I had studied the damped forced vibration and had come to know that the angular speed of force which creates resonance is   Which is counter-intuitive to me that the angular frequency should match with the damped frequency so that the force is in phase with the motion.  Therefore I am confused of the reasons behind this, I know how to differentiate a function, yet I cannot come up with a reasonable interpretation.
123244
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.
54091
I struggle to find a satisfying reason and formulation, why for a point mass,  that is fixed in a rotational system and the radius is changed (like pulling on a string with a small ball rotating around it), the speed of this point mass would increase due to the conservation of angular momentum.  that follows a trajectory with decreasing radius (like a Roller Coaster on a Spiral shaped rail), the speed of this point mass would not increase, since conservation of angular momentum is not applicable in this case. (I already started doubting, if it really does not increase).   The only reason I found so far is that in the second case, the rail acts as an external force to the mass, while in the first case the radius change is an internal change of the system that comes along with putting energy into the system by working against the centrifugal forces. While I expect this is a somewhat correct answer, I still find this is a little bit hand waving argument. I expect that I just miss a proper clean definition of external force vs internal force. I also constructed the following thought experiment for myself, that involves an external rail, but I would expect to react like the first case: When a point mass rotates on a circular stretchable rail and the radius of that rail would be changed by applying some force against the centrifugal force of the point mass, I'd also expect the velocity of the point mass increases, since some force applied over a distance (energy change) is involved. But I doubt that the applied force over a distance is the underlying fundamental reason and I expect that there is another more satisfying fundamental difference between the fixed predetermined trajectory and the thought experiment / the first case mentioned in the beginning.
10312
Suppose we have a Hamiltonian that depends on various real parameters.  When tuning the values of these parameters, the energy eigenvalues will often avoid crossing each other.  Why?  Is there a physically intuitive justification for  and ?  It would be nice to see a general argument.
14637
While trying to understand the second law of Newton from "An Introduction to Mechanics" by Kleppner and Kolenkow, I came across the following lines that I don't understand:      "It is natural to assume that for three-dimensional motion, force, like acceleration, behaves like a vector. Although this turns out to be the case, it is not obviously true. For instance, if mass were different in different directions, acceleration would not be parallel to force and force and acceleration could not be related by a simple vector equation. Although the concept of mass having different values in different directions might sound absurd, it is not impossible. In fact, physicists have carried out very sensitive tests on this hypothesis, without finding any variation. So, we can treat mass as a scalar, i.e. a simple number, and write $\vec{F} = m\vec{a}$."   The lines above lead me to question:   Why is it not" obviously true" that force behaves like a vector?  Why is it not impossible for mass values to be different in different directions?
69621
Let me start by explaining my particle physics background is very patchy, so this question may not be as coherent as  I would like it to be.   In general terms, what is the difference between a resonance found during a particle experiment and a particle?   From reading Wikipedia it seems to be mostly based on timescales. If a heavy particle decays quickly, how can we distinguish it from a resonance?  I would not be posting this question except for the fact that the Wikipedia page says "This page has some issues".        and I would be interested in knowing more about the subject.
24646
In this equation:  $$ \mathbf a_i\overset{\rm def}{=}\left(\frac{d^2\mathbf r}{dt^2}\right)_i=\left(\frac{d\mathbf v}{dt}\right)_i=\left[\left(\frac{d}{dt}\right)_r+\boldsymbol\Omega\times\right]\left[\left(\frac{d\mathbf r}{dt}\right)_r+\boldsymbol\Omega\times\mathbf r\right] $$ (), why is  $$\left(\frac{d}{dt}\right)_r \boldsymbol{\Omega} \times \mathbf{r}=\frac{d\boldsymbol{\Omega}}{dt}\times\bf{r}+\bf{\Omega}\times \bf{V_r}$$  In particular, I have qualms with the term  $$\frac{d\bf{\Omega}}{dt}\times\bf{r}$$  Why are we deriving the angular velocity? Why is it a derivative not of the rotational type (Namely $(\frac{d}{dt})_r$ )  Other sources do not point out that term I have problems with. In any case, I want to know how you evaluate that derivative.
76715
So according to me in this situation there are two forces acting on A at the moment, one is the reactive force of 70 N by newton's third law and the other is the force of 60 N exerted by B. Now don't you think the tension should be the sum of the two pulling forces experienced by A (i.e. 70+60=130 N)...Please tell me where my thinking process is going wrong.
92255
Referring to unidimensional motion, it is obvious that it doesn't always make sense to write the speed as a function of position. Seems to me that this is a necessary condition to derive formulas like:  $$v^2=v_0 ^2 +2\int_{x_0}^{x}a\cdot dx$$  In fact, in the first step of the demonstration (the one I saw, but I think that this step is crucial) it's required to write $a=dv/dt=(dv/dx)(dx/dt)$, that doesn't make sense if $v$ isn't a function of $x$.   When can one rigorously write $v=v(x)$?
