Simple harmonic motion: why cant you divide by cos?

In summary, the displacement x= Acos(\varpit+\phi) repeats for every increase in 2pi. However, trying to make it equal to Acos(\varpit+\phi + 2pi) and dividing by cos is not a valid operation, as dividing by a trigonometric function can lead to undefined results. Angular frequency is not limited to circular motion, but can be defined for any periodic motion. It is called angular because it gives the frequency with which phase changes.
  • #1
connor415
24
0
The displacement, x= Acos([tex]\varpi[/tex]t+[tex]\phi[/tex]), repeats for every increase in 2pi. Why can't I make the above equal to Acos([tex]\varpi[/tex]t+[tex]\phi[/tex] + 2pi), and divide by cos. This gives [tex]\phi[/tex]]=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.
 
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  • #2
no sorry it gives 2pi=0.
 
  • #3
no sorry it gives 2pi=0.
 
  • #4
connor415 said:
The displacement, x= Acos([tex]\varpi[/tex]t+[tex]\phi[/tex]), repeats for every increase in 2pi. Why can't I make the above equal to Acos([tex]\varpi[/tex]t+[tex]\phi[/tex] + 2pi), and divide by cos. This gives [tex]\phi[/tex]]=0. This is clearly wrong. Why?
connor415 said:
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

[tex]\frac{\cos\left(\omega t + \phi\right)}{\cos\left(\omega t + \phi + 2\pi\right)} = 1[/tex]

Which doesn't help you at all. I think that your getting a little confused with the maths.
connor415 said:
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[itex]\pi[/itex], so if a system has a frequency, one can define and angular frequency.
 
  • #5
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?
 
  • #6
connor415 said:
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 [itex]f\left(x\right)[/itex] and [itex]g\left(x\right)[/itex]:

[tex]\frac{f\left(x\right)}{g\left(x\right)} \neq \frac{x}{x}[/tex]

Specifically,

[tex]\frac{\cos\theta}{\cos\phi} \neq \frac{\theta}{\phi}[/tex]

Or for a numerical example:

[tex]\frac{\cos\left(2\pi\right)}{\cos\left(\pi\right)} = \frac{1}{-1} = -1 \neq \frac{2\pi}{\pi} = 2[/tex]

Do you see your mistake now?
connor415 said:
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?
connor415 said:
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.
 
  • #7
ah ok. I see my schoolboy errors now. Been a while ha
 

Related to Simple harmonic motion: why cant you divide by cos?

1. What is simple harmonic motion?

Simple harmonic motion is a type of periodic motion, where a body oscillates back and forth around a fixed equilibrium point. It occurs when the net force acting on the body is directly proportional to its displacement from the equilibrium point and is directed towards the equilibrium point.

2. Why is it important to understand simple harmonic motion?

Simple harmonic motion is a fundamental concept in physics and is used to explain the behavior of many systems, including pendulums, springs, and sound waves. It also has applications in fields such as engineering, astronomy, and music.

3. Can you divide by cos in simple harmonic motion equations?

No, you cannot divide by cos in simple harmonic motion equations. This is because the cosine function is used to represent the displacement of the body from the equilibrium point, which is not a constant value. Dividing by a variable value would result in an incorrect solution.

4. Are there any exceptions to this rule?

While you cannot divide by cos in simple harmonic motion equations, there are some cases where it may be used in a different form. For example, in the equation for kinetic energy in simple harmonic motion, the cosine function is squared, so dividing by cos would be equivalent to dividing by its squared value, which is a constant.

5. What other mathematical operations should be avoided in simple harmonic motion equations?

In addition to dividing by cos, it is also important to avoid taking the square root of negative values, as this would result in imaginary solutions. It is also important to use the correct trigonometric functions, as using the wrong one can lead to incorrect solutions.

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