Simple harmonic motion: why cant you divide by cos?

[connor415]

The displacement, x= Acos($$\varpi$$t+$$\phi$$), repeats for every increase in 2pi. Why can't I make the above equal to Acos($$\varpi$$t+$$\phi$$ + 2pi), and divide by cos. This gives $$\phi$$]=0. This is clearly wrong. Why?

And how does angular frequency apply to a mass on a spring anyways, it doesn't move in a circle.

 

[connor415]

no sorry it gives 2pi=0.

 

[connor415]

no sorry it gives 2pi=0.

 

[Hootenanny]

The displacement, x= Acos($$\varpi$$t+$$\phi$$), repeats for every increase in 2pi. Why can't I make the above equal to Acos($$\varpi$$t+$$\phi$$ + 2pi), and divide by cos. This gives $$\phi$$]=0. This is clearly wrong. Why?

no sorry it gives 2pi=0.

Well firstly, it's never a good idea to divide by a trigonometric function (unless you restrict the domain) since you are dividing by a function that is sometimes zero. I don't see why you would want to divide by cosine in any case since doing so would yield

$$\frac{\cos\left(\omega t + \phi\right)}{\cos\left(\omega t + \phi + 2\pi\right)} = 1$$

Which doesn't help you at all. I think that your getting a little confused with the maths.

And how does angular frequency apply to a mass on a spring anyways, it doesn't move in a circle.

Angular frequency isn't simply restricted to circular motion, one can define an angular frequency for any periodic motion. Angular frequency is defined as the product of the frequency and 2$\pi$, so if a system has a frequency, one can define and angular frequency.

 

[connor415]

Thank you very much Hootenanny! Ok well when i divided by cos, I got the top bracket equals the bottom bracket, this simplifies to zero. My mistake is dividing by cos. I don't understand why the fact that it is sometimes zero means that I can't divide by it?

I understand that angular frequency=2pi times f. Why is it called angular then? Is it just a theoretical quantity?

 

[Hootenanny]

Thank you very much Hootenanny! Ok well when i divided by cos, I got the top bracket equals the bottom bracket, this simplifies to zero.

That is not true. For example, if we have two functions $f\left(x\right)$ and $g\left(x\right)$:

$$\frac{f\left(x\right)}{g\left(x\right)} \neq \frac{x}{x}$$

Specifically,

$$\frac{\cos\theta}{\cos\phi} \neq \frac{\theta}{\phi}$$

Or for a numerical example:

$$\frac{\cos\left(2\pi\right)}{\cos\left(\pi\right)} = \frac{1}{-1} = -1 \neq \frac{2\pi}{\pi} = 2$$

Do you see your mistake now?

My mistake is dividing by cos. I don't understand why the fact that it is sometimes zero means that I can't divide by it?

What is one divided by zero?

I understand that angular frequency=2pi times f. Why is it called angular then? Is it just a theoretical quantity?

Angular frequency is no more a theoretical quantity than frequency. Angular frequency is so called because gives the frequency with which phase changes.

 

[connor415]

ah ok. I see my schoolboy errors now. Been a while ha