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Excellent!

@@heythere8318 glad it helped!

Of course!

@@JohnDoe-xc2nzIf the gravitational field is uniform then think of this: the force vector distribution will be solely dependent on the *mass distribution* of your object. Define a “center of gravity” using this distribution as the point on the object where the net torque about it is 0. The center of mass location is *also* solely dependent on the mass distribution of your object. Thus, what you will prove is that the center of gravity is identical to the center of mass point in a uniform gravitational field. Thus, there is no torque about the center of mass just like in this example in the video.

​@@madaydude_physics​ Thank you very much for your time and response. What I don't understand is that the reason you gave in this video to justify the fact that the rigid body was rotating around its center of mass was that if it was not the case, then that would mean that the center of mass has an acceleration not equal to zero, which contradict the initial hypothesis. However, since now we have a gravitational field, center of mass has an acceleration so we cannot use this argument anymore. What argument should I chose to justify that rigid body is still rotating around center of mass despite the fact that now the center of mass is accelerated by gravitational acceleration. Sorry if I misunderstood something trivial.

@@JohnDoe-xc2nz Here’s the difference: the gravitational field would cause a net *linear* acceleration on the center of mass. This is different than the requirement to have an object’s center of mass move in a circle (like in an object spinning at a point other than its center of mass) which would require a *centripetal* acceleration.

​@@madaydude_physicsThank you very much!! Not all heroes wear capes!! Have a good one.

@@JohnDoe-xc2nz Glad this helped! :3

Thanks :)

While there's no real difference, the choice followed from the physical example which we used Hz. Naturally in experiment and real world applications I see f more, and in problem solving w. I structure my videos to be consistent throughout, including units, so if I'm going to use Hz in the example, the Fourier transform is written the same

Of course, happy to hear this helped

Excellent! Love to hear when these ideas just “click” for you all :)

Use a u substitution of r - Rcos(theta). Don't forget to update boundaries. This will recast the integral in a more common form: udu/(R^2 - r^2 + 2ru) ^(3/2) If you want to suffer, now do the integral manually with integration by parts, or again reference a table/integral calculator. If you have further questions, let me know, but I'm guessing this will be enough for you.

Did you have the value of g at the poles? How did you get it?

Yup, in both cases, during and after poking the stick, the center of mass motion must obey Newtons 2nd law

The rod is spinning on a flat surface like on the surface of a table, not with or against gravity.

Glad you enjoy! At the end of the day I make these videos because I enjoy it, I think of each one as a little project to create. I like to think of the basic goal as a physicist or just a scholar in general is in 2 parts: building new knowledge (research) and spreading knowledge, so this lets me start working towards the latter early on in life :)

Nice question, I think this is a great idea to make a little video on those proofs. I’ll add this to my video plans

Thanks! Glad to hear you enjoy hydro :)

@@madaydude_physics I might make my own simulation. I made a numerical damped pendulum simulation, but this would be really interesting. Do you think it'd make sense to make a "coil" shape by taking a sine function with a fixed number of periods and plotting it in space while shifting it and rotating it as the spring expands and contracts? Because I'm pretty sure a coil from a side view is just a trig function

@@hydropage2855 Yup, there are different periodic shapes people will use for their springs, but that’s the right idea. I’d be happy to see a video of your simulation if you end up making it :3

@@madaydude_physics What program did you use? I think I’ll use Processing. Also, I’m really curious, how can damping be incorporated into a Lagrangian? I’m not sure how damping would work for a spring-dulum in general, I’m struggling to imagine that

​@@hydropage2855 Hi again hydro-- one nice way to incorporate damping is with the Rayleigh Dissipation Function: these links will explain the basics phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/10%3A_Nonconservative_Systems/10.04%3A_Rayleighs_Dissipation_Function#:~:text=The%20Rayleigh%20dissipation%20function%20R(q%2C%CB%99q)%20provides,both%20Lagrangian%20and%20Hamiltonian%20mechanics.&text=Consider%20the%20two%20identical%2C%20linearly,%CE%B2)%20shown%20in%20the%20figure. en.wikipedia.org/wiki/Rayleigh_dissipation_function

Thank you! Will do :)

Measured the signal received by the solar cell from the home lights, then subtracted the dc offset leaving the oscillations only.

a. Use conservation of energy law. I assume that the object is released from relaxed spring. Then, the total energy at the beginning is E0 = -mg l0 cos(θ). Maximim elongation happens when kinetic energy is zero and θ=0 (so all energy is used to elongate the spring) Then by energy conservation law, we get a quadratic equation: E0 = V + T = -mg(l0 + ρ) + 1/2 k ρ² + 0, or: -mg l0 cos(θ) = -mg(l0 + ρ) + 1/2 k ρ² which gives two solutions: 1. ρ = 0 2. ρ = 2mg/k Since we are looking for maximum elonagtion, the accept the second solution, ρ = 2mg/k. Note that this is an upper bound on elonagtion and might not be reached. But I suspect that unless there is some weird resonance, the spring will come arbitrary close to this elonagtion. b. I am almost sure that you cannot calculate this. You can only calculate pairs of speed and elonagtion that are possible. You could also do some asymptotic analysis to approximate the solution, if approximation is good enough for your application.

