Same here. My goodness, the amount of "so we pull out the solution out of a magical hat, and you can see it works!" crap that goes on with harmonic motion pisses me off. This is elegant and pretty straightforward.
This is the clearest exposition of this topic I have seen, and I have seen many! I'm hoping you cover phase angle in subsequent videos. Also, KA does not cover this topic. Thanks
Well the argumentation goes pretty much like this: since x is a variable that has at each point in time t a certain value you can describe its value for every moment in time by an euler function- because every number can be described by an euler function! Therefor you can substitute x by the expression e^rt with r being a constant needing to be specified!
i is just a constant...you can combine all constants into single constant. Ex. When we integrate both sides of equation, that should result to two constants, one for each side, but it can combine into one.
Thing to note here is that A and B can be complex. To get the real solution you must specificly choose A and B to be complex conjugates of each other. E.g. A = c + di, B = c - di. Then A + B = 2c, which is real and A - B = 2di becomes real when you multiply it with the i that was outside the brackets.
It's just kind of hidden inside the C1-C2. While its definitely there if you decompose the problem, it really doesn't matter since we are just choosing arbitrary constants. So just replace it with B. That's the huge benefit of Euler's formula. It lets you jump from complex to real plane.
He made a booboo there. Thing to note is that A and B can be complex. To get the real solution you must specificly choose A and B to be complex conjugates of each other. E.g. A = c + di, B = c - di. Then A + B = 2c, which is real and A - B = 2di becomes real when you multiply it with the i that was outside the brackets.
He made a booboo. A and B can be complex numbers and to get a real solution to the whole equation you must choose A and B so that they are not just complex, but also complex conjugates of each other. Then in the 1st term imaginary parts get substracted out and in the 2nd term they get multiplied out, i*i = -1.
How would you go about finding A and B? I understand you need some initial conditions but what form would they come in, in a question? Excellent video by the way
+George Initial conditions could include many parameters, but for a scenario like this, velocity and position are common initial conditions. A and B represent the amplitude of the oscillations of the mass on the spring. The approach to finding A and B would be similar to determining the highest point of a projectile's path based on its initial launch angle and velocity, except instead of gravity acting on the mass, in this situation the spring force is acting on the mass... but the spring force depends on the position of the mass... hence the second-order DiffEQ. If you're curious about some examples, check out Walter Lewin's Lectures of MIT Physics 8.03. The whole lecture set (More than 20 lectures) is really well done, but I think lectures 1 and 2 will address your question effectively.
Same here. My goodness, the amount of "so we pull out the solution out of a magical hat, and you can see it works!" crap that goes on with harmonic motion pisses me off. This is elegant and pretty straightforward.
Ikr
Yes after so many searches and asking teachers i finally found this wonderful video
Perfectly illustrated, broken down into the simplest bits of information, extremely helpful. Thanks so much for demystifying this!
best simple explanation i can found
This is the clearest exposition of this topic I have seen, and I have seen many! I'm hoping you cover phase angle in subsequent videos. Also, KA does not cover this topic. Thanks
i don't understand the part where you add the test function "X=e^rt" part
Oh my god! this is so beautiful!!! thankyou
Well the argumentation goes pretty much like this: since x is a variable that has at each point in time t a certain value you can describe its value for every moment in time by an euler function- because every number can be described by an euler function! Therefor you can substitute x by the expression e^rt with r being a constant needing to be specified!
Okay, but how do you get rid of the complex part of the solution.
My question as well.
You can just ignore it
It doesnt exist don't worry 😂
i is just a constant...you can combine all constants into single constant. Ex. When we integrate both sides of equation, that should result to two constants, one for each side, but it can combine into one.
Thing to note here is that A and B can be complex. To get the real solution you must specificly choose A and B to be complex conjugates of each other. E.g. A = c + di, B = c - di. Then A + B = 2c, which is real and A - B = 2di becomes real when you multiply it with the i that was outside the brackets.
Holy shit this video is magical. You saved my grade!
It sounds like there are crickets in the background, it's very relaxing 🦗🎻
At the latter part of your solution, where you represented the variables by A and B, what happened to the conplex number, í?
It's just kind of hidden inside the C1-C2. While its definitely there if you decompose the problem, it really doesn't matter since we are just choosing arbitrary constants. So just replace it with B. That's the huge benefit of Euler's formula. It lets you jump from complex to real plane.
He made a booboo there. Thing to note is that A and B can be complex. To get the real solution you must specificly choose A and B to be complex conjugates of each other. E.g. A = c + di, B = c - di. Then A + B = 2c, which is real and A - B = 2di becomes real when you multiply it with the i that was outside the brackets.
how would you go about making a programme to solve these equations if anyone had any info it would be much obliged
Can you solve one in which the differential equation does not equal zero but a constant
Awesome video! Thank you!
Thanks Dave, this helped a lot.
So B is comlex or not? how can you not explain that?
how would you solve a differential equation for an inverse cube force?
is that equation at 9.51 the equation of Motion for SHM?
Hey, maybe you know the form of the solution dE/dt=0 for the same case? i mean, i want to find the same answer parting of dE/dt=0 you know how?
nice presentation and helpful
Wonderfully, you made it simple! Thank you!
You are awesome! keep it up dude :)
Good and clear explaination
thank you soooo much, much clear than mine stutter prof
Thank you very much, really ope up my brain like a canned sardine! Awesome
Thank you very much for your help.
An example for a simple number would go like this: 2=e^x with x being ln2!
Thank you very much! You helped me a lot
Since B must be i times some real number how can Bsin(wt) be a real function?
im curious as well :S
He made a booboo. A and B can be complex numbers and to get a real solution to the whole equation you must choose A and B so that they are not just complex, but also complex conjugates of each other. Then in the 1st term imaginary parts get substracted out and in the 2nd term they get multiplied out, i*i = -1.
thanks for this wonderful explanation:)
Like the video, but the sound quality is a bit bad.
thank you very much :)
really helpfull
How would you go about finding A and B? I understand you need some initial conditions but what form would they come in, in a question? Excellent video by the way
+George Initial conditions could include many parameters, but for a scenario like this, velocity and position are common initial conditions. A and B represent the amplitude of the oscillations of the mass on the spring.
The approach to finding A and B would be similar to determining the highest point of a projectile's path based on its initial launch angle and velocity, except instead of gravity acting on the mass, in this situation the spring force is acting on the mass... but the spring force depends on the position of the mass... hence the second-order DiffEQ.
If you're curious about some examples, check out Walter Lewin's Lectures of MIT Physics 8.03. The whole lecture set (More than 20 lectures) is really well done, but I think lectures 1 and 2 will address your question effectively.
Thanks a lot mate
helped alot ... thank you
Really helpfull
Thanks!
you forgot to add the "i" to the sin function during the last conversion.
thank you
I think I love you
based
Your not Khan.
You are correct. Khan is trash
gatosexo