... 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.
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.
this is a weird way to think about it. Displacement and velocity are not independent, so there are still many ways how you could trace a circle over time.
However, if (d^2+v^2) is constant over time, then the time-derivative is 0, ie. d * d' = - v * v'. Since d' = v, you get d = -d''. From there, you get to sines through differential equations or taylor series expansion.
... 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!
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.
this is a weird way to think about it. Displacement and velocity are not independent, so there are still many ways how you could trace a circle over time.
However, if (d^2+v^2) is constant over time, then the time-derivative is 0, ie. d * d' = - v * v'. Since d' = v, you get d = -d''. From there, you get to sines through differential equations or taylor series expansion.
I think your first point is fair, so I added a section at the end explaining in simple terms.
As to why I'm trying *not* to lean on prior knowledge of calculus, it's basically this: https://lcamtuf.substack.com/p/why-are-sine-waves-so-common/comment/99637703
Just FYI, when I view this in the Substack iOS app, some of the equations are missing plus signs. Looks fine on the web.