# What is Chain: Definition and 976 Discussions

A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A chain may consist of two or more links. Chains can be classified by their design, which can be dictated by their use:

Those designed for lifting, such as when used with a hoist; for pulling; or for securing, such as with a bicycle lock, have links that are torus shaped, which make the chain flexible in two dimensions (the fixed third dimension being a chain's length). Small chains serving as jewellery are a mostly decorative analogue of such types.
Those designed for transferring power in machines have links designed to mesh with the teeth of the sprockets of the machine, and are flexible in only one dimension. They are known as roller chains, though there are also non-roller chains such as block chain.Two distinct chains can be connected using a quick link, carabiner, shackle, or clevis.
Load can be transferred from a chain to another object by a chain stopper

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1. ### The chain rule for the Stratonovich integral

Professor says one way to do this is to convert the equations to Itô form and back. ##dX_t=bdt+\sigma\circ dW_t## converted to Itô's SDE is \begin{align*} dX_t=&\left(b+\frac{1}{2}\sigma\frac{\partial}{\partial x}\sigma \right)dt+\sigma dW_t. \end{align*} We use Itô's formula to compute...
2. ### Timing Marks on Engine, Timing Chain - Conceptual Question

On an engine, the timing chain has colored links, and you have to line up these colored links on timing marks on the engine. There's typically marks on sprockets, bearing caps, gears. A understand that you have to line these up in order to get proper timing. I understand that is the reason why...
3. ### I Using the Chain Rule for Vector Calculus: A Tutorial

This is probably a stupid question, but I have never realised that there's an order things should be done in the chain rule , so for example ## \nabla(\bf{v}.\bf{v})=2\bf{v} (\nabla\cdot \bf{v}) ## and not ## 2 \bf{v} \cdot \nabla \bf{v} ## Is there an obvious way to see / think of this...

14. ### Derivatives and the chain rule

I originally thought you’d have to use the chain rule to get h’, as in: f’(g(x))*g’(x). Plugging in 1 for x, I got an answer of 10. An online solution, however, said that you only had to get f(g(1)), which was f(-1), then look up f’(-1) in the table. Both approaches seem logical to me, but they...
15. ### Doubt In Explanation of Proof of Chain Rule

In Chapter 3 of Thomas’s Calculus, they give the following proof of the Chain Rule. After the proof, the text says that this proof doesn’t apply when the function g(x) oscillates rapidly near the origin and therefore leads delta u to be 0 even when delta x is not equal to 0. Doesn’t this proof...
16. ### A discrete-time queue Markov Chain problem

The following problem is seriously tricky and I urgently need help with it, thanks. For part a: we have the following transition probability matrix P = a0 a1 a2 a3 a0 a1 a2 a3 0 a0 a1 b2 0 0 a0 b1 Now, is a0 = a1 = a2 = a3 =...
17. ### The infinite limits of the probability transition matrix for Markov chain

Consider a Markov chain with state space {1, 2, 3, 4} and transition matrix P given below: Now, I have already figured out the solutions for parts a,b and c. However, I don't know how to go about solving part d? I mean the question says we can't use higher powers of matrices to justify our...
18. ### Why can the potential energy at any point be chosen to have any value?

The force is -Mg and distance is L/3. So W=-MgL/3. Not right! Maybe it’s done using COM but there is no additional information is given.
19. ### Work per particle of a NaCl chain

The problem states to find the work per particle to assemble the following NaCl chain. I just want to post my work here to verify I have the correct answer. My work is attached in the image provided.
20. ### Morin's mechanics problem 2.3 (motionless chain)

In his solution, Morin solves the problem as the hint suggests: cutting the chain into small pieces, taking the component of the external forces along the curve (which is just the component of gravity here) and summing up an in integral, obtaining 0. He then claims that because the "total...
21. ### Weight of gold chain when dropped on a weighing scale

The force exerted downwards on the scale by the chain when it is kept on it would be Fg= Mg =λLg where λ is the linear mass density However when the chain is dropped onto the scale it exerts an additional force due to its change in momentum The force exerted by each part of the chain would...
22. ### "Doubling up" effect when pulling a chain on a table

