Oscillations of a horizontally driven pendulum

Using Lagrangian mechanics to solve for the time evolution of a pendulum which is forced to vibrate horizontally at its base.
About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.
My website: benyelverton.com/
#physics #mathematics #pendulum #resonance #forcedoscillations #drivingforce #dynamics #lagrangian #mechanics #lagrangianmechanics #eulerlagrange #calculus #differentialequation #simpleharmonicmotion #oscillations #action #stationaryaction #leastaction #physicsproblems #maths #math #science #education

[@r2k314]

Thanks for the analysis of the amplitude. What comes out of the maths never ceases to amaze.

[@r2k314]

Please do a lagrangian that's really hard to set-up with non-Lagrangian methods if you have time.

[@MissPiggyM976]

Well done, thanks!

[@DrBenYelverton]

Thank you!

[@AmanSharma-fh1uj]

Amazing video!!!❤
What would happen if the displacement would have been vertically instead of horizontally?🤔
Also, I had a question which I'm having trouble solving. So, how can I send it to you so you can help me out🥺.

[@DrBenYelverton]

Thanks! Actually the behaviour of a vertically driven pendulum is quite unexpected and interesting - it oscillates about θ = π, i.e. the stable equilibrium configuration has the mass directly above the pivot. It's a bit harder to solve and I'm planning to cover this on the channel some time soon. I can't guarantee that I can help with your question but feel free to type it out as a reply to this comment!

[@AmanSharma-fh1uj]

@@DrBenYelverton My problem is the same problem discussed in the video just that instead of horizontal displacement there's vertical one. If you could make a video on it ASAP that would be great😇.

[@DrBenYelverton]

@@AmanSharma-fh1uj As it happens, I just worked through all the maths yesterday in preparation for recording the video. It will definitely be posted by the end of this month!

[@AmanSharma-fh1uj]

@@DrBenYelverton Thanks a lot for the update!!!😇

[@AmanSharma-fh1uj]

@@DrBenYelverton Can't wait for it!!!!😇

[@Francesco_Langella_]

Thanks, if i have a cycloidal or a polinomial function?

[@DrBenYelverton]

You mean what happens if we change the acos(ωt) to a different type of function?

[@Francesco_Langella_]

Yes

[@DrBenYelverton]

@@Francesco_Langella_ I think the method would be very similar but the equation of motion at 9:30 would have a different forcing term on the right hand side. For the solution, the complementary function would be the same but you'd need to guess a particular integral of the appropriate form, e.g. if the displacement of the support varies as a polynomial in time, you'd guess a polynomial of the same order and substitute it in to find the coefficients.

[@Francesco_Langella_]

Are there any methods to derive these integrals?

[@DrBenYelverton]

After using Euler-Lagrange to get the equation of motion, you just have to guess something sensible including some unknown coefficients, and substitute it into the equation to solve for the coefficients. If the forcing term is sinusoidal, the particular integral will be of the form Acos(ωt)+Bsin(ωt); if it's a polynomial of degree n, you'd try another polynomial of degree n where the coefficient of each power of t needs to be determined; if it's proportional to e^(kt) you'd guess Ce^(kt) and substitute to find C, etc.