How to know whether motion is simple harmonic motion or not?

[Beelzedad]

Homework Statement

(1) By just looking at the time period of the oscillation, can we know whether the motion is simple harmonic or not? How?

(2) Is the same true for angular (circular) SHM?

Relevant Equations

A simple harmonic motion is described by:

$\dfrac{d^2 x}{dt^2}=-\omega^2\ x$

For angular:

$\dfrac{d^2 \theta}{dt^2}=-\omega^2\ \theta$

I am reading "Coulomb and the evolution of physics and engineering in eighteenth-century France". There it is said in page 152 para 1 that "Coulomb found that within a very wide range, the torsion device oscillated in SHM".

My questions are:

(1) By just looking at the time period of the oscillation, how can we know whether the motion is simple harmonic or not?

(2) Is the same true for angular (circular) SHM?

 

[epenguin]

I guess that sort of a question plus the rarely seen effort of a student reading around the subject justifies the "show some effort" principle this forum imposes. But to make it perfect, maybe still a bit more effort?
What do you imagine, guess? After all, there are a limited number of things here that you could observe at all easily.

PS Checking up he does not look like a beginning student, sorry if that sounds condescending. But anyway, any ideas?

 

[PeroK]

Homework Statement:: (1) By just looking at the time period of the oscillation, can we know whether the motion is simple harmonic or not? How?

(2) Is the same true for angular (circular) SHM?
Homework Equations:: A simple harmonic motion is described by:

$\dfrac{d^2 x}{dt^2}=-\omega^2\ x$

For angular:

$\dfrac{d^2 \theta}{dt^2}=-\omega^2\ \theta$

I am reading "Coulomb and the evolution of physics and engineering in eighteenth-century France". There it is said in page 152 para 1 that "Coulomb found that within a very wide range, the torsion device oscillated in SHM".

My questions are:

(1) By just looking at the time period of the oscillation, how can we know whether the motion is simple harmonic or not?

(2) Is the same true for angular (circular) SHM?

To focus on (2) how would you determine whether a particle was moving in a circle at uniform speed?

 

[Beelzedad]

After all, there are a limited number of things here that you could observe at all easily.

So how did Coulomb ensure that motion was SHM?

Like for your RSVP. But, I also request you to not go away from this discussion before a conclusion is reached.

 

[epenguin]

I've now looked at your source. It seems to me all answer is given at the end of the first paragraph and is exactly what I would have said.

 

[Beelzedad]

To focus on (2) how would you determine whether a particle was moving in a circle at uniform speed?

If the body had experienced a centripetal acceleration.

 

[PeroK]

If the body had experienced a centripetal acceleration.

I was expecting an answer like: record its path and check whether the path is a circle.

 

[Beelzedad]

I was expecting an answer like: record its path and check whether the path is a circle.

Ok, then how shall we know the angular acceleration (in my question) is directly proportional to the angle ($\theta$)

 

[PeroK]

Ok, then how shall we know the angular acceleration (in my question) is directly proportional to the angle ($\theta$)

You have to start thinking for yourself.

 

[Beelzedad]

To focus on (2) how would you determine whether a particle was moving in a circle at uniform speed?

We can know the path (circle) by just looking. We can also know what $\theta$ is at a certain time, again by looking.

I have no idea of how to know angular velocity or acceleration by only looking at it.

 

[PeroK]

We can know the path (circle) by just looking. We can also know what $\theta$ is at a certain time, again by looking.

I have no idea of how to know angular velocity or acceleration by only looking at it.

Clearly it's harder to determine whether something is moving in SHM than in a circle. You could map position against time and check that it looks like a sine function. You can get the angular frequency from the maximum displacement and the midpoint. You can get the amplitude from the maximum displacement etc.

 

[Beelzedad]

Thanks, I get it... A little more help is needed if you can...

Can you help to decipher that wierd equation $3$ on page 153.

 

[PeroK]

Thanks, I get it... A little more help is needed if you can...

Can you help to decipher that wierd equation $3$ on page 153.

Sorry, I think I'd have to read about that experiment in some detail.

 

[epenguin]

Odd comments:
the equations you give in #1 are using a 'redimensioned' $x$, that is not measured in metres but incorporating the elastic constant and the mass (?). I am all for that simplification when teaching the mathematical theory, in fact for that purpose I would reduce it further to d²θ/dt² = -θ , but descending to experimental physics you have to express it in metre lengths and incorporating all the physical constants of the situation.

There appears to be a little misprint in the text in the first equation with a second derivative.

For experimental checking, I is calculable from first principlesfor an object of uniform density whose shape is known. So one could check the effect of varying that. n on the other hand is a non-fundamental constant of a non-fundamental empirical law, applying to only certain materials, not predictable from any elementary theory.The smaller the angle, the better we expect this linear law should be obeyed.It should be possible to determine it also in a static force measurement.The full theory of torsion gives a dependence on the length of the wire which could be checked experimentally.

 

[epenguin]

Clearly it's harder to determine whether something is moving in SHM than in a circle. You could map position against time and check that it looks like a sine function.

I guess you could map position against time in a modern teaching laboratory or even at home, with a mirror on the bar reflecting a laser beam, or the bar could even be a laser, and a TV camera recording over time or something like that. More difficult to think of a way of doing it in 1784!

 

[epenguin]

What is the point? I mean this is just a boring physics teaching lab type experiment and calculation. I am guessing that the period gave Coulomb a convenient method to measure the elastic constant n, which he then used for measuring electrostatic force somehow with the same torsion balance to establish his famous laws of electrostatics?

 

[vela]

I am reading "Coulomb and the evolution of physics and engineering in eighteenth-century France". There it is said in page 152 para 1 that "Coulomb found that within a very wide range, the torsion device oscillated in SHM".

I don't think the text is very clear here. Very wide range of what?

(1) By just looking at the time period of the oscillation, how can we know whether the motion is simple harmonic or not?

I think what the book meant is that Coulomb found that the period of oscillation was independent of the amplitude of the oscillation, which is consistent with simple harmonic motion. If you have simple harmonic motion, you know that the period is independent of the amplitude. If you observe oscillatory motion where the period is independent of amplitude, it wouldn't be unreasonable to assume that the motion is simple harmonic.

 

[haruspex]

I don't think the text is very clear here. Very wide range of what?I think what the book meant is that Coulomb found that the period of oscillation was independent of the amplitude of the oscillation, which is consistent with simple harmonic motion. If you have simple harmonic motion, you know that the period is independent of the amplitude. If you observe oscillatory motion where the period is independent of amplitude, it wouldn't be unreasonable to assume that the motion is simple harmonic.

True, but x=Af(t) has a period independent of the amplitude A if f is periodic and independent of A. I see the text gives a footnote reference (39).