Harmonic Motion and Springs Question

[Mattara]

I'm trying to derive the formula for the period T as a function of the mass of the object. Here is my attempt. Note that I cheated and passed the section I had trouble with, without fully understanding it.

http://www.filehive.com/files/0722/image.jpg

Much of this is quite straightforward for me.

[tex]y = A sin \alpha[/tex] (basic trigonometry)
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Here is the part I'm not sure I understand fully:

[tex]\alpha = \omega t[/tex]

Why does that equation work?
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The angle alpha is replaced by [tex]\psi t[/tex] so the result is:

[tex]y = A sin \omega t[/tex]

The speed in the y direction is as follows:

[tex]v(t) = \frac {dy} {dt} = \omega A cos \omega t[/tex]

The acceleration in the y direction is:

[tex]a(t) = \frac {d^2x} {dt^2} = -\omega^2 A sin \omega t[/tex]

The above is simple calculus.

When a net force is acting on a body an acceleration will show.

The force that is acting on the body is (via hookes law):

[tex]F = -ky[/tex]

If we replace y with the expression we derived earlier we get

[tex]F = ma = -m \omega^2 A sin \alpha t[/tex]
[tex]F = -ky = -k A sin \alpha t[/tex]

ie.

[tex]m \omega^2 = -k \Leftrightarrow \omega = \sqrt{k / m}[/tex]

Combining the above expression with the commonly known

[tex]\omega = 2 \pi / T[/tex]

and you get the final result

[tex]T = 2 \pi \sqrt {m / k}[/tex]

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My question is:

Why is [tex]\alpha = \omega t[/tex]?

Thank you for your time. Have a nice day.

 

[arildno]

They assume the angular velocity is a constant.
It probably isn't, so they've hidden away an order of magnitude argument that would show it is a good approximation.

(That is, they've hidden away everything that physics is about)

 

[Mattara]

So it is basically because

[tex]\alpha = \omega t[/tex]

Unit:

rad = rad/s x s

with the approximation that angular velocity is constant?

 

[arildno]

Yep, that should be it.

 

[neutrino]

I was just wondering why you used psi instead of omega. Nothing wrong with it, but it's not written that way, usually.

alpaha = omega.t is just the rotational counterpart of s = vt (linear motion with constant velocity).

 

[Mattara]

I was just wondering why you used psi instead of omega. Nothing wrong with it, but it's not written that way, usually.

alpaha = omega.t is just the rotational counterpart of s = vt (linear motion with constant velocity).

Yes, I noticed that, so I changed it. The "how-to-latex" got me confused for a bit before i realized it.

Thank you arildno and neutrino! I really should check the units (as in rad = rad/s x s) more often

 

[nrqed]

They assume the angular velocity is a constant.
It probably isn't, so they've hidden away an order of magnitude argument that would show it is a good approximation.

(That is, they've hidden away everything that physics is about)

The question is not explicitly provided but I was under the impression that that goal was to relate the motion of a mass attached to an ideal spring to circular motion. In that case, using a constant angular velocity is not an approximation or a guess. It follows from the fact that the projection along one of the axis must represent simple harmonic motion. And that implies a constant omega.

Just a comment.

Regards

Patrick