# Harmonic Motion and Springs Question

1. Jul 22, 2006

### 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.

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

$$\alpha = \omega t$$

Why does that equation work?
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The angle alpha is replaced by $$\psi t$$ so the result is:

$$y = A sin \omega t$$

The speed in the y direction is as follows:

$$v(t) = \frac {dy} {dt} = \omega A cos \omega t$$

The acceleration in the y direction is:

$$a(t) = \frac {d^2x} {dt^2} = -\omega^2 A sin \omega t$$

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):

$$F = -ky$$

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

$$F = ma = -m \omega^2 A sin \alpha t$$
$$F = -ky = -k A sin \alpha t$$

ie.

$$m \omega^2 = -k \Leftrightarrow \omega = \sqrt{k / m}$$

Combining the above expression with the commonly known

$$\omega = 2 \pi / T$$

and you get the final result

$$T = 2 \pi \sqrt {m / k}$$

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My question is:

Why is $$\alpha = \omega t$$?

Thank you for your time. Have a nice day.

Last edited: Jul 22, 2006
2. Jul 22, 2006

### 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)

3. Jul 22, 2006

### Mattara

So it is basically because

$$\alpha = \omega t$$

Unit:

with the approximation that angular velocity is constant?

4. Jul 22, 2006

### arildno

Yep, that should be it.

5. Jul 22, 2006

### 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).

6. Jul 22, 2006

### Mattara

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

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

7. Jul 22, 2006

### nrqed

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