Chain sliding off an inclined plane

Рет қаралды 1,116

7 ай бұрын

A chain of total length l lies along an inclined plane (or table) in the direction of greatest slope, with a length x hanging vertically off the edge of the plane; it is released from rest at time t = 0. The plane makes an angle θ with the horizontal. In this video we use Newton's second law and energy conservation to find x(t), v(t) and v(x), where v is the velocity of the chain.
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 #hyperbolictrig #integration #calculus #force #newtonslaws #energy #conservationofenergy #conservationlaws #table #inclinedplane #resolving #mass #gravity #physicsproblems #maths #math #science #education

Пікірлер: 24

@mujahidali6988 20 күн бұрын

Thanks Ben, clear physics and clear mathematics,

@DrBenYelverton 20 күн бұрын

Glad to hear it, thanks!

@elibrahimi1169 7 ай бұрын

i've worked on a problem recently that resulted in a similar differential equation , except q was equal to 1. i used laplace transforms , but your method was elegant as well great job sir Ben ! However , if you're interested in which problem i was studying here it is ; assume a helicopter going at a constant velocity v , and an object is attached to it's rotary wing ; what is the trajectory equation ? ( in polar coordinates )

@DrBenYelverton 7 ай бұрын

Nice! It's a good, reliable method for second order ODEs like this. I'm not sure if I understood your problem correctly, but you can probably write down parametric equations for the trajectory as a function of time without using differential equations.

@elibrahimi1169 7 ай бұрын

@@DrBenYelverton sorry my explanation was misleading ; the object isn't "attached" but more like loosely clinged to the rotary wing , so it is capable of moving up and down the rotary wing without friction ( it's frictionless just to simplify the problem )

@DrBenYelverton 7 ай бұрын

Ah, so you do need to set up a differential equation as the centrifugal force is changing over time. I'll probably be covering some similar problems on the channel at some point.

@elibrahimi1169 7 ай бұрын

@@DrBenYelverton tysm for considering that Dr Ben

@MissPiggyM976 7 ай бұрын

Well done, thanks!

@harinachiappansubramanian9206 7 ай бұрын

Hi so i tried to attempt this question after taking a long break in physics and im a little confused about my solution when i solve for the second order ode as the form of x(t) = Ae^pt + c where p is the same p in your video and c is the particular solution of the ode. I get a weird result where A just becomes 0 when i try to find for V(0) = 0. I think you understand where im coming from. What is it that im doing wrong and is this not an exponential function?

@DrBenYelverton 7 ай бұрын

Exponentials will work just as well as hyperbolic functions, but when you substitute a trial solution of x(t) ∝ e^(rt), you find that r²-p²=0 and hence r = ±p. That means your complementary function is Ae^(pt) + Be^(-pt), so you were just missing that second term.

@harinachiappansubramanian9206 7 ай бұрын

I realise now. Thanks, subbed

@velike 7 ай бұрын

Dr. Ben, why do we project the weight of the chain, that is on the inclined plane, when it is collinear with the weight of the hanging chain?

@DrBenYelverton 7 ай бұрын

It's because the chain is always accelerating along its own length, so only the components of the forces which are parallel to the part of the chain they're acting on contribute to the acceleration. The component of the weight which is perpendicular to the plane doesn't do any work and is balanced by the normal contact force. You could also try the method suggested at around 5:03, i.e. setting up equations of motion for the two parts of the chain separately, including the internal tension force, and adding the equations to make the tension cancel. You'll end up with the same overall equation of motion, which justifies the usage of the method in the video.

@velike 7 ай бұрын

@@DrBenYelverton thank you. I get it now

@mxminecraft9410 7 ай бұрын

Sir I didn't get the way you added up the forces ? Why did a factor of sine come up as the weight of chain on slope and hanging part is in same direction so couldn't we have directly added them ?

@DrBenYelverton 7 ай бұрын

Please see my response to another commenter, @velike, and let me know if that helps!

@mxminecraft9410 7 ай бұрын

@@DrBenYelverton thx