In the case of forced, undamped oscillations, why is the amplitiude of the steady state oscillations bigger, when the frequency of the driving force gets closer to the natuaral frequency of the oscillating system. I know, that we can get this result by solving appropriate equation, but I am looking for an intuituive explanation. I know the "kid on a swing" explanationa, but it relates to driving in a form of short impulses, rather than a sinsuoidally changing force.
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The steady-state motion of a driven oscillator is given by;$$x =\underset{\text{amplitude}} {\dfrac{F_0}{m({\omega_0}^2 - {\omega}^2)}} \cos\omega t.$$ As we see, the amplitude becomes maximum when $$\underset{\mbox{driven frequency}}{\omega} = \underset{\mbox{natural frequency}} {\omega_0}.$$  Coming now to the forced oscillator with damping, in order to maintain constant amplitude, the average power-input is given by $$\bar{P}(\omega) = \frac{F_0 ^2 \omega_0}{2kQ} \dfrac{1}{\left(\dfrac{\omega_0}{\omega} - \dfrac{\omega}{\omega_0}\right)^2 + \dfrac{1}{Q^2}}.$$ This also passes through maximum when $\omega = \omega_0$ i.e. at resonance.  So, these formulae tell that during resonance i.e. when the driven frequency is equal to the natural frequency, the amplitude &amp; the power consumed by the oscillator is maximum.    What is actually happening physically that makes the amplitude &amp; the power consumption maximum during resonance?  What is the reason that the oscillator stores maximum energy only when driven at natural frequencies?    I want to know what is happening actually that maximses the power absorbed &amp; amplitude during resonance as is evident from the equations above.
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Why do different wavelength get impeded more or less when in different materials? Moving with the same speed, but a longer physical distance would imply that the fields oscillate less times in the material, but I don't know why a difference in the number of oscillations would impede the wave- I don't even know why things slow down in general. Why some electromagnetic wave would slow down just because it's entering other electromagnetic fields... It would seem to me that the only factor would be time taken to physically move some electron or something in the direction of the fields... But that seems to simple of an explanation to me.
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I have been looking for an answer to 'Why is the laboratory frame energy always greater than the center of mass frame energy during collisions?'.  A lot of resources provided mathematical explanations. I wanted to get some perspectives in terms of physics.  Could you explain this?
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As a start I assumed  which is the reciprocal of  as the frequency upper limit, but not so sure. On the other hand, is there a lower limit? Can we consider the reciprocal of our universe as a theoretical limit which is not yet limited, or just assume 0? I am just curious to know if you have any ideas with thanks.
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Suppose I have two counter-propagating travelling waves of the same frequency but of different amplitude, like the transverse waves produced in a string by partial reflection off an endpoint.  I read that we will get a mixture of standing and traveling waves, and I can see the theoretical part, but what does this mean, exactly? What are the nodes now? Also, can I now say that energy between loops will be non-transmitting? Because now we have both a traveling and a standing wave.  If possible please provide a picture for better visualization.
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I studied electromagnetism and I am currently working with Inductors.  I could not figure out the physical reason based on electromagnetics on why inductor current lags the applied voltage by 90 degrees?  Everywhere I find the Formula as the starting point of explanation which is partially satisfying.    Induced EMF =L di(t)/dt    From the above equation, I understood that when there is a change in current there will EMF induced in the coil terminals. But How this induced EMF makes the current to lag the supply voltage V(t) = Vm*sin(wt)? (Please don't just differentiate and say there is a lag, I am looking for a physical sense on why this happens)  Lets us consider the AC voltage is applied across the terminal at time t=0; At this time there is almost no current. From this transient state how the steady-state is reached where the current is delayed with respect to the voltage?  Can someone help me to figure out why the current lags the Supplied voltage by using electromagnetism, the motion of Charges or any other basic concepts?