So the end result of this set of videos was an equation sheet with all the coordinate essential formulas summarized for polar, spherical, cylindrical coordinates- I post it and reference it in physics videos where I need a given formula for the coordinate system I’m working in. You can find it linked in the last video I posted about Spring Pendulums.

I’m glad you enjoy the videos here! I’m currently an undergraduate going into a Physics PhD, doing experiment based research, not theory, so generally speaking my videos are naturally going to have a bit more of a utilitarian flavor to them. To answer your question: it’s probable I will *eventually* cover such concepts, but regardless I would want to make more foundational content with undergrad level E&M, Quantum, Thermo etc before I get to that (assuming I don’t drown in grad level work and research first haha).

Glad to hear it! Now I’m no expert on grasshopper landing *ahem* so I might not be of too much use, but I’d be curious to hear about your work, sounds interesting!

@@madaydude_physics oh, don’t worry too much, the model is simple. It’s almost acts as a three link manipulator, i am mainly concerned about my (derivation, kinematics, and how i add an input), thanks in advance, so where can i send you the file?

@@husamaltalhi8579 Ok, try emailing it to me: use the email attached on my channel page under channel details... I'm going to use you as my guinea pig also to make sure that's set up right ;)

@@madaydude_physics i sent the files, 🙏

@@husamaltalhi8579 Excellent, I will check it out when I have spare time

You’re 100% spot on- in this we use the observation of the ball’s circular motion and focus on extracting out the period of the circling. To PROVE that the motion is indeed circular, instead of plugging in expression for centripetal force directly in, you would more generally set F = ma (r double dot in this context) then have to “solve” the equation of motion (or just guess a circular motion solution which indeed will satisfy the equation- see paper linked in description, this is how they approach it).

@@madaydude_physics Thanks! I will look at the paper, you and Steve Mould have piqued my interest. :)

Wow, what an honor, much appreciated! :)

Thanks, glad to hear it :)

Thank you! Oooh, that’s really tough >.> Hamiltonian Mechanics really appears everywhere, particularly foundational in Quantum… so that’s hard to beat + I happen to like thinking in terms of phase space! Both are amazing though!

I don’t have a very strong opinion on this either way for this problem, as you would know from your simulation work formalisms like Lagrangian and Hamiltonian mechanics are so nice due to their form invariance under coordinate transformations (canonical transformations at least). I would at the very least say in problem solving if you save on the number of coordinates in a different coordinate system, you should use it. For example, for a simple pendulum it would be far more efficient to use a single angle instead of tracking both x and y (or having to write out the dependence of y on x using your constraint since you’d be using more coordinates than your degrees of freedom otherwise). But yeah, of course coordinate systems are a choice, we can convert between them easily as well etc etc. Cool simulations by the way, I checked a few out :)

@@madaydude_physics Alright, thanks for the input. I've been curious for a while why derivations (textbooks and otherwise) almost always use polar coordinates. The angle is indeed the simplest choice for the 1DOF simple pendulum, and also for the zero gravity case of the spring pendulum that reduces to 1DOF from conservation of angular momentum. For the simple pendulum, it is actually more efficient to use two Cartesian coordinates than a single angle (about a factor of 2 or 3 in C. Trigonometric functions are just that slow) - although this requires constraints that generally make derivations non-trivial. Of course you may disregard numerical performance for tasks involving analytical treatment. Thanks, you too. Apparently there was a reason why my feed picked up on spring videos just now.

@@Zymplectic Yes, thank you as well, having this nice numerics perspective with an idea of the differences between those computation times will be nice for others to note as well

Of course!

Good luck! :3

Thanks :)

Excellent, glad you enjoyed!

Good question! I haven't tried myself, but presumably it would be a bit longer-- without the assumption you would go through and instead of immediately substituting equation for centripetal force in, you would more generally use F = ma (r double dot with reference to the picture), and then with an initial condition (say the ball starts at some position r0 and has some initial velocity v0) you'd have to solve the equation of motion (again assuming no slippage) to first prove the circular motion (at this stage it would still be easiest to assume the final answer is uniform circular motion, then plug into differential equation and show it satisfies it, which is the approach the paper takes, but there might be other differential equation methods to go about solving it).