric circuit?
We begin by considering a mass-spring system that can behave as a simple harmonic oscillator. When the mass is pulled on or pushed, thus extending or compressing the spring, the spring exerts a restorative force back in the direction of equilibrium. In other words, the spring exerts a force in order to go back to its natural length.
Once the spring is stretched and then released, the mass-spring system undergoes Simple Harmonic Motion - the spring exerts a force on the mass to bring the spring back to its natural length. The system oscillates back and forth symmetrically about the equilibrium position. Simple Harmonic Motion is defined by an object experiencing a force (technically acceleration) that is proportional to the object's displacement, and in the opposite direction to the displacement. We look at how this can be described by a simple differential equation by equating the net force on the system (F = ma) with the force exerted by the spring. We also see how the acceleration, a, of the system can be described as the second time derivative of the system's displacement or position.
Next, we look at damping, i.e. a force that resists the motion of the system. The simplest way to model this is that the damping force is proportional (and in the opposite direction) to the velocity of the system. We add this term to our differential equation, and convert the velocity term into the first derivative of displacement with time.
Finally, we add a driving force. This driving force can be any external force we can exert on the system, but we choose a sinusoidal force. The frequency with which the driving force varies is arbitrary. If it matches the system's "natural" frequency then we see "resonance", where the oscillation gets bigger and bigger in amplitude.
At this point we have our final differential equation - a driven, damped, harmonic oscillator. We take this equation and modify a few terms (though the equation still keeps the same form). And after doing this, we find an equation that can be used to describe a different kind of oscillator entirely - an electric oscillator! This time, the stuff that's oscillating is a series of charged particles within an electric circuit! The derivatives are now of charge with respect to time, rather than position.
The driving term now describes the voltage provided by the power source connected to the circuit we are studying (which is a series RLC circuit with an alternating voltage source).
The original "damping" force term now describes the resistive behavior of the circuit. In other words, any resistor in the circuit acts like the damping fluid for the mechanical (mass-spring) oscillator. The term in our equation describing the resistive effects actually is equivalent to Ohm's law (voltage across resistor = current x resistance).
The original "net force" term now describes how inductors behave in the circuit. The voltage across any inductor is given by the inductance multiplied by the rate of change of current, which is also the second derivative of the charge w.r.t. time.
The original "spring force" term is now given by the charge divided by the total capacitance in the circuit. In other words the capacitance behaves almost like an "inverse spring".
And so the equation for the circuit basically says that the voltage supplied by the power source is equal to the sum of the voltages across the inductor, resistor, and capacitor. But it also looks at how the system behaves like a driven, damped, harmonic oscillator!
Thanks for watching, please do check out my socials here:
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Here are some affiliate links for things I use! I make a small commission if you make a purchase through these links.
Introduction to Elementary Particles (Griffiths) - the book used in this video: https://amzn.to/3I3ld71
Quantum Physics Book I Enjoy: https://amzn.to/3sxLlgL
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My video about Ohm's Law: youtu.be/Zao9JV1BLg8
Extra Reading:
Voltage - https://en.wikipedia.org/wiki/Voltage
Timestamps:
0:00 - A Beautiful Parallel Between Two Areas of Physics
0:29 - Mass-Spring (Mechanical) Simple Harmonic Oscillator
4:38 - Damping the Mechanical Oscillator
6:33 - Driving the Mechanical Oscillator
8:40 - The Oscillator Equation and the Electrical Oscillator
9:16 - Charge, Current, and Electrical Stuff
11:14 - Driving the Electrical Oscillator (Power Source)
11:42 - Damping the Electrical Oscillator (Resistor)
12:52 - Inductance and Capacitance Terms
13:51 - Mechanical vs. Electrical Oscillator
What links a mass-spring system to an RLC electric circuit?
We begin by considering a mass-spring system that can behave as a simple harmonic oscillator. When the mass is pulled on or pushed, thus extending or compressing the spring, the spring exerts a restorative force back in the direction of equilibrium. In other words, the spring exerts a force in order to go back to its natural length.
