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I tried searching for proofs here and there but couldn't find.. Some people used $a= w^2 x$ to prove this.. But how do you prove that?

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    $\begingroup$ The picture is not legible. Please write out specifically what your question is, and use MathJax. $\endgroup$ – kaylimekay Jan 9 at 15:00
  • $\begingroup$ My question is what's the derivation of k = mw^2 or what's the derivation of a = w^2x $\endgroup$ – Yash Gabra Jan 9 at 15:10
  • $\begingroup$ And I thought that $g=\ell \omega^2$. :) $\endgroup$ – Bill N Jan 9 at 16:03
  • $\begingroup$ @yashgabra Those are two different questions. Plus you haven't given a specific context. We can read between the lines, but that's not a good way to pose a question. You should state what the specific physical system is. Also, the character is omega, $\omega$ not $w$. $\endgroup$ – Bill N Jan 9 at 16:06
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Newton's second law states the force on a system is proportional to its acceleration. For linear restoring force $-kx$, we have

$$F = ma$$ $$ -kx = m\ddot{x}$$

where $\ddot{x}$ is the second time-derivative of position, i.e. acceleration. Then you have

$$ \ddot{x} = a = -\frac{k}{m}x = -\omega^2 x$$

where we define $\omega = \sqrt{\frac{k}{m}}$.

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  • $\begingroup$ Is omega another constant in this and not angular frequency? $\endgroup$ – Yash Gabra Jan 9 at 15:27
  • $\begingroup$ $\omega$ is the angular frequency, but for a simple harmonic oscillator, that happens to be a constant value. $\endgroup$ – zhutchens1 Jan 9 at 15:29
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In Zhutchens1's force equation, the second derivative of x is a negative constant times x. The simplest solution for this differential equation is x = sin(ωt + φ) where ω is the angular frequency and φ depends on when you start the clock. If you put this function for x into the equation, you find that $ω^2$ = k/m.

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I think what is bothering you is how did we end here: $$a=-\omega^2 x$$ form here: $$a=-(k/m) x$$

This is matter more of Mathematics than Physics. In mathematics, the standard solution of the below differential equation: $$\frac{d^2 y}{dx^2}= - p^2 y\tag1$$ is this: $$y= C_1\sin(px + C_2)$$where $C_1$ and $C_2$ are arbitrary constants.

In Physics, so we assign the value $\omega=k/m$ on our own "arbitrarily". And further obtain the solution: $$y= A \sin(\omega t + \phi)$$ where A and $\phi$ are arbitrary constants. It just so happens that we physically interpret A as amplitude and $\phi$ as initial phase.


Now if you want the solution of equation (1) , that was nothing to do with Physics. Try referring to a Mathematics text for the same.

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    $\begingroup$ Having two $k$'s, with $k=mk^2$, is not great for clarity $\endgroup$ – fqq Jan 9 at 15:42
  • $\begingroup$ It's not an arbitrary assignment. In fact, what you call $k^2$ is, in the careful physics analysis, $k/m$ for a spring-mass oscillator system. $\endgroup$ – Bill N Jan 9 at 16:00
  • $\begingroup$ @BillN I used the word "arbitrarily" in quotes for the same purpose. Are you suggesting me to edit my terminology? $\endgroup$ – Tony Stark Jan 9 at 17:32
  • $\begingroup$ @fqq Point noted. $\endgroup$ – Tony Stark Jan 9 at 17:35
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    $\begingroup$ Please don't mess up with variables, try to make it lucid. Solve the differential equation to show the asker how it's derived without just writing it as a standard solution. $\endgroup$ – Roger Michealson Jan 9 at 17:40

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