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. Brett Royale

    [Enders Game] Split molecules and cause a chain reaction

    during my reading of ender's game, there was a device that could split molecules and cause a chain reaction of destruction. this seems possible yet impossible to me. can such a weapon be created and if so what causes the reaction to be so destructive.
  2. Alettix

    Beads on a thread - What stops the acceleration?

    Hello! I would like to ask for your help with understanding a few things connected to the following problem: 1. Homework Statement There is an (infinitly) long thread, on which small beads can move without friction. The beads with mass m are lined up on the thread with a constant distance d...
  3. AHashemi

    Heavy chain moving upward at constant velocity

    Homework Statement We have a heavy chain with length of L and weight of m placed on a table. we take an end of it and move it with constant velocity of v upwards. so each moment it gets heavier. If the force needed for this uniform speed is called F, and length of chain which is higher than the...
  4. Math Amateur

    Rotman's Remarks on Modules in Context of Chain Conditions

    I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 7.1 Chain Conditions (for modules) ... I need some help in order to gain a full understanding of some remarks made in AMA on page 526 on modules in the context of chain conditions and...
  5. Math Amateur

    MHB Rotman's Remarks on Modules in the Context of Chain Conditions and Compostion Series

    I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 7.1 Chain Conditions (for modules) ... I need some help in order to gain a full understanding of some remarks made in AMA on page 526 on modules in the context of chain conditions and...
  6. wololo

    How Can dv/dx Be Determined to Solve for dv/dt?

    Homework Statement Homework Equations Chain rule, partial derivation The Attempt at a Solution dv/dt=dv/dx*dx/dt+dv/dy*dy/dt dx/dt=-4t -> evaluate at (1,1) =-4 dv/dt=-4dv/dx+4(-2) dv/dt=-4dv/dx-8 How can I find the missing dv/dx in order to get a value for dv/dt? Thanks!
  7. C

    Chain rule and multivariable calculus

    Say I have a function of three variables, ##F=F(s_{12},s_{23},s_{13}) = F(s,t,-s-t)##, where ##s_{12}=s,s_{23}=t## and ##s_{13}=u = -s-t##. I want to compute the differential operators $$\frac{\partial}{\partial s}, \frac{\partial}{\partial t}\,\,\text{and}\,\,\frac{\partial}{\partial u}.$$...
  8. C

    Using chain rule to obtain the derivative dz/dt

    Homework Statement Homework Equations dz/dt = dz/dx⋅dx/dt + dz/dy⋅dy/dt The Attempt at a Solution [/B] I am getting : =[-sin(x+7y) ⋅ 10t] + [-sin(x+7y) ⋅ 7 ⋅ (-1/ t2)] then changing x and y terms: =[-sin((5t2)+7(1/t)) ⋅ 10t] + [-sin((5t2)+7(1/t)) ⋅ 7 ⋅ (-1/ t2)]
  9. Math Amateur

    MHB Noetherian Rings and Modules: Theorem 2.2 - Cohn - Section 2.2 Chain Conditions

    I am reading P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ... I am currently focused on Section 2.2: Chain Conditions ... which deals with Artinian and Noetherian rings and modules ... ... I need help with understanding a feature of the Theorem...
  10. B

    Finding the rate of convergence for a markov chain

    Homework Statement For the following Markov chain, find the rate of convergence to the stationary distribution: \begin{bmatrix} 0.4 & 0.6 \\ 1 & 0 \end{bmatrix} Homework Equations none The Attempt at a Solution I found the eigenvalues which were \lambda_1=-.6 or \lambda_2=1 . The...
  11. M

    Can the 1D Random XX Spin Chain Model Be Solved Exactly?

    Hi, Consider model of one dimensional spin chain with a random couplings J. The Hamiltonian is the following: $$ H = \sum_i J_i (S_i^x S_{i+1}^x+ S_i^y S_{i+1}^y)$$, Which by Jordan-Wigner transformation we can transform it to the fermionic representations. $$ H = \sum_i J_j/2 (c_i...
  12. T

    Find the Tension in a Flexible Chain Resting on a Cone

    Homework Statement A loop of flexible chain, of total weight W, rests on a smooth, frictionless right circular cone of base radius r and height h. The chain rests in a horizontal circle on the cone, whose axis is vertical. Find the tension in the chain. Homework Equations Virtual work, but...
  13. A

    Understanding the Application of Chain and Product Rules in Calculus

    Im stuck on theorem 5 where the book used chain rule then used product rule then again using the chain rule. How in the world does it work? I don't get product rule used and chain rule used after.
  14. C

    Solving N Bead Gaussian Chain Partition Function

    Homework Statement Consider a system made up of joining together ##N## beads and ##N-1## springs. The positions of the beads is indicated by ##N## real numbers ##\left\{x_i\right\}_{i=1,...N}.## The Hamiltonian which characterises it is $$\mathcal H =...
  15. C

    Markov chain: finding a general solution

    1. The problem statement Given a stochastic matrix P with states s_1...s_5: P = \begin{pmatrix} 1 & p_2 & 0 & 0 & 0\\ 0 & 0 & p_3 & 0 & 0\\ 0 & q_2 & 0 & p_4 & 0\\ 0 & 0 & q_3 & 0 & 0 \\ 0 & 0 & 0 & q_4 & 1 \end{pmatrix} and the matrix A (which is obviously related to P, but I can't see...
  16. J

    Tension at the lowest point of a hanging chain

    Homework Statement Chain of mass M hangs between 2 walls with its ends at the same height. The chain makes an angle θ with each wall. Find the tension at the lowest point of the chain. a) By considering the forces on half of the chain. b) By using the fact that the height of the chain is...
  17. F

    Chain rule and Kinematic quantities x,v,a

    Hello Forum, I have a couple of kinematics questions. The position of a point object is given by the position vector x(t). Speed is v(t)=dx(t)/dt and the acceleration a(t)= dv(t)/dt. What if we wanted to know the velocity and/or the acceleration as a function of position, i.e v(x) or a(x)...
  18. L

    What is the definition of the degrees of freedom in a kinematic chain?

