Physics 68 Lagrangian Mechanics (18 of 32) Two Mass - Two Spring System
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- Опубликовано: 12 сен 2024
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We will find the equations of motion of motion of a two mass-two spring system Lagrangian equations.
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At 7:10 , you're taking the partial derivative of the PE, not of the lagrangian. But actually you should take the partial of the Lagrangian with respect to x1, which would mean that there is a negative sign infornt of the expression(since it is KE - PE) that you are taking the partial derivative of. So in the end you'd have a different derivative. Can you explain that?
same doubt bro
I too believe there is some mistake with signs
Yes the -ve was left out of the lagrangian derivative. Correct e.o.m. are mx1''+kx1-kx2=0 and mx2''-kx1+2kx2=0
@@Doc-1510 Yep, I have the same answer. This should be correct
I truly cannot express how helpful these videos are..
Thank you very much.
We are glad you find these videos helpful. Thanks for sharing. 🙂
Hello, I suspect you have a small mistake with signs: you have to take derivative from Lagrangian (where PE has minus before it) but for simplicity you took derivative from PE itself, but did not take minus in L=KE-PE, so your equations will be correct with other signs. Also you have a typo: you wrote x squared where it should be x-sub-2, but after that you added it correctly to 2x-sub-2
Please correct me if I'm wrong, thanks!
There is no error with the sign, but you did find the typo.
@@MichelvanBiezen , thanks for your answer! But I insist on the fact that you've mixed the sign in your derivations. Please, can you double check it and follow my explanation: your PE for the system is: PE = 1/2 * k * (x1^2 - 2*x1*x2 + x2^2) + 1/2 * k * x2^2, when you substitute it into Lagrangian, you have: L = KE - PE (pay attention to this minus!). When you substituted PE into Lagrangian, you did not expanded (a - b)^2 into (a^2 - 2ab + b^2): L = KE - [1/2 * k * (x1 - x2)^2 + 1/2 * k * x2^2]. And when you were finding derivatinves, you took expanded value of PE for derivation, but you did not count the "-" sign in Lagrangian itself: L = KE - PE, so your equasions are incorrect because of this minus. I'm kindly asking you to double check it, and if I'm wrong anyway, please please explain! Thank you, sir, for your videos!
Ah yes. The partial derivative of L with respect to x1 has the sign reversed. Good catch.
As a Physics Educator, I highly appreciate your efforts professor.
Thank you. Glad we are able to give you some ideas to try yourself. (We all learn from one another). 🙂
Do the double pendulum with springs not strings. Once you get these down you can get pretty wild.
We will be showing more complicaed examples in the near future. 🙂
can u also do for spherical and cylindrical coordinate?
Thank you so much sir! I would like to clarify on calculation of the partial derivative of L with respect to x2. I think only -kx1 will remain.
Thank you for your video, which helped me solve a problem at work!
Glad it helped.
Thank you for your videos. At 10:09 you say correctly "x 2" but then you write "+ k x ^ 2". It is a "hand lapsus" because you complete correctly the equation, this just to inform you.
velocity supposed to be relative to each other V1 - V2 because velocity of each one of them will affect either one and the spring tensionnwon't have affectiont
حينما اوجدنا البارشل كان يجب ان تستخدم من دالة لاكرانج، وليس من قبيل الطاقه
لان دالة لاكرنج اشارتها سالبه لذلك ستتغير الاشارات
هل حينما نستخدم الطاقه لايجاد البارشل يعطي نفس القيم المطلوبه؟
When we found the Barcel, it should have been used from the Lakrang function, and not from the energy, because the Lakrang function has a negative sign, so the signs will change. Is it when we use the energy to find the Parcel that it gives the same required values?
Sorry, but we didn't understand the question. ☹
Love this method. I use to do this for systems I made up on my own. I'm gonna try and find the other videos for this problem with different masses and/or different k's. I wish my handwriting was a nice as yours. 🙂 ✏
If the masses of k's are different you would use the exact same method. Just replace m by m1 and m2 and k by k1 and k2.
@@MichelvanBiezen That's what I thought. I haven't done these in a long while. Thank you
Really like these videos! I can review what I learned before and what I will learn soon or later!
Glad you like them!
