## Simple Harmonic Motion!?

Yes a particle can pursue SHM in a linear fashion. An example would be the individual particles or elements of a stretched string that is vibrating. The individual elements move up and down at right angles to the string with SHM.

One interpretation of a wave along the string is as an assembly of elements coupled so that the SHM of each individual element is transmitted from element to element in an orderly fashion.

The interesting thing about SHM is that the equation of motion is sinusoidal (sin or cos). Now both the differential and integral of a sinusoid is another sinusoid.
So the displacement /time graph is sinusoidal
The velocity time graph is sinusoidal (differentiate once)
The acceleration/time graph is sinusoidal (differentiate twice)

So a particle moving along a straight line with SHM shows a sinusoidal variation of velocity (and acceleration). That is it starts with some velocity, slos down to zero, reverses and speed up in the other direction - like the pendulum but along a line not an arc.
A machine that does this is called reciprocating it turns circular motion into linear and vice versa. A simple example is the scotch yoke.

I think perhaps there was a misprint in your book? It sometimes happens even in the best ones.

The units of frequency are cycles per second. The dimension of frequency is T-1.
You could indeed use f, but then you would have to multiply it by 2π to convert it to radians because sin tables are drawn up in terms of radians or degrees. ω is already in terms that you can enter directly into the sin(ωt) equation.

It always pays to try to distinguish between things that are done for arithmetical convenience and those that are done for fundamental physics reasons when learning a subject.

go well

 Ok so because the Simple Harmonic Motion when described graphically is sinusoidal, you can use the same equations as you do for circular motion? So the error in my book is it saying ω is in the units /s? It should be in rad/s, which in this case is angular frequency right? And even though the actual movement in real space is linear, in terms of a graph, it is sinusoidal so angular frequency can be used? (Im trying to string it together in my head now, please correct me if im wrong there.) So when observing SHM, my ω is going to always be rad/s (angular frequency) and my f is going to always be /s (frequency), and I use the equations to convert between the two if necessary? This is where I was struggling, if that is where the book is wrong I can move on, hopefully, and just correct it in the book. I'm hoping (partly for your sake) that I'm right in everything I've said above? Thanks again for your time!
 I prefer frequency in cycles per second rather than Hz - it is more descriptive but otherwise OK. Just out of interest circular motion is the combination of two simple harmonic motions at right angles. This is similar to the (vector) addition of two linear motions at right angles giving a resultant motion vector somewhere between the two. For linear motion this resultant is at 45degrees or π/4 rads if the motions are equal. For two shm the motion is elliptical if they are unequal and exactly circular if equal.

 Quote by Hunt4Higgs Ok so because the Simple Harmonic Motion when described graphically is sinusoidal, you can use the same equations as you do for circular motion? So the error in my book is it saying ω is in the units /s? It should be in rad/s, which in this case is angular frequency right? And even though the actual movement in real space is linear, in terms of a graph, it is sinusoidal so angular frequency can be used? (Im trying to string it together in my head now, please correct me if im wrong there.) So when observing SHM, my ω is going to always be rad/s (angular frequency) and my f is going to always be /s (frequency), and I use the equations to convert between the two if necessary? This is where I was struggling, if that is where the book is wrong I can move on, hopefully, and just correct it in the book. I'm hoping (partly for your sake) that I'm right in everything I've said above? Thanks again for your time!
I can't vouch for technician's approach, though I endorse much of what Studiot has written. The way I always keep this straight is to make sure I call "f" the "cyclic frequency" and ω the "angular frequency". Both have literal dimensions of T-1, but have different units. I usually just denote the units of ω with "s-1" (pronounced "inverse seconds" or "per second", depending on the context), while I denote the units of f with "Hz" (Hertz) or "cycles per second". It is vitally important to make sure you know which frequency you are talking about. A frequency of 10 Hz works out to about 62.8 s-1, and you really don't want people to think you mean 62.8 Hz.... Hertz are never used for angular frequency, only for cyclic frequency.

