Simple Harmonic Motion- From Uniform Circular Motion

[chantalprince]

Homework Statement

I don't have a homework question exactly, but I need help with an equation please.


Angular frequency: W= 2 pi/T = 2(pi)(f) f= frequency

And- W = square root of (k/m) k = spring constant m= mass

So, wouldn't T = 2 pi / square root of (k/m) ??



My instructor has given us the following equations in class a few times. I cannot figure out what the heck is going on!

W = (2 pi) x the square root of(k/m)
T = (2 pi) x the square root of (m/k) -------> m/k this time

Any help is appreciated. I am so confused right now.

Homework Equations

The Attempt at a Solution

 

[Kurdt]

This is simply because of the fact that:

$$\frac{1}{\frac{a}{b}}=\frac{b}{a}$$

 

[chantalprince]

Ok- but in the book it gives: W = sq. root of (k/m)

Instructor gives: W = 2pi x sq. root (k/m)

Whats with the 2 pi??

Thanks-

 

[timmay]

What's the difference between measuring angular frequency in Hz and measuring frequency in radians per second? That might give you an idea where the conversion comes from.

 

[chantalprince]

Ok...I'll sit down with that thought. So, either one works right? They are the same thing??

 

[timmay]

Ah I see what you're getting at now. There's a mistake in the equation for frequency:

$$\omega_{n} = \sqrt{\frac {k}{m}}$$ (1)

where frequency is in radians per second.

But what if you want to express the frequency in Hertz? Well, we know that 1 Hz is equal to one cycle per second. In the case of circular motion, one cycle is equal to $$2\pi$$ radians.

So to convert from radians per second to Hertz, one must divide by $$2\pi$$. Hence:

$$\omega = \frac {1}{2\pi} \sqrt{\frac{k}{m}}$$ (2)

where frequency is now in Hertz.

Now let's express this in terms of the period of one cycle, T. Bear in mind that if you were simply to reciprocate the expression for frequency when expressed in radians per second (equation 1), you would be stating the length of time of rotation for one radian alone. Hence you have to multiply the expression by $$2\pi$$ now to obtain the period for a single cycle. This is now the same equation as you would obtain by reciprocating equation 2.

Hope this helps.