... which of course raises the question of "But why sine waves in particular?"
There are infinitely many possible pairs of curves that trace out a circle, and it's not obvious why the angle should increase linearly when the force from the spring isn't constant.
But, it's not that hard to show why this happens: (still assuming k and m are 1 for simplicity)
On our circle, the x-axis is velocity, which is how fast the displacement changes, or how fast the system is moving along the y-axis.
The y-axis, displacement, per F=k*d, is the acceleration, or how fast the velocity changes and how fast the system is moving along the x-axis.
From here, we can apply some Pythagoras to find that the rate the system moves away from a point is sqrt(x^2 + y^2), which is the same as the distance of that point from the center of the circle. But on a circle, all points are the same distance from the center. Therefore, the angular velocity must be constant -- yielding sinusoids.
The math-y explanation is that "the derivative/integral of a sinusoid is a sinusoid", so it arises when a variable is linearly, but indirectly effecting itself. Like how position affects spring force (linear), which affects velocity acceleration (integral), which comes back to position (integral). If you want to be precise, each integration/differentiation introduces a 90° phase shift, and the reverse direction of the spring's force introduces a 180° shift: 180° + 90° + 90° = 360°, so the overall effect is to keep the weight moving on a sinusoid.
The math graphics erroneously render d^2+v^2 as "d^2v^2" implying multiplication where there should be summation, starting somewhere around total energy summation
Thanks! Really weird, it's not reproducing for me on mobile. It looks like Substack rendered the LaTeX stuff on the server for you, and I can definitely reproduce *that*, but the images then render normally for mr:
Another way to think about this is that the solution of basic first order differential equations is always an exponential function (with complex or real argument). Thus we see exponential growth, exponential decay and harmonic oscillation everywhere, often in combination. y = A * exp(B * x) + C
There's a simple reason I'm trying not to do that: even among geeky audiences, I think that most people have not done a whole lot of calculus, and if they did, they don't remember most of it after 10+ years of non-use.
Further, even for folks who do remember it, they often no longer remember how the rules were derived... if they were even taught that first place. So it's kinda intellectually unsatisfying to say "this <physical thing> is explained by the Laplace transform" or something like that.
Don't get me wrong, it's definitely an expedient way to do this if you're teaching a college course and all the students have taken the prerequisite calculus a year earlier, but there's no shortage of articles targeted at non-student audiences that still read that way.
Yeah, I totally agree that different ways to approach the question make sense to different people. On the other hand, my mother did remember some calculus, but had never realized the connection to decaying sine waves.
Can I be the 93rd reader to question why gravity does not come in to the equations? The value of g and k set the neutral stationary length of the spring, but is the work done m*g*h changing the height of the block not also a factor?
The work done by the spring accelerating the block is done against g upwards, and releasing this energy again on the way down. So it seems odd that the max velocity of the block would be symmetric in each direction? It would seem that the minimum and maximum displacements would not be an equal distance from the line of zero tension.
If oscillating in the horizontal plane on a friction-less surface , g obviously plays no part and we have the math given.
This was probably worth covering - I asked for it by drawing the setup vertically and not horizontally.
But the TL;DR is that gravity is a constant force acting on the mass, and including it in the equations just shifts the spring's equilibrium point - the "neutral" location is now no longer "the spring exerts no force", but "the spring is extended a bit to exert force equal to gravity". Everything else remains symmetrical, including spring deflection / amplitude.
... which of course raises the question of "But why sine waves in particular?"
There are infinitely many possible pairs of curves that trace out a circle, and it's not obvious why the angle should increase linearly when the force from the spring isn't constant.
But, it's not that hard to show why this happens: (still assuming k and m are 1 for simplicity)
On our circle, the x-axis is velocity, which is how fast the displacement changes, or how fast the system is moving along the y-axis.
The y-axis, displacement, per F=k*d, is the acceleration, or how fast the velocity changes and how fast the system is moving along the x-axis.
From here, we can apply some Pythagoras to find that the rate the system moves away from a point is sqrt(x^2 + y^2), which is the same as the distance of that point from the center of the circle. But on a circle, all points are the same distance from the center. Therefore, the angular velocity must be constant -- yielding sinusoids.