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Consider a forced, damped harmonic oscillator  $$\ddot{\phi} + 2\beta \dot{\phi} + \omega_0^2 \phi = j(t) \, .\tag{1}$$  If I pick a sinusoidal driving force $j(t) = A \cos(\Omega t)$, I find  $$\phi(t) = \text{Re} \left[ e^{-i \Omega t} \frac{-A}{\Omega^2 - \omega_0^2 + 2i\beta \Omega} \right] \, .\tag{2}$$  From here, how do I define the ? Is it the point where $\langle \phi(t)^2 \rangle$ is maximized?  Things I do know: The frequency at which $\langle \phi(t)^2 \rangle$ is maximized is $$\omega_r ~:=~ \omega_0 \sqrt{1 - 2(\beta/\omega_0)^2},\tag{3}$$ but I thought I read/heard that the resonance frequency of a damped oscillator is just $\omega_0$.  I also calculated that the free oscillation frequency is $$\omega_{\text{free}} ~:=~  \omega_0 \sqrt{1 - (\beta / \omega_0)^2},\tag{4}$$ but I don't think that's the same thing as the resonance frequency under steady driving.
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Can you explain  to a layman? Could someone explain this famous and important equation with "plain words"? If my question is too broad for an answer, I will also be very thankful for some introductory words.
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This is really bothering me for a long time, because the math is easy to do, but it's still unintuitive for me.  I understand the "law of the lever" and I can do the math and use the torques, or conservation of energy. or whatever... And I can see that a lever can amplify a force you apply to it  if you apply a force on the longer side of the beam.  If I were to look at the molecular lever and see what actually happens when I push on the lever, and I give acceleration to the molecules, how does it actually happen that more force is transmitted to the other side?  Thank you all  p.s I'm looking just for an explanation in terms of forces and acceleration, it's clear to me how to do this in terms of energy or torques
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Consider a string under tension, for example, a string on a guitar. When a guitar string is plucked, it vibrates at a certain frequency. When the tension on the string is increased by twisting the tuning peg at the end of the neck, the frequency of the string increases. According to , increasing the tension on a string also increases the speed of the wave moving through it. According to  the speed of a wave is higher in a more dense medium.  Does this mean that increasing the tension also increases the density?  This does not seem very logical to me, since density is the "compactness" of a substance, or how close together the individual molecules are. By twisting the tuning peg on a guitar, you rotate the cylinder that the string is fixed to, wrapping it around more tightly. This pulls more of the string around that cylinder, similarly to when you reel in a fishing rod. Since more of the string's molecules are now wrapped around the peg, there should be fewer molecules over the same distance between the bridge and the nut (the two ends of the vibrating portion of the string), therefore bringing the density down.
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Bit of a strange question, but what is it? My physics teacher said it was kind of like a "push" that pushes electrons around the circuit. Can I have a more complex explanation? Any help is much appreciated.
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In my physics notes, it has been given that the damping force increases the period of oscillation. I am unable to understand this part. How is this possible? The only relation I know is that as the damping force increases the amplitude decrease. But how does this contribute to increased time period of oscillation?
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So this Youtube video has been around for a while now  . Also this  may be related.  I just wanted to know if there is any standard hydrodynamic terms to explain it. Derek clearly states it is not about Bernoulli principle, as in this case the ball rotation is important.  Has any recent paper/videos discussed what happens if you do the same experiment with hydrophobic spheres? Can it be predicted with laminar flow theories or does it need turbulence?  Too many questions have been posed, but not so many responses on the internet yet.  Edit: I said Derek stated "it wasn't Coandă effect", yet what he says is that it is not "just Bernoulli's principle". Coandă effect is the typical hairdryer ping-pong ball levitation which is explained by Bernoulli's principle.   Edit2: rephrasing.  Additional comment: There seems to be two explanations "water adhere to the sphere and is ejected conserving horizontal momentum" and "Magnus effect" (ME). The later is not so obvious to me as the system is in equilibrium and there are two fluids, water and air, (does ME means air pushes the ball horizontally?).   Edit3: DESCRIPTION To avoid rot links and people that do not want to see the video, I'll try to explain the phenomena. You have a sphere and a water jet. The water jet is vertical (opposite to gravity), and touches the sphere almost tangentially. The ball not only rotates due to the water jet but also stays at constant height from the ground. The angular velocity can be defined in the direction of the cross product between the position of the contact point between the jet and the sphere (origin in the center of the sphere) and the direction of the jet. Some water drops fly away at the top of the sphere away from the jet.  Edit4: New question, would this phenomenon occur if there was no air?
