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Guru Vishnu
Guru Vishnu
04:37
Hi sir. Good morning :-)
Guru Vishnu
Guru Vishnu
05:26
@JohnRennie: Hi sir. Good morning :-)
John Rennie
John Rennie
@GuruVishnu hi :-)
Guru Vishnu
Guru Vishnu
Are you free now sir? If yes, may I ask a doubt from SHM?
John Rennie
John Rennie
Yes
Guru Vishnu
Guru Vishnu
Question:
A particle under the action of a simple harmonic motion has a period $3~\mathrm{s}$ and under the effect of another it has a period $4~\mathrm{s}$. Its time period under the combined action of both the SHM's in the same direction is $0.4x$ seconds. Find the value of $x$.
Answer: $x=6$
I know how to arrive at the answer. It's just assuming $F_1=-k_1x$ produces the SHM with a period $3~\mathrm{s}$ and $F_2=-k_2x$ produces the SHM with a period $4~\mathrm{s}$. Then both of these in the same direction is represented by this equation: $F_1+F_2=-(k_1+k_2)x$. Using $T=2\pi\sqrt{\frac{m}{K}}$ and a few manipulations, we can arrive at the answer.
My doubt:
How can two SHM's with two different angular frequencies give rise to an SHM? Shouldn't the motion be complicated? Are there any other constraints in this question to prevent this from happening (like same amplitude)? Thinking this in terms of superposition of the equations $x_1=A_1\sin\omega_1 t$ and $x_2=A_2\sin\omega_2 t$ is confusing for me. Do I want to simply proceed with the trigonometric expansion?
(End of message)
John Rennie
John Rennie
I think it's a really badly worded question.
As you've done in your working it means there are two central forces, one with a force constant $k_1$ and one with a force constant $k_2$, then it asks about the motion if both forces are applied at the same time i.e. the total force constant is $k_1 + k_2$.
This is different from asking what happens if we superpose two SHM motions.
Guru Vishnu
Guru Vishnu
05:33
Can you please tell why is it different from superposition sir? This is what I can interpret from "under the combined action of both the SHM's in the same direction".
The problem is, I think, the result of two SHM's with two different periods is complex/chaotic.
John Rennie
John Rennie
If you add two sinusoidal motions you get something like:
$$ \sin(\omega_1 t) + \sin(\omega_2 t) = 2\sin((\omega_1 + \omega_2) t) \cos((\omega_1 - \omega_2) t) $$
Guru Vishnu
Guru Vishnu
Fine. So that's under the assumption the amplitudes of the two SHM's are same.
John Rennie
John Rennie
So as you say you get a result that is not SHM.
@GuruVishnu true, ut if the amplitudes are different you get that plus a bit left over e.g. if $A_1 > A_2$ you'd get:
$$ A \sin(\omega_1 t) + \sin(\omega_2 t) = 2\sin((\omega_1 + \omega_2) t) \cos((\omega_1 - \omega_2) t) + (A-1) \sin(\omega_1 t) $$
which is also not SHM.
Guru Vishnu
Guru Vishnu
Ok sir. I understood so far. But by the addition of the two forces, we get $F=-(k_1+k_2)x$ which suggests the combined motion is SHM. Why is there a contradiction when we consider this and the regular equation $A\sin\omega t$?
John Rennie
John Rennie
They are two different situations.
Suppose you have a mass on a two springs in parallel, and you can disconnect either spring so you can make the mass move with either spring A connected, spring B connected or both springs connected.
This is (I think) the sort of thing your question is asking.
In all three cases it's just a mass on a spring, just with three different force constants.
You aren't superposing two different simple harmonic motions, you're just changing the spring constant.
Guru Vishnu
Guru Vishnu
05:45
Ok sir. So the constraint (in our case) is having a fixed value of $x$ for all the three SHM's by some other means. If so how is this different from the sinusoidal equation? Which constraint are we relaxing there so as the resultant motion is not an SHM?
John Rennie
John Rennie
Well suppose we added two electric fields $E_1 = \sin(\omega_1 t)$ and $E_2 = \sin(\omega_2 t)$
Guru Vishnu
Guru Vishnu
Ok sir.
John Rennie
John Rennie
This would be supurpsition of two different SHMs.
Guru Vishnu
Guru Vishnu
Agreed. And as per our previous discussion the result is not an SHM.
John Rennie
John Rennie
Yes.
Incidentally this sort of thing is done a lot in electronics. For example heterodyne circuits do this.
Guru Vishnu
Guru Vishnu
05:49
Fine. I haven't learnt about heterodyne circuits. I thought this to be the emf across an AC source.
Heterodyne
Heterodyning is a signal processing technique invented by Canadian inventor-engineer Reginald Fessenden that creates new frequencies by combining or mixing two frequencies. Heterodyning is used to shift one frequency range into another, new one, and is also involved in the processes of modulation and demodulation. The two frequencies are combined in a nonlinear signal-processing device such as a vacuum tube, transistor, or diode, usually called a mixer. In the most common application, two signals at frequencies f1 and f2 are mixed, creating two new signals, one at the sum f1 + f2 of the tw...
I think it's incorrect to use $x=A\sin\omega t$ here because the $x_1$ and $x_2$ provided by the two equations with $\omega_1$ and $\omega_2$ give two different values whereas the object can only remain at one particular point at a given instant. This is why this method gives the result is not an SHM.
John Rennie
John Rennie
@GuruVishnu yes. With an electric field the total field can be the sum of two different fields.
But an object can only be in one place at a time. It doesn't make a lot of sense to say that its position is the sum of two different positions.
Guru Vishnu
Guru Vishnu
Ok sir. It seems I understood why the $A\sin\omega t$ method fails. Let me go though this once and ask further if I've any doubts.
Thank you for your time :-)

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