Undamped Mechanical Vibrations & Hooke's Law // Simple Harmonic Motion

Consider a mass on a spring moving horizontally. The only force on the mass is the spring itself which we can model using Hooke's Law. Then by Newton's Second Law that F=ma, this gives the differential equation mx''+kx=0. This is a second order, constant coefficient, homogeneous differential equation which we can study! We use our techniques from ODEs to solve this system and then do a bit of trigonometry to convert the answer to the standard form. This type of oscillation is often called either undamped mechanical vibrations or more generally simple harmonic motion as this occurs in a myriad of places in physics beyond just this scenario.

0:00 Mass on a Spring
1:18 Newton's 2nd Law & Hooke's Law
2:51 Solving the ODE
4:28 Rewriting into standard Form

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[RahulSharma-ocqd]

At 3:25, Equation has minus sign error. Only by considering plus sign you could get imaginary value of r.

[ogunsadebenjaminadeiyin]

Calling 'g' shift rather than 'phase' makes huge sense. Well done Prof.

[miaalexanderthegreat]

Thank you so much Dr. Bazett!!!! You have no idea how much your works benefit students from all around the world😭😭🙇🙇🙏🙏🙏

[bendavis]

Thanks for the awesome video, I have a quick question though. I understand everything up to the point around 5 minutes in where you introduced gamma into the equation. In my textbook at this point in the explanation, the author solved for the leading constants A and B at t = 0, and then he got B=0 and A=x_0 where x(0) = x_0 is the starting position of the spring. The final equation he got was x(t)=(x_0)*cos(a*t) where a:=sqrt(k/M). I didn't see anything resembling gamma after this and I'm confused what it's purpose is in the equation. Thanks again for the help and let me know if you need me to clarify anything.

[georgesadler]

Professor Trefor Bazett, thank you for introducing Undamped Mechanical Vibrations and Hooke Law in Differential Equations. DR. Bazett, please correct the sign error in the characteristic equation. Professor Bazett, you have mr^2 - k equal zero instead of mr^2 +k equal zero.

[branndn_]

For god the third time, thank you for being out here as a life savior to my finals

[joelschneider]

you are one the most underrated YouTube professors! Thank you!

[mohfa]

Thank you Dr. For all your great educational videos, and for sure you have talent to deliver education and information.

I have a question please :
In your derivation you assumed that acceleration is in the positive x direction and thats why when we apply newton's law projecting on the positive x direction we get the equation m.r(2) + k =0 with two imaginery solutions, but what if we initially assume that the acceleration is in the negative x direction then when we apply newton's law projecting on the positive x direction we will get m.r(2) - k =0 with two real solutions ??, what is wrong with my logic ??, i am sure that iam missing something here !..

Thank you in advance

[Icglez]

Really great explanations! Can you make a video on the method of variation of parameters?

[PunmasterSTP]

Undamped mechanical? More like "Phenomenal lectures that are unparalleled!" 👍

[Abhi-mucy]

good video sir but how you have written x= Acoswt + B sinwt we got the roots -+iw right so on substitution using the eulars formula we get real part on adding (A+B)coswt, (A-B)sinwt will be imaginary

[lucieneyvrard]

I’m really confuse
When we talking about the Characteristic Equation why there is a sign minus before the « k » ?
Same for the i we know there some complex because there no y’ right ?
Thanks for your help guys

[serkanbasatlk]

Characteristic equation has a minus mistake 03:35

[ay-kgcy]

Thanks for the explanation! My textbook always confuses me with so much ugly theory when it covers real world applications.

[mustangpolygraph]

Thanks for this video ! Would love to see more of dyanamics videos

[LucasDimoveo]

Is there a course that goes into the theory of Controls without having to take a boatload of mechanical engineering courses?

[utuberaj]

@ 3.22- you have cleverly manipulated the real & distinct roots r1 and r2 into 'imaginary' ones and gone ahead. But actually the roots are real is'nt it? That is r1= +Sqrt(W_0 t) and r2= - Sqrt(W_0 t) for which the general solutions will be of the type x= c1. e^r1t + c2.e^r2t. The solution is purely real and there are no Cosines/Sines here. In fact the solutions are exponential type.
This is the method shown in your earlier video (569- "Constant Coefficient ODEs: Real & Distinct vs Real & Repeated vs Complex Pair". Can you kindly explain please Professor?

[shankylezapanta]

where did the pythagorean theorem came from?? I mean why did the 2 undetermined coefficients had a geometric relationship suddenly being perpendicular?? just came out of nowhere

[fuzailhamid]

Just wondering. How/Why did you guess x=e^rt?

[v-doc]

Why can I not just use either cos or sin from the start? Both are oscillating anyway?