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I'm not looking for how it's derrived. I'm wondering why it works out this way that doubling the speed quadrouples the KE.   Here's my thought process:   A car is moving down the road (ignoring air resistance and any other external force reliant on speed) at 10 m/s. To get to 20 m/s the engine should burn x amount of gas, containing x amount of energy. To do the same acceleration over again it should burn x gas again, resulting in 2x gas burned total and resulting in another 10 m/s increase in speed. This would leave it at 30 m/s.   Say the amount of gas burned to increase speed by 10 m/s is 100 energy units, and the car's mass is 2. Then increasing from 10 to 20 with ke = 1/2 mv2 should increase the KE by 100. Therefore from 20 to 30 it should also be 100, resulting in 200 total energy units and 2x gas burned. But using the equation and plugging in the same mass value with the total acceleration from 10 to 30 (30 -- 10 = 20 and plugging that in for v rather than 10) you get the total ke as 400 units rather than 200, but you theoretically should have still used 2x gas.  If someone could explain to me what the issue is in my reasoning and explain why speed is squared, that would be much appreciated.
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I recently saw this picture posted on  which shows a so-called  rising from an ice cube tray.    I have read the Wikipedia page, but it doesn't mean much to me. My instinct was that it is caused by vibrations from the freezer setting up a resonant frequency on the surface of the water, and the spike has gradually "built up" as the water freezes. But I'm not a physicist, so I don't know if this makes sense.  I'm ideally looking for a layman's explanation of why this happens.
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A sound wave is essentially air particles oscillating parallel to the direction of travel of the wave.  We learnt that $v = f\lambda$, where $v$ is the speed of the wave, $f$ is the frequency of the wave and $\lambda$ is the wavelength of the wave.  Suppose the air particles oscillate with a frequency of $f_{particle}$. Is there any relationship between $f_{particle}$ and $v, f, \lambda$ of the entire sound wave?   I thought about this because intuitively, if the air particles vibrated faster, then the wave should travel faster, but I am unable to come up with any formula that describe this relationship (if it is even true!).
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Well, it will be a broad schematic. To summarize, there my approach is:   Yellow part is for oscilations at three different freqs(around 0.8 - 2 kHz, different for each of the three signals), sinusoidal, I can get nice 6 Vpeak sine waves, observed with oscilloscope, no noise, smooth. In the orange part, I use classical potentiometers to change the amplitudes of the signals. For this part, I suspect from the potentiometers, I know they are very unstable but I somehow have to use them. Then, red part, adds the signals together, nothing fancy. Finaly, the green part is my design to drive an 8 ohms speaker.   Now, when I connect the speaker, I get a noticable noise on the speaker output. It is both understandable from sound and also checked by oscilloscope. I have heatsinks for transistors, and I suspected from them for the reason I explained in my .     when I inspect the signals in the circuit via oscilloscope, I see there is some noise on waveforms, while trying to understand where it comes from, I have touched the heatsink (...) It becomes much smooter.   However, then, I thought it might be resulting from something else and wanted to get your opinions about it. What can I do to minimize the noise ? Would soldering things on something like  solve things?  Things to note:   I got satisfying results in simulations, everything was as I wanted. However the reality spoiled me. I implement the circuit on a breadboard for now, while prototyping, I suspect it may be bad in terms of noise. I don't know it is relevant or not but the noisy parts are mostly in the upper parts of sinusoidal waves.   Thanks in advance.