The math-y explanation is that "the derivative/integral of a sinusoid is a sinusoid", so it arises when a variable is linearly, but indirectly effecting itself. Like how position affects spring force (linear), which affects velocity acceleration (integral), which comes back to position (integral). If you want to be precise, each integration/differentiation introduces a 90° phase shift, and the reverse direction of the spring's force introduces a 180° shift: 180° + 90° + 90° = 360°, so the overall effect is to keep the weight moving on a sinusoid.
There's a nice geometric explanation of the calculus in one of the electronics articles: https://lcamtuf.substack.com/p/primer-core-concepts-in-electronic
I incorporated your proof as an addendum and made an illustration. Thanks!
The math graphics erroneously render d^2+v^2 as "d^2v^2" implying multiplication where there should be summation, starting somewhere around total energy summation
https://pasteboard.co/OOl0KSc5Cpk8.png
https://pasteboard.co/pDLNVmTDWVFa.png
Thanks! Really weird, it's not reproducing for me on mobile. It looks like Substack rendered the LaTeX stuff on the server for you, and I can definitely reproduce *that*, but the images then render normally for mr:
https://substackcdn.com/image/fetch/f_auto,q_auto:best,fl_progressive:steep/https%3A%2F%2Fsubstack.com%2Fapi%2Fv1%2Flatex%2Fjpeg%3Fexpression%3Dd%255E2%2520%252B%2520v%255E2%2520%253D%2520%255Ctextrm%257Bconst%257D%26version%3D9
Hrmpf...
If you double-urldecode the url you have sent (without the cdn part) you get this:
https://substack.com/api/v1/latex/jpeg?expression=d^2 + v^2 = \textrm{const}&version=9
This renders the wrong jpeg, probably because the parser sees "+" as a url encoded empty string token.
So there might be an extra url decode step somewhere that is done from the ios substack app and not web app that you are using
As a url the above results in this url (not sure if converted by ios or substack server):
https://substack.com/api/v1/latex/jpeg?expression=d%5E2%20+%20v%5E2%20=%20%5Ctextrm%7Bconst%7D&version=9
As you can see, the plus is kept as "+", which urldecodes to en empty string
Ah, clever! Yeah, that's probably it. I'm increasingly annoyed with Substack for all the bugs they have around formula rendering specifically...
I'm not sure I'm seeing it - can you send me a screenshot?
Another way to think about this is that the solution of basic first order differential equations is always an exponential function (with complex or real argument). Thus we see exponential growth, exponential decay and harmonic oscillation everywhere, often in combination. y = A * exp(B * x) + C
There's a simple reason I'm trying not to do that: even among geeky audiences, I think that most people have not done a whole lot of calculus, and if they did, they don't remember most of it after 10+ years of non-use.
Further, even for folks who do remember it, they often no longer remember how the rules were derived... if they were even taught that first place. So it's kinda intellectually unsatisfying to say "this <physical thing> is explained by the Laplace transform" or something like that.
Don't get me wrong, it's definitely an expedient way to do this if you're teaching a college course and all the students have taken the prerequisite calculus a year earlier, but there's no shortage of articles targeted at non-student audiences that still read that way.
When I'm sneaking in elements of calculus, it's usually by explaining it from first principles, as in the case for the derivation of capacitive reactance here: https://lcamtuf.substack.com/p/primer-core-concepts-in-electronic
Yeah, I totally agree that different ways to approach the question make sense to different people. On the other hand, my mother did remember some calculus, but had never realized the connection to decaying sine waves.
Can I be the 93rd reader to question why gravity does not come in to the equations? The value of g and k set the neutral stationary length of the spring, but is the work done m*g*h changing the height of the block not also a factor?
The work done by the spring accelerating the block is done against g upwards, and releasing this energy again on the way down. So it seems odd that the max velocity of the block would be symmetric in each direction? It would seem that the minimum and maximum displacements would not be an equal distance from the line of zero tension.
If oscillating in the horizontal plane on a friction-less surface , g obviously plays no part and we have the math given.
This was probably worth covering - I asked for it by drawing the setup vertically and not horizontally.
But the TL;DR is that gravity is a constant force acting on the mass, and including it in the equations just shifts the spring's equilibrium point - the "neutral" location is now no longer "the spring exerts no force", but "the spring is extended a bit to exert force equal to gravity". Everything else remains symmetrical, including spring deflection / amplitude.