Sir, it's a wonderful explanation. You solve a difficult problem in the easiest way. It helped me a lot.
Glad you found our videos. 🙂
Why would newton's law not work here? It would simply be ma=k(x1-x2). Same as your answer
You can use Newton's laws as well. For mor complicated sytems the Lagrangian methods become easier to execute compared to Newton's laws.
How do you go about it if the spring constants a different?
You would work it out exactly the same way, just replace k by k1 and k2.
I really love all your video, very useful, very easy to understand, they're all great, you're a great teacher. I just want to ask and confirm if a/ax1=kx1-kx2 is correct or not?
are you referring to the equation: ma = kx1 - kx2 ? If yes, they are not the same.
@@MichelvanBiezen yes, sir. In the last result i get ma=-kx1+kx2 but i see you get ma=kx1-kx2
Yes, you are correct. I got lazy and skipped a step, so I didn't carry over the negative sign.
@@MichelvanBiezen It's okay, sir. Thank you for your respond and I hope you will upload more of these content in the near future, it has helped me alot. Thank you so much, sir.
Really cool! Since the solution ended up being to differential equations relating the motion functions of the masses, would you then find the equation of motion for each mass by solving a system of differential equations?
The two equations are in their final form. (they have the form of ma = kx for a single spring). You need 2 equations, one for each of the masses.
@@MichelvanBiezen pretty clear👍
what will be the angular frequency of the system?
Did we write angular frequency of both blocks seperately according to equations or there is single angular frequency of the system
In Lagrangian mechanics, we tend to assign a separate equation for each mass in the system.
@@MichelvanBiezen yes I get that, but my question is whether both blocks have same angular frequency or different angular frequency. And what would be the angular frequency of the system
gracias pelao, me hiciste la tarea de sld
Omg thanks sir. I've been waiting for 1year 😂
We'll be publishing more videoa on Lagrangian soon.
prof, IF AFTER SPRING 2 WE HAVE PULLY AND THEN WE HAVE MASS 2 MOVE ON Y-Axis
(so the system comes like The video but mass 2 moves on the y-axis )
is it have the same solution or i must add potential energy for mass 2 and Kenitek energy just the same?
If there is no spring holding the hanging mass, then yes, you must add the potential energy.
.
Dr.
I meant if the same example in the video.
just there is a pully between mass2 and spring2 so just mass2 moves over the y-axis ( spring 2 as it before the pully so it moves over x-axis as in the video).
so is it the same solution?
It is my understanding that each of these equations of motion you have boxed isn't particularly useful on its own, correct? It is the system of equations of motions that allows us to actually determine the motion of each object. But a substitution for either x1 or x2 in terms of the other would allow us to actually specify either object's motion independently with a single equation?
yes, but typically during these problems solving the coupled differential equation is either tedious or impossible analytically, so its solution is beyond the scope of the video
2:35. Thanks for the video. Why is this not x1-x2-l? I thought PE is 0.5k*extension squared and that extension is new length minus old
Likewise for the second spring, why not x2-l?
When the left block moves to the right a distance x2, it reduces the elongation of the right spring by a distance x2, therefore we use for the elongation of the right spring = k1 - k2
@@MichelvanBiezen thanks. I just realised our reference points are difference. All of my measurements are from the wall, not the respective mass equilibrium points
Hi sir I am from Pakistan and I really appreciate your cooperation 🎉❤❤❤❤❤ thanks for teaching us ❤
Glad you found our videos. Welcome to the channel! 🙂
@@MichelvanBiezen I am really appreciative of what you did
Thank you professor
You are very welcome
Hello Dr. There is a simple mistake in 6:53 minute in partial derivative with respect to x1
It's correct
Superb sir❤
Thank you
Thank you son much professor
You are very welcome
How to find the period of oscillation
Thanks sir❤🙏
Most welcome
Very useful in robotics.
Glad you think so!
I dare say at 10:15 you mixed superscript (^2) and subscript (#2) of x_second
I probably did.
Nice one thnku
Glad you liked it. 🙂
Cfbr
?🙂
@@MichelvanBiezen commenting for better reach
Aselash
? 🙂
Choti bacchi ho kya!?!?!
That seems like an odd question........