I am a high school student and i am little confused between superposition of Simple harmonic motions{SHM's}, suppose a spring of spring constant $k_1$ has time period $T_1$ and another spring of spring constant $k_2$ and time period $T_2$ when connected with a block separately, if both are now connected in parallel or series, we say that the resultant motion will also be SHM, but why? we have studied that if angular frequencies of two SHM are different{which is the case here as time periods are different for both} and they get superimposed then their resultant motion will not be a SHM, then why do we say that, here it is SHM, is it different from SHM? if yes then why? and please also give some practical examples other than this where superposition of SHM can be seen?
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2$\begingroup$ what does SHM mean? simple harmonic motion? $\endgroup$– UmaxoOct 22, 2020 at 9:20
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1$\begingroup$ I would suggest reformulating the question for broader audience. It feels to me like this question is addressed to people who took the same class you are taking. $\endgroup$– UmaxoOct 22, 2020 at 9:27
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$\begingroup$ i apologize, i will write it in its full form $\endgroup$– Arun BhardwajOct 22, 2020 at 12:33
1 Answer
You seem to be confused with the superposition of two oscillators. The following discussion will be useful for you :
Waves are usually described by variations in some parameters through space and time—for example, height in a water wave, pressure in a sound wave, or the electromagnetic field in a light wave. The value of this parameter is called the amplitude of the wave, and the wave itself is a function specifying the amplitude at each point.
The point that should be noted is that for superposition you must have at least two waves (at least two degrees of freedom) otherwise you can't apply the superposition principle.
In an abstract way, If $\psi_1$ and $\psi_2$ satisfy the equation for linear differential equation so does $$\psi=c_1\psi_1+c_2\psi_2.$$
Take a simple example to digest this
If you go about solving this equation, you will find out that $$M\frac{d^2\psi_a}{dt^2}=k(\psi_b-\psi_a)-k\psi_a$$ $$M\frac{d^2\psi_b}{dt^2}=-k(\psi_b-\psi_a)-k\psi_b$$
The general solution for such a system is the superposition of two-equation, like
$$\psi_a=A_1\cos(\omega_1 t+\phi_1)+A_2\cos(\omega_2 t+\phi_2)$$
That's what means to be a superposition of two oscillators, Note that you can decouple these oscillators into two independent systems.
On OP's problem, you have a differential equation which looks like $$M\frac{d^2}{dt^2}(x_1+x_2)=-(k_1x_1+k_2x_2)$$
which seem to have two variable (two degrees) but this is not true you need to fulfill the condition on joint (Newton's third law) so that $x_1k_1=x_2k_2$. Substitute this to your differential equation so $$M\frac{d^2x_1}{dt^2}=-\frac{k_1k_2}{k_1+k_2}x_1$$ You see actually there is only one independent degree of freedom in the system and so you don't have the superposition of two waves. You just need to tell $x_1$ to find out where the mass is (state of the system).
The answer is already too long but I'm adding one more example of a parallel spring system on OP's request.
The following is the system in my mind. It's obvious that displacement for both the spring will be the same, say $x$.The force equation for the mass
$$F=m\ddot{x}=-k_1x-k_2x=-(k_1+k_2)x=-Kx$$
The fact remains the same. Until you don't have two degrees of freedom, you can not have two different frequencies. I will recommend OP to look on JAlex answer on this question, When you let there is mass, you see there was a superposition solution but as you let that one mass is zero, the second frequency vanishes.
I hope this will help you. Best Wishes!
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$\begingroup$ i am a high school student, and i was able to understand only a little bit of it, could you please explain that the thing that I mentioned in my question i.e when two spring{constants ki and k2} oscillate separately they will have very different time periods and angular frequency ,but when they are now connected to a same block in parallel ,then is it also a superposition example?if not,then why? or if is it superposition then how the resultant motion of spring will be a Simple harmonic motion,it shouldn't be here superposition of two SHM's of different frequncies is getting superimposed..... $\endgroup$ Oct 22, 2020 at 12:38
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$\begingroup$ I can understand. Let me put this. In a nutshell, to apply the superposition you need atleast two moving parts ( two degrees of freedom) in your system. In your example, You have only one, so you just need one parameter to specify the position of mass. The spring constant are different but that would just change the way this mass will oscillate (frequency). Two superpose these two masses you need to couple them with the help of another spring and joint them like the figure. $\endgroup$ Oct 22, 2020 at 12:46
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$\begingroup$ , i am confused because in my textbook, it is written that, when a particle is acted upon by two separate forces each of which can produce a simple harmonic motion,,then the resultant motion is the sum of these two simple harmonic motions ,that"s why i thought that it will be a superposition? because here also two forces from different springs are acting on it and each of them can cause SHM, i am now more confused in it,why do we need two oscillators for superposition? why we cannot superpose here the two SHM's performed by spring separately? $\endgroup$ Oct 22, 2020 at 14:15
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$\begingroup$ @ArunBhardwaj I extendend the solution ,please look at this if this solve your doubt. Free to ask. Cheers. $\endgroup$ Oct 23, 2020 at 9:20
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$\begingroup$ I did not understand the point in which you have written that k1x1=k2x2 why they are equal? $\endgroup$ Oct 23, 2020 at 11:49