Ball and chain projected up a rough inclined plane
Рет қаралды 1,406
7 ай бұрынA long chain is coiled up at the bottom of a rough inclined plane with a ball attached to one end. The chain has linear density λ, the ball has mass m, the plane makes an angle θ with the horizontal, and the coefficient of friction between the ball and chain and the plane is μ. The ball is projected up the slope with initial velocity u. In this video we find the maximum distance up the slope reached by the ball. A difficult Physics problem in which energy is not conserved!
To support the channel: ko-fi.com/benyelverton
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 #dynamics #mechanics #chain #differentialequation #integratingfactor #equationofmotion #chainrule #integration #calculus #force #newtonslaws #friction #inclinedplane #particle #ball #energy #collisions #inelastic #table #mass #gravity #physicsproblems #maths #math #science #education
@DrBenYelverton 7 ай бұрын
Note: I forgot to draw the friction arrow on the diagram, but don’t worry, this doesn’t affect any of the working as I did include the frictional term in the equation of motion. For completeness there should of course be an arrow labelled μN pointing down the plane!
@mingmiao364 7 ай бұрын
Brilliant problem as usual! Seems like a lot of the chain problems give rises to Bernoulli DEs. I wish in DE courses could mention them, otherwise these strange forms of DEs often seem to come out of nowhere
@DrBenYelverton 7 ай бұрын
Thanks! That's true, I suppose it's because if the chain is uncoiling, we always get that quadratic velocity term from the product rule when differentiating the momentum, and because we can always write the acceleration as ½dv²/dx.
@darwinvironomy3538 7 ай бұрын
cool, what about a truck holding of a coiled chain that's falling and depleting over time with the case of a truck holds at some velocity or the second case where the truck initially at some velocity (so the depletion of the chains out from the trunk going to increase the truck velocity somehow). and the last thing the idea about this is probably a ball and a chain act like a pendulum, but at first the chains are loose until it hits the level where the chain acts as a *non* massless string that gives tension.
@DrBenYelverton 7 ай бұрын
Thanks for the suggestions!
@Mtmonaghan 7 ай бұрын
Fantastic stuff, thanks Ben❤
@edcoad4930 7 ай бұрын
Glorious!
@haricotvert7310 7 ай бұрын
Hello ! This is a very cool problem ! But i didn't understand at 7:21, how do you project to get a cos ? Is it that when you project on the perpendicular axis you get a cos and when you project on the parallel axis you get a sin ? Because I've never learned this way of projection.
@DrBenYelverton 7 ай бұрын
Imagine a right angled triangle whose hypotenuse is the W arrow on the diagram, with one side in the direction of the dashed red line and the other side parallel to the plane. You can then see that the side in the direction of the dashed line is adjacent with respect to θ, and it follows from trigonometry that cosθ = adj/hyp = (perpendicular component)/W. For the parallel component, you're then interested in the opposite side of the triangle instead of the adjacent, so sinθ is the relevant trig function.
@press2701 7 ай бұрын
Following your vids closely as they are educational. I tutor AP and college intro physics in the USA (encore career, after 30yrs in chem engineering research) now. Am always looking to "up my game". I tell my students to "follow the process". To do well in physics, airbus pilot or brain surgery, you don't need to be brilliant...but you do need to have the mental discipline to "follow the checklist". IF you follow the checklist, follow the process, you will, more likely, not kill the patient, crash the plane, or get the wrong answer. In dynamics, that means 1. diagram, 2. FBD(s), 3. coordinate system, 4. eqns of eq'm, and 5 "math". If you were my student (!) I would gently admonish you for (a) no friction force on the diagram and (b) no coordinate system. But I suspect you're not 16yrs old, not a junior in, say, Ohio, trying to get into a 'good university' via 5.0 in AP physics C.
@Mtmonaghan 7 ай бұрын
You missed one. Degrees of freedom
@velike 7 ай бұрын
When integrating 2l/(m+lx)dx, shouldn't the answer be 2l*ln(m+lx), Dr? Cos l (lambda) is a constant
@DrBenYelverton 7 ай бұрын
If you differentiate 2ln(m+λx) you get [2/(m+λx)]×λ, where the extra λ comes from applying the chain rule. For the integral you can just use the same argument in reverse, so the λ on the numerator disappears. That's how I usually do this sort of integral, but to see it more explicitly you can also try integrating using the substitution y = m+λx so that dy = λdx.
Gufee.medalin
Рет қаралды 9 МЛН