33453
For a constant force, $P=Fv$. I understand the mathematical derivation of this, but this seems to me, intuitively, to be nonsense. I feel that my discomfort with this comes from a fundamental misunderstanding of force and Newton's Second Law, so I'm not really looking for any mathematical explanation. So, to begin:  How is it that a constant force does not add energy to a system at a fixed rate? Consider a rocket burning a fuel at a constant rate. The chemical potential energy should be converted to kinetic energy at a constant rate, that is, $(1/2)mv^2$ should be increase linearly. The magnitude of the velocity of the rocket would then increase at a less than linear rate, implying a nonconstant acceleration and therefore, a nonconstant force/thrust (F=ma).  If force is indeed a "push or a pull," shouldn't that constant rate of burning of fuel yield a constant "push or pull" as well? Clearly not, so I would have to think that, somehow, a given force applied to a certain object at rest would in some way be different than that a force of the same magnitude being applied to that same object in motion. In this sense, is force merely a mathematical construct? What does it tangibly mean, in physical terms? Would a given force acting upon me "feel" differently to me (in terms of tug) as I am moving at differing velocities?  Force being defined as a "push or pull," which is how it has been taught in my high school class, seems rather "handwavy," and maybe that's the issue. It's been troubling me for a couple of weeks and my teacher hasn't really been able to help, so thanks!
66433
Why does a body rotate faster if force is applied away from the pivot? I need an intuitive answer to this. Like why does a door rotate slower if I push at it closer to its hinge, or faster if I push at it farther away from its hinge? What actually makes that happen?
22550
While reviewing Newton's laws of motion I came across the statement which says Newton's laws exist only in inertial reference frames except the third one. Why is it like that?
6243
I've seen a  here on the Magnus force, which explains why balls with a spin will curve.  However, my intuition is still telling me that the Bernoulli's principle would push it the opposite way and I need help understanding why my reasoning is flawed.  Imagine that you kick a soccer ball on the left side so that it's spinning clockwise (viewing the ball from above) and the ball will curve to the right.  Since the left side of the ball is spinning against the air, wouldn't this mean faster relative speeds and thus a lower pressure than the right side?  And wouldn't this lower pressure on the left side cause it to curve left instead of right?
101700
Lets say an object is tied to a string and it is being swung in a circle:    In this case, the tension of the string acts as a centrifugal force that keeps the object moving in a circle. However, if centrifugal force is the only force acting on the object, why doesn't it collapse into the center? I am pretty sure this has to do with the object's inertia (or the centrifugal pseudo-force), where it moves slightly forward in a tangent before falling, restoring the orbit.  Also, what keeps the string taut during the swinging? There has to be some force countering the centripetal force that keeps the string tight. I think this might again have to do with the intertia/centrifugal pseudo-force but can a pseudo-force balance a real force? Also, if centripetal were to be balanced by the pseudo-force, wouldn't the object not move at all (since the net force would be 0)?
9422
When you energise a taut string, the following resonant modes of vibration occur:    Plotting on the frequency domain, you can see their corresponding frequencies:    But what is the underlying physical principle? Why does this happen?  Is there any way of explaining it that could be understood by a smart 15 year old?  EDIT: I'm going to give my best attempt so far. Here goes:   We can start with sympathetic resonance. Sounding a particular frequency, a pure sinewave.  And noticing the string resonates sympathetically at the frequency of each harmonic. Say this is explained and understood. Now imagine that plucking a string is equivalent to a burst of white noise, which contains frequencies all the way through the spectrum. This could be approached backwards, by starting with random frequencies and noticing that resultant wave produced looks like white noise.   If the above is scientifically correct, then it restricts the domain of the question.  I would really like to be able to understand it scientifically and also be able to explain it intuitively.  PS Images from
53833
When I got more into physics, I started asking myself if just like acceleration represents the growth of speed, something else could also represent the growth of acceleration itself. And it came that is exists and is called jerk. Before I thought about this, I thought that acceleration was the highest derivative that made sense on physics.  My question is, why can we perform accurate calculations without account for jerk? I never needed it in a single physics class! I've also read that it is seldom used.  Also, while position, speed and acceleration seem familiar to me due to everyday experience, why doesn't jerk seem familiar to me? I never tell someone when driving a car "you're jerking too much", whereas I could say "you are accelerating too much".  Is it that jerk has little influence because it is almost constant? Even if it is not, what about higher order derivatives? Why don't derivatives higher than acceleration have much influence on calculations? We perform a lot of reasoning in everyday life without resorting to it!
72400
A cylinder at rest lying on a rough ground is given an impulse which imparts a translational velocity (no angular velocity) to it. The question goes ahead with finding time after which rolling starts etc. etc.  But I have another question: The friction from the ground will decrease the translational velocity but increase the angular velocity (as the torque of friction will be in the direction of rolling). After some time, the translational velocity will become equal to angular velocity and hence pure rolling will start.  After this time work done by friction will be 0 as the point of contact is not moving and hence translational velocity will be constant. Now my doubt is that, when translational velocity is constant, it implies angular velocity will also be constant (as translational velocity is a multiple of angular velocity). But the torque from friction (about the centre of mass) will not be zero which means angular velocity will not be constant. In some book where there was the same problem it was written friction will vanish. How can friction vanish? How can anguluar velocity be constant even when there is a torque by friction?
8610
When dragging an object, there is a greater start-up force than the force it takes to keep it moving. Why is this? Why are there two different values for friction?