In the book, it is stated that if your hand move a distance x, then x/2 is the length of the moving part of the chain because the chain gets “doubled up.” as in the image below. I don't get the meaning of this. For example, if our hand move L metres from initial position, shouldn't the moving...
23. ### I Proving P(X_4|X_1,X_2)=P(X_4|X_2) for Markov Chain with Finite Possibilities

Given events ##X_i## and the following: ##P(X_3|X_1,X_2)=P(X_3|X_2)## and ##P(X_4|X_1,X_2,X_3)=P(X_4|X_3)## prove ##P(X_4|X_1,X_2)=P(X_4|X_2)##? Special case proof: Finite number of possibilities for each ##X_i##. Transition matrix from ##X_2## to ##X_4## is product of transitions from...
24. ### Error calculation in a measurement chain

[Mentor Note -- thread moved to the schoolwork forums from the technical forums] Summary:: I would like to understand how to calculate the error of a measurement provided by a measuring chain Hello to everyone, I'm going to explain the problem I would like to understand better. Thank you for...
25. ### How to prove the chain rule for mutual information

Hi, I was attempting the following problem and am a bit confused because I don't quite understand the structure/notation of how mutual information and entropy work when we have conditionals (a|b) and multiple events (a, b). Question: Prove the chain rule for mutual information. I(X_1, X_2...
26. ### Chain rule (multivariable calculus)

##f_x=3*x^2+y## ##f_y=2*y+x## ##(3*(t^2)^2+e^{t-1})*2*t+(2*e^{t-1}+t^2)*e^{t-1}## Well, I am not sure how to evaluate it. I got a wrong result by multiplying by 0.1, i.e. ##((3*(t^2)^2+e^{t-1})*2*t+(2*e^{t-1}+t^2)*e^{t-1})*0.1## I guess it is trivial but I am lost. :(
27. ### A Russian Commentary on Chain Wheel Experiment

Just a nice experiment; comment in Russian is obvious
28. ### Calculating Chain Link Strength: Which Method Should You Use?

Hi, I am interested in the topic of hand calculations of chain link's strength. I am talking about a regular industrial chain with hanging weight. From what I've read, there are 3 potentially possible approaches: - Lame's problem (circular cross-section has to be replaced with equivalent...
29. ### Why is the tension in a falling chain not equal to ρgy?

Firstly, There is something I want to clarify. When the system starts moving, parts of the chain that still lies on the table, which have mass ## \frac {(L- y_0)M} {L}##, will be pulled by the force that the hanging chain's weight exert,right? If yes, then : As far as I know, the formula ##F=...
30. ### Understanding the Chain Rule in Multivariable Calculus

But, If I use chain rule than, I get that. ##\vec v_i = \frac{dr_i}{dt}=\sum_k \frac{\partial r_i}{\partial q_k} \cdot \frac{\partial q_k}{\partial t}## But, they found that?
31. ### The work that is necessary to pull a hanging chain

This is the solution from my textbook, and I have some questions about the method The mass of hanging chain : $$m_h =\frac m 5$$ the center of mass of the hanging chain : $$h_1 = - \frac{1} {2} \cdot \frac L 5 = - \frac L {10}$$ (the minus sign here means that it is under the table surface)...
32. ### Implicit differentiation: why apply the Chain Rule?

Hi, PF ##y^2=x## is not a function, but it is possible to obtain the slope at any point ##(x,y)## of the equation without previously clearing ##y^2##. It's enough to differentiate respect to ##x## the two members, treat ##y## like a ##x## differentiable function and make use of the Chain Rule...
33. ### Solve T(y)-T(y+dy)=ug(dy) | Easier Method?

T(y)-T(y+dy)=ug(dy) is what I have got. How would I solve this? Or is there a simpler method.
34. ### U-235 Chain Reaction: Exploring Nuclear Fission

Today I learned that U 235 is not a German Uboat
35. ### Using chain rule when one of the variables is constant

So first thing I tried was to separate the variables then differentiate by parts, setting u = E and v = 1/ln(E) (and the other way around) but I couldn't do the integral it gave. Then I tried to reason that because dx was constants then dE/dx is equal to E/x but I was told that's not the case...
36. ### A Ising model open chain and periodic boundary conditions

One dimensional Ising model is often treated as open chain system with free ends. Then when external field is added it is treated with cyclic boundary condition. Can someone explain me are those methods equivalent, or not?
37. ### B Proof of Chain Rule: Understanding the Limits

