Understanding Simple Harmonic Motion and its Equations | Physics Explained

[Hunt4Higgs]

Simple Harmonic Motion!?

Hi All,
New to the forum, sorry if this is a thread that has been discussed before, however I've had a quick look and have been killing myself for days over this.
Approached my college tutor today and he couldn't explain it!?

I'm almost sure that it is something easy and simple, but as it stands at the moment, I can't see the light!

Ok, the question...

I understand that SHM can be thought of as circular motion, I have read through the derivations, beginning with the centripetal acceleration equation a=ω^2r (r as radius)...and ending with with the SHM equation of motion a=-ω^2x

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?

At what point and how does the ω change from rad/s to s^-1?

(I know there is a relation between the two, however as far as I could see on the internet, a hertz is a cycle per second, but that means per cycle there is 2∏ rad/s? So they arent equivalent...)

Also, I found the equation ω=√k/m is the equation to find angular frequency?
Whats the relation between angular frequency, angular velocity, hertz considering they are all s^-1?

The other problem relating to this I have is the fact that when looking for the period of motion of a circle the equation is T=2∏/ω...where ω is the angular velocity.
and...the period of motion of an oscillation is T=2∏/ω but here ω is in hertz?

Very sorry if this is confusing or that its a ridiculous and stupid question, I'm sure you've all had that Math/Physics block at some point where you just can't see it at first...
Please explain all you can, bare in mind I'm fairly new to Physics, so don't hit me with hugely complicated stuff if it can be helped.

Much appreciated if anyone can help!
Thanks.

 

[technician]

I think your problem here is to do with units.
Frequency Is measured in Hertz but the unit of Hertz is s^-1
this is because part of the unit is a pure number. 1 Hertz means 1 cycle per second
In the same way 'radian' is not a 'unit'...it is a pure number because it is a ratio of 2 distances.
So an angular velocity of 1radian per sec has units of 'per sec' or s^-1

 

[Studiot]

Welcome to Physics Forums H4H.

My problem is that my book says the ω in the SHM equation is "a positive constant with units of s^-2",

Are you sure your book says this and not that

ω² has units s^-2?

The dimensions of ω are the reciprocal of time ie T^-1

Note I have said dimensions, not units.
You should distinguish between these.

One foot and one metre are both units of length, which has dimension - L.

 

[Hunt4Higgs]

Ok so are they essentially the same thing it just depends on the system in which you're observing?

Radian being the ratio of angle turned through in relation to the radius?

So you could effectively write that the angular velocity of a car is 6 s^-1 but the radians is put there to give more information?

So "omega" has different dimensions, all of which are reciprocals of time?

Sorry if I'm completely not getting the obvious, it's frustrating hitting a mental block on something I can see is probably simple.

 

[technician]

You have got the idea! When you say 'angular velocity is 5 radians per second' it means something different to saying ' a frequency of 5 hertz' (this means 5 revolutions per sec)
BUT both have units of s^-1 (and dimensions of T^-1if you find yourself involved in dimensional analysis)

 

[Hunt4Higgs]

Ok feeling closer to the truth now.

But I thought there were 2pi radians in a revolution, therefore the rad/s will be higher than the Hz?

Does this not mean that the two can't be equivalent as one has to be 2pi x more than the other?

Most obvious example for me is the equation for the period of motion/oscillation, how can they both have the same equation, but different "omega" (angular and Hz) values with one 2pi x more than the other?

That looks confusing now I've written it out.

You're helping a lot more than my tutor did though so thanks!

 

[technician]

What you have met here is common in physics.
A good example is force x distance which has units Nm
Force x distance can mean 'work done' and then Nm are given the special name 'Joules'
In equilibrium applications Force x distance can be called 'turning effect' or 'moment' or 'torque' or 'couple' the units are Nm but are not given a special name.
Having a feel for units ...( or dimensions)... Can be a great help in physics.

 

[technician]

What you have just written is covered by
T = 2∏/ω
Can you see it?
And f = 1/T = ω/2π

 

[Hunt4Higgs]

I kind of see what your saying, I hadn't thought of it in that way.
But with work done and turning moment, there is no difference in size like there is between a radian and a cycle?

All force is measured in Newtons regardless of where it is...

If you have a question asking for the omega value of an oscillating system, you use the omega=2pi/T
You get s^-1 as your unit?

You use the same equation to find the omega value given a question asking about circular motion, omega=2pi/T and your given units in rad/s?

How can that be when rad/s is 2pi x bigger than Hz?

Sorry about this dude, must be boring going over basics like this.

 

[technician]

I take your point about force x distance !
You say rads/sec is 2pi x bigger than Hz is very wise...you have got it
'rads' is not a unit ! So 'rads/sec' is only /sec or s^-1
and Hz is cycles per sec...cycles is not a unit so Hz is only /sec or s^-1

So omega = 2pi x f

( it is not boring!)

 

[Studiot]

Sorry to be boring but rads or radians are a unit - they are just dimensionless, like all angles.

 

[Hunt4Higgs]

(Pre-warning: I'm not even sure these questions are making logical sense anymore.)

But if in circular motion ω=2∏/T (when looking at the period of motion) the omega is defined as angular velocity which is the change of radians over time ... ω=Δθ/t...you see the circle, you see the number of radians that are turned through per second...

However in SHM ω=2∏/T (again when looking at period of motion) the omega here is in cycles per second? A cycle which is larger than a radian...