Once the spring is stretched and then released, the mass-spring system undergoes Simple Harmonic Motion - the spring exerts a force on the mass to bring the spring back to its natural length. The system oscillates back and forth symmetrically about the equilibrium position. Simple Harmonic Motion is defined by an object experiencing a force (technically acceleration) that is proportional to the object's displacement, and in the opposite direction to the displacement. We look at how this can be described by a simple differential equation by equating the net force on the system (F = ma) with the force exerted by the spring. We also see how the acceleration, a, of the system can be described as the second time derivative of the system's displacement or position.
Next, we look at damping, i.e. a force that resists the motion of the system. The simplest way to model this is that the damping force is proportional (and in the opposite direction) to the velocity of the system. We add this term to our differential equation, and convert the velocity term into the first derivative of displacement with time.
Finally, we add a driving force. This driving force can be any external force we can exert on the system, but we choose a sinusoidal force. The frequency with which the driving force varies is arbitrary. If it matches the system's "natural" frequency then we see "resonance", where the oscillation gets bigger and bigger in amplitude.
At this point we have our final differential equation - a driven, damped, harmonic oscillator. We take this equation and modify a few terms (though the equation still keeps the same form). And after doing this, we find an equation that can be used to describe a different kind of oscillator entirely - an electric oscillator! This time, the stuff that's oscillating is a series of charged particles within an electric circuit! The derivatives are now of charge with respect to time, rather than position.
The driving term now describes the voltage provided by the power source connected to the circuit we are studying (which is a series RLC circuit with an alternating voltage source).
The original "damping" force term now describes the resistive behavior of the circuit. In other words, any resistor in the circuit acts like the damping fluid for the mechanical (mass-spring) oscillator. The term in our equation describing the resistive effects actually is equivalent to Ohm's law (voltage across resistor = current x resistance).
The original "net force" term now describes how inductors behave in the circuit. The voltage across any inductor is given by the inductance multiplied by the rate of change of current, which is also the second derivative of the charge w.r.t. time.
The original "spring force" term is now given by the charge divided by the total capacitance in the circuit. In other words the capacitance behaves almost like an "inverse spring".
And so the equation for the circuit basically says that the voltage supplied by the power source is equal to the sum of the voltages across the inductor, resistor, and capacitor. But it also looks at how the system behaves like a driven, damped, harmonic oscillator!
Thanks for watching, please do check out my socials here:
Instagram - @parthvlogs
Patreon - patreon.com/parthg
Music Chanel - Parth G's Shenanigans
Merch - https://parth-gs-merch-stand.creator-spring.com/
Here are some affiliate links for things I use! I make a small commission if you make a purchase through these links.
Introduction to Elementary Particles (Griffiths) - the book used in this video: https://amzn.to/3I3ld71
Quantum Physics Book I Enjoy: https://amzn.to/3sxLlgL
My Camera: https://amzn.to/2SjZzWq
ND Filter: https://amzn.to/3qoGwHk
Microphone (Fifine): https://amzn.to/2OwyWvt
Gorillapod: https://amzn.to/3wQ0L2Q
My video about Ohm's Law: youtu.be/Zao9JV1BLg8
Extra Reading:
Voltage - https://en.wikipedia.org/wiki/Voltage
Timestamps:
0:00 - A Beautiful Parallel Between Two Areas of Physics
0:29 - Mass-Spring (Mechanical) Simple Harmonic Oscillator
4:38 - Damping the Mechanical Oscillator
6:33 - Driving the Mechanical Oscillator
8:40 - The Oscillator Equation and the Electrical Oscillator
9:16 - Charge, Current, and Electrical Stuff
11:14 - Driving the Electrical Oscillator (Power Source)
11:42 - Damping the Electrical Oscillator (Resistor)
12:52 - Inductance and Capacitance Terms
13:51 - Mechanical vs. Electrical Oscillator