First I quote the text, and then the attempts to solve the doubts: "Proof of the Chain Rule Be ##f## a differentiable function at the point ##u=g(x)##, with ##g## a differentiable function at ##x##. Be the function ##E(k)## described this way: $$E(0)=0$$...
38. ### Analyzing the Fall of a Chain: Problem 103 of 200 Puzzling Physics Problems

My attempt: At first, only a small part of the chain has fallen through. Let that part have mass m, speed v, and length x. Suppose the chain has a mass per unit length of u. To accelerate a small length of chain on the table to speed v, Force needed = v dm/dt = v (dm/dx) * (dx/dt) = uv^2...
39. ### Why is m*dv/dt 0 in the Falling Chain Problem?

I have seen the solution for this problem but still there are some things I do not understand and would like clarification. In the equations below I understand that we use the chain rule on m and v but what I don't understand why m*dv/dt is 0, I don't think is because the acceleration of dm is...
40. ### MHB Using Chain Rule to Solve Questions - A Step-by-Step Guide

Hey everyone, could anyone help me figure out how to use chain rule to solve these questions in the attachments below?
41. ### Verifying Chain Rule for Partial Derivatives

I have no answer or solution to this. So I'm trying to seek a confirmation of whether this is correct or not: ##df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial t}dt ## ##\frac{df}{dt} = \frac{\partial f}{\partial x} \dot x + \frac{\partial f}{\partial t} ## Therefore, ##...
42. ### I Why does the summation come from?

I want to take the derivative of a composite function that looks like $$f( g(x), h(x) ).$$ I know from Wolfram that the answer is $$\frac{ df( g(x), h(x) ) }{ dx } = \frac{ dg(x) }{ dx }\frac{ df( g(x), h(x) ) }{ dg(x) } + \frac{ dh(x) }{ dx }\frac{ df( g(x), h(x) ) }{ dh(x) }.$$ We can...
43. ### Equation of motion of a chain with moving support

In the figure assume the "ceiling" moves with motion ##Y(t)##, i.e. it is a point support. Applying Newton's law in the vertical direction ##T(y).\hat{y}=\rho y[g+\frac{d^{2}Y}{dt^{2}}]## If ##\theta## is the angle between ##T## and ##\hat{y}## that means ##|T|\cos\theta=\rho...
44. ### Solving Derivatives with the Chain Rule

Hello! Now this is not really a physics problem of the usual kind but I'd say you could consider it one.Still I'd like to post my problem here because here I always get great help and advice.Now for this problem in particular,it is in the section of the book that deals with derivatives so I...
45. ### Designing a Pallet Chain Conveyor: Calculations and Approaches

Hi, I'm not sure how to approach the design calculations for chain conveyor. It's rather easy to find formulas for belt conveyor but it's not the same. I also thought about the use of procedure applicable to chain drive but in case of conveyor load is placed directly on the chain (or a pair of...
46. ### Solving the Puzzle of Chain Energy: Tips & Tricks

I don't need the whole answer just a few tips to do it. I think it's something with the two different radiuses, but I'm really bad at this. We did a similar one in class, but there is a trick in this one that I can't figure out.
47. ### Linear chain of oscillators and normal coordinates

Hello, I hope the equation formatting comes out right but I'll correct it if not. So far, I have attempted to write ##\ddot{a}_k(t) = \sum_{n}(u^{k}_n)^*\ddot{q}_n(t) ##. Then I expand the right hand side with the original equation of motion, and I rewrite each coordinate according to its own...
48. ### MHB Using Chain rule to find derivatives....

y = (csc(x) + cot(x) )^-1 Find dy/dx
49. ### I Derivative using the chain rule

I'm coming back to maths (calculus of variations) after a long hiatus, and am a little rusty. I can't remember how to do the following derivative: ## \frac{d}{d\epsilon}\left(\sqrt{1 + (y' + \epsilon g')^2}\right) ## where ##y, g## are functions of ##x## I know I should substitute say ##u = 1...
50. E

### A chain falling off of a table

This question came up in a lecture, and I wasn't really satisfied with how it was solved. Specifically, they assumed that the hanging part of the chain has zero horizontal velocity. What they did was essentially write down the equation ##m\ddot{x} = \frac{mg}{l} x##, which under the above...