Achintya
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This was not hard to find:Achintya said:Summary:: While coming across the general equation of SHM i wondered as to how the concept of Sin fn came into this equation...and how have we replaced k/m with (w^2). ?
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.PeroK said:
It's not a guess, it's just shorthand notation.Achintya said:.
but sir my question is on what basis are we assuming this fact.
how would someone guess this...![]()
PeroK said:It's not a guess, it's just shorthand notation.
Sir that is an example of SHM, so that will definitely follow the general equation of the SHMetotheipi said:What's the equation of motion for a block on the end of a horizontal spring?
Achintya said:can you be more clear...because i read many textbooks and i couldn't find the reason behind this assumption...we just assume it out of no where...
It's not an assumption, it's a shorthand notation. Physics texts often write:Achintya said:can you be more clear...because i read many textbooks and i couldn't find the reason behind this assumption...we just assume it out of no where...
sir i have seen the derivation but that does not contain the omega term unless and until we take this assumptionetotheipi said:It's not assumed, it's derived. Write down the full ##F_x = m\ddot{x}## relation for the spring and see where it gets you!
It's not an assumption.Achintya said:just check this out ...the solution of the differential equaltion came in the form of k/m...but it is only after the assumption that k/m=w^2 that we can see the w term in shm general equation
Achintya said:just check this out ...the solution of the differential equaltion came in the form of k/m...but it is only after the assumption that k/m=w^2 that we can see the w term in shm general equation
but sir this is something we know that shm is a one dimensional motion, so i think it makes sense to write F=-kx..since it was observed experimentally that the force is varying linearly with displacement..PeroK said:Let me try to answer this a different way. Suppose you were telling me about SHM and you wrote down your equation:
$$F = -kx$$
And I asked: "why are you assuming that the motion is in the x-direction?" What happens if the motion is not in the x-direction? Then your equation and everything you are doing is invalid.
Then, I might ask, "why are you assuming the spring constant is ##k##"? What happens if the spring constant is ##2k##? Then all your equations don't work.
How would you answer that?
What happens if motion is in the ##y## direction? How do you know it's in the x-direction?Achintya said:but sir this is something we know that shm is a one dimensional motion, so i think it makes sense to write F=-kx..since it was observed experimentally that the force is varying linearly with displacement..
it hardly makes a difference even if the motion were in the y-direction...what we are sure about is that its a one dimensional motion...isn't it?PeroK said:What happens if motion is in the ##y## direction? How do you know it's in the x-direction?
Exactly, the motion could be in the y-direction. I want you to justify your assumption that motion is in the x-direction. Why use ##x## instead of ##y##?Achintya said:it hardly makes a difference even if the motion were in the y-direction...what we are sure about is that its a one dimensional motion...isn't it?
so what's the conclusion?PeroK said:Exactly, the motion could be in the y-direction. I want you to justify your assumption that motion is in the x-direction. Why use ##x## instead of ##y##?
The answer is that it's not an assumption, in the sense of something to assume without justification. The motion is assumed to be one-dimensional. It's not an additional assumption that the motion is in the x-direction. We take ##x## to be the direction of motion.Achintya said:so what's the conclusion?
PeroK said:The answer is that it's not an assumption, in the sense of something to assume without justification. The motion is assumed to be one-dimensional. It's not an additional assumption that the motion is in the x-direction. We take ##x## to be the direction of motion.
Likewise, we take the quantity ##\omega## to be the quantity ##\sqrt{k/m}##. This is not an assumption.
It's so fundamental to mathematical physics that you can replace a complicated expression with a single letter that it's difficult to explain.
Anyway, if you don't like ##\omega##, then whenever you see ##\omega## just replace it with ##\sqrt{k/m}##. They are equivalent ways of writing the same thing.
Achintya said:Sir actually the source of confusion here was that the omega already has some physical significance of its own...its not an arbitrary variable...so u see this gives a wrong impression
Wow..nice .i understand it nowGaussian97 said:Ok let's see, if you derive the equation of motion you get
$$\ddot{x}=-\frac{k}{m}x$$
wich has as general solution:
$$x(t)=A\sin{\left(\sqrt{\frac{k}{m}} t + \phi_0\right)}$$
Ok until here? Let's find now the period ##T##, that is defined as the smallest non-zero time that fulfils ##x(t)=x(t+T)## for ANY value of ##t## so, by definition
$$x(t+T)=A\sin{\left(\sqrt{\frac{k}{m}} t+ \sqrt{\frac{k}{m}} T + \phi_0\right)}=A\sin{\left(\sqrt{\frac{k}{m}} t + \phi_0\right)}$$
$$\sin{\left(\sqrt{\frac{k}{m}} t+ \sqrt{\frac{k}{m}} T + \phi_0\right)}-\sin{\left(\sqrt{\frac{k}{m}} t + \phi_0\right)}=0$$
Applying some trigonometric identities
$$2\sin{\left(\sqrt{\frac{k}{m}} \frac{T}{2}\right)}\cos{\left(\sqrt{\frac{k}{m}} t+ \sqrt{\frac{k}{m}} \frac{T}{2} + \phi_0\right)}=0$$
The only way this can be true for ANY ##t## is
$$\sin{\left(\sqrt{\frac{k}{m}} \frac{T}{2}\right)}=0 \Longrightarrow \sqrt{\frac{k}{m}} \frac{T}{2}=k\pi\Longrightarrow T = 2k\pi \sqrt{\frac{m}{k}}$$
Because we want the smalles non-zero time
$$T = 2\pi \sqrt{\frac{m}{k}}$$
This is the period, then the angular frequency is defined as
$$\omega = \frac{2\pi}{T}=\sqrt{\frac{k}{m}}$$