You are right about one thing: this is potentially very confusing. There are hardly any examples where we use two different symbols to describe the same physical quantity. Imagine if your textbook used F to denote force in Newtons and G to denote force in pounds. Even that's not quite as confusing as ω and f, because we rarely mix systems of units (SI and English) but we often have to deal with both ω and f in the same calculation.

A side note: just as a matter of practicality, I usually don't bother to track radians. Radians are funny units, because they don't really exist. You can add radians and square radians, for example, and not get a contradiction (the same is not true of degrees and square degrees!). For example, the Taylor series expansion for cos() is

cos(x) = 1 - x2/2! + x4/4! - x6/6!...

Since "x" is implicitly in radians, we see we are adding a pure number (1) to square radians to radians^4.... Now, we usually want to track units to make sure our algebra is correct -- if one side of an equation has kg2 and the other has kg4, you can usually conclude you've made a mistake. But you can't conclude you've made a mistake if you wind up with rad2 and rad4 on the two sides of an equation. Further, we think of radians as a unit of angle measure, but because of the definitions of the trig functions we have De Moivre's Theorem

exp(i*x) = cos(x) + i*sin(x)

so if we really believed radians were worth tracking we would have to say the argument of exp() should be in radians. But there are lots of contexts where exp() is used without reference to angular measure.

BBB

p.s. if you go back and look at your first post, you will see that you wrote
 My problem is that my book says the ω in the SHM equation is "a positive constant with units of s^-2", which I understand is Hertz^2 right?
which has caused a lot of confusion, because what you should have written is
 My problem is that my book says the ω^2 in the SHM equation is "a positive constant with units of s^-2", which I understand is Hertz^2 right?

 p.s. if you go back and look at your first post, you will see that you wrote My problem is that my book says the ω in the SHM equation is "a positive constant with units of s^-2", which I understand is Hertz^2 right? which has caused a lot of confusion, because what you should have written is My problem is that my book says the ω^2 in the SHM equation is "a positive constant with units of s^-2", which I understand is Hertz^2 right?
Thanks, BBBeard did you see my post #3?

 Quote by Studiot Thanks, BBBeard did you see my post #3?
Yes, and you were right. It's just that the OP is still confused about the "error in [the] book". The OP wrote in the most recent post:

 So the error in my book is it saying ω is in the units /s?
which suggests ongoing confusion And as you and I both suspect, the "error" is not in the book at all, but was an error of transcription.

BBB

 Ok BBB the first half of what you said made sense, after that it was pretty much white noise but I'm sure thats only because I am only at the beginning of this journey! So even though technically they both have units of s^-1, I have to just seperate them both, pretty much mentally, by saying to myself that ω can be denoted by the s^-1, but I should never write the cyclic frequency, f, with units of s^-1, I should write Hz? Does ω in angular frequency when written with units s^-1 just mean the same thing as if it was written with rad/s? (Just to check is this only done with SHM, for instance we were taught to write the units as rad/s if you were using ω in angular velocity?) Also, again to check, is this the only time units would be written as s^-1? When looking at Angular frequency? Im glad you said its potentially very confusing because everything I've done up until now I've done great in, and this has just frustrated the hell out of me because I can see it isnt exactly a complicated physics concept or anything. Sorry for any confusion with anything in the thread.

 Also, again to check, is this the only time units would be written as s^-1? When looking at Angular frequency?
Time units would never ever ever be written as seconds-1.

By the way it is time to introduce you to the two icons in the main reply box superscript and subscript.

These are the ones labelled X2 and X2 - the fifth and sixth in from the right on the second row.

They are great, to get there just use the reply button at the top of the page, don't just paste into the open field at the bottom.

go well

 Sorry Studiot, what I was saying is this the only "instance" in which you would see units written as s-1 , when looking at Angular Frequency? Not time units, apologies for the mix up. Thanks for the tip.
 The further you go in technical subjects the more important attention to detail becomes. Even a simple slip can come back and bite you in the *** bigtime. As regards your question, there are other phenomena in the physical world where the dimensions s-1 are used. You probably have not met these yet. An example would be the 'relaxation rate' of a system. Another would be the count of radioactivity in counts per second (or Bequerel)

 Tags angular, classical, frequency, harmonic, motion