But during the derivation of the SHM equations, you use circular motion equations, the ω starts out in rad/s (a=ω^2r) and ends up as s^-1 (a=-ω^2x) without anything being done to it?
How when rad/s ≠ s^-1 in terms of actual values, 1 rad/s ≠ 1 s^-1?
Or is it that I shouldn't be looking at numbers?
(Argh head is jumbled!)

In terms of units, I see that... rad/s = /s because a radian is not a unit in the sense that it has no definite value, if that's right? (Always dependent on the size of the cirlce yes?)
Likewise I assume the same is true for a cycle/s = /s because a cycle again will be dependent on the size of the circle? Not definite?
Whereas something like a m/s ≠ /s because a metre is a unit, has a definite value?
Or looking at N/m again, definite values?
Am I on the right track with that?

(Just as a note, I think you've pretty much made me understand it, I'm just trying to clear things up in my head now...and test you at the same time!)
Thanks for your time so far!

 

[Studiot]

In terms of units, I see that... rad/s = /s because a radian is not a unit in the sense that it has no definite value, if that's right?

I'm sorry no definite value?

You are confusing 'units' which are a system of measure of a particular quantity and 'dimensions' which have (fundamental) physical significance. There are many quantities in Physics that are dimensionless, but they all have units and if we change these units we change the number.

So for instance 1 radian is about 57.3 degrees. Clearly neither are the same number but both definite and both refer to the same angle.

Equally I metre is about 3.3 feet, both refer to the same length, but the units are different so the numbers are different.

 

[Ken G]

However in SHM ω=2∏T (again when looking at period of motion) the omega here is in cycles per second?

No, ω is radians per second, we just don't usually bother to write the "radians". There is a cycles per second, usually denoted f, but that's not what goes into the formulas you've been writing. For example, the solution usually looks something like cosine(ωt) or cosine(2∏ft), but not cosine(ft). The cosine function is cyclic over 2∏, not over integer entries.

 

[Hunt4Higgs]

I'm clearly not getting this the way I should be.

How were you guys taught it?
Should I try and look at the "dimensions and "units" you mention?

Should I just accept that when looking at SHM the ω is given in the units /s?
...and when looking at circular motion the ω is given in the units rad/s?

Is it stupid to keep asking why when I'm at the low level that I am?

 

[Studiot]

I suppose what you are really asking is

Why do we bother to introduce ω at all?

Well look at the attached sketches.

The first shows something going round and round in a circle with constant velocity.
The second shows a pendulum swinging back and fore between A and C. Note the velocity is not constant.

They have, however, something in common.

Both are repetitive motions. Each pursues the same path over and over again in the same time interval. The time interval for each is not necessarily the same, but we can make it so.

The repetition occurs when the object passes through the same point going the same way. The time interval between successive occurrences is called the period.

So the circular motion passes through A once every complete circle.
This is 360 degrees or 2π radians. If this takes T seconds we say that is has an angular speed of 2π/T radians per second.

So the equation of motion connecting distance and time is

d = QPt where Q is a constant, equal to 2π/T, P is a constant (=radius)

The pendulum starts from B, goes through A and back to B in the opposite direction and then to C and then back to B again in the original direction, in its period T.
Because it is traveling with SHM the distance it travels in some time t is given by, ie the equation of motion for SHM connecting distance and time is

d = Psin(Qt), where P and Q are constants.

Now to make the connection.

It turns out that the constant Q is also given by 2π/T; Pdetermines the amplitude of the SHM and is equivalent to the radius in the circular formula.

If we compare this with the formula for circular motion we can see that they are the same so we say that SHM proceeds with and angular velocity of 2π/T.
However because we are lazy and don't want to write this fraction every time and because it occurs in lots more useful formulae we give it its own letter ω.

Depending upon your level of maths you may be able to see that this coincidence is no accident but occurs because sin(t) is a function which repeats every 2π radians or 360 degrees.

If you then ask why do we use radians at all, well one justification is that there are lots of formulae in physics involving sin(θ) where we can make the approximation that for small angles sin(θ) = θ so long as we measure angles in radians.

Does this help?

 

[Hunt4Higgs]

Ok I think that's cleared it up a lot more. Thanks.

So just to check, if the system is exerting SHM but the movement is linear (rather than angular like the circle or pendulum), is the ω still rad/s?
As isn't the frequency, denoted f, in the units /s?

Is the ω just another letter in place of f? As both are looking at SHM, both looking at cycles, and both have the units /s in SHM?

The acceleration equation describing SHM...a=-ω^2x
As you said earlier Studiot, the book says ω^2 ...units = (s^-2) but the derivation of this equation starts with circular motion and the centripetal acceleration a=ω^2r where the omega has units (rad/s).

 

[Studiot]

Glad we are getting somewhere.

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

 

[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 I am 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!

 

[Studiot]

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.

 

[bbbeard]

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 I am 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?

 

[Studiot]

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?

 

[bbbeard]

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

 

[Hunt4Higgs]

Ok BBB the first half of what you said made sense, after that it was pretty much white noise but I'm sure that's 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 separate 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 isn't exactly a complicated physics concept or anything.
Sorry for any confusion with anything in the thread.

 

[Studiot]

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.

Think about it.

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

 

[Hunt4Higgs]

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.

 

[Studiot]

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)