    As I know, when a link in a kinematic chain is fixed, the kinematic chain will become a mechanism. And, I only know the definition of the DOF of a mechanism. Is there the definition of the DOF of a kinematic chain ?
  19. Jordan&physics

    What is the force exerted by a falling chain on a table?

    Homework Statement 1. (30 points) A very flexible uniform chain of mass M and length L consisting of very small links is suspended from one end so that it hangs vertically, the lower end just touching the surface of a table. The upper end is suddenly released so that the chain falls onto the...
  20. C

    Confused About the Chain Rule for Partial Differentiation

    Hey all, I am reading Goldstein and I am at a point where I can't follow along. He has started with D'Alembert's Principle and he is showing that Lagrange's equation can be derived from it. He states the chain rule for partial differentiation: \frac{d\textbf{r}_i}{dt}=\sum_k \frac{\partial...
  21. J

    How Does the Chain Rule Apply in Polar Coordinates?

    Homework Statement z = ƒ(x,y), x = rcos(θ), y = rsin(θ) Use the chain rule to show that: \frac{1}{r^{2}}\frac{\partial ^{2} z}{\partial \theta ^{2}} = sin^{2}(\theta)\frac{\partial ^{2} z}{\partial x^{2}}-2sin(\theta)cos(\theta)\frac{\partial ^{2} z}{\partial x \partial...
  22. nettle404

    Modeling a hanging chain as a PDE

    Homework Statement A flexible chain of length \ell hangs from one end at x=0 but oscillates horizontally. Let the x axis point downwards and the u axis point to the right. Assume that the force of gravity at each point of the chain equals to the weight of the part of the chain below the point...
  23. A

    Conservation of energy for a system

    Homework Statement Figure shows a massless wheel of radius R on which at a point a mass m is fixed and a uniform chain of mass 2m is tied to it which passes over the rim of the wheel and half of its length is hanging on other side as shown in the figure. When a small clockwise jerk is given to...
  24. P

    What Is the Expected First-Passage Time in a Markov Chain?

    Homework Statement Markov process has probabilities p_{j,j+1} = 1-p_{j,0} = (\frac{j+1}{j+2})^k for j=0,1,2,... If T_j = min[n>1 : X_n=j] What is E[T(j)|X_0=j] for j=0,1,2,...? Homework EquationsThe Attempt at a Solution I figured that T(j)|X_0=j is j+1 but don't know how to work out the...
  25. B

    Proton-proton I chain and mass defect reactants and products

    Hello there, I'm starting S382 astrophysics with the OU. The course book says "The proton-proton chain converts four hydrogen nuclei (protons) into a ^4_2He nucleus, two positrons that quickly collide with electrons and are annihilated, and two neutrinos. Hence, branch I of the p-p chain may be...
  26. S

    Chain Rule Problem (Partial derivatives)

    Homework Statement Homework EquationsThe Attempt at a Solution I have the solution to this problem and the issue I'm having is that I don't understand this step: Maybe I'm overlooking something simple but, for the red circled part, it seems to say that ∂/∂x(∂z/∂u) =...
  27. synMehdi

    Automotive Can You Convert a Trike Differential Gear to a Chain Drive for a Kart?

    Hi, I'm a beginner in mechanics so I need some help with a project I started. My project consists of a small kart-like car. The car will be Rear-Wheel-Drive and Rear-Engine too (RR layout). and I am willing to use a small 110cc moped engine. I have a trike Differential Gear for the rear shaft...
  28. N

    Can a Bullet Break a Bike Chain at 6m Underwater?

    < Mentor Note -- OP has been reminded to use the HH Template and show their work in future posts... > A man fires a bullet into a swimming pool. There is a distance of 6m from the gun to the bottom of the pool. Does the bullet have enough force through the water to break an average bike chain...
  29. Thor90

    Eigenstate solution for a spin chain with Hubbard model

    Homework Statement I am trying to solve the model analitically just for 2 sites to have a comparison between computational results. The problem is my professor keeps saying that the result should be a singlet ground state and a triplet of excited states, but when I compute it explicitally I...
  30. C

    How Does the Chain Rule Explain Acceleration in Terms of Distance?

    Hi I'm learning about The chainrule, and I understand how to apply the chain rule on various problems, but there is a problems I don't understand how works: 1) The book I'm reading writes acceleration as a=v*(dv)/dt And IT argues that v=ds/dt and a=dv/dt (which i understand) So therefore by...
  31. Adoniram

    A chain falling off a rotating disc

    Homework Statement A uniform chain of length L = πR and mass M is placed on the upper half of a uniform thin disc of radius R and mass M. The disc is placed vertically and can rotate freely about its center that is fixed in space. With a small disturbance the chain starts to fall. Find out the...
  32. evinda

    MHB Countable Chains of Subsets in an Infinite Set

    Hello! (Wave) A family of sets is called chain if any two sets of the family are comparable, i.e. the first set contains the second or the second the first. How many sets can a chain of subsets of an infinite set contain? More specifically, is there an infinite set, each chain of which is...
  33. Kinta

    Understanding Partial Derivatives in $$\frac {\partial f}{\partial \alpha}$$

    Homework Statement I'm unable to see fully how the following equality is determined: $$\frac {\partial f(y + \alpha \eta, \, y' + \alpha \eta', \, x)}{\partial \alpha} = \eta \frac {\partial f}{\partial y} + \eta' \frac {\partial f}{\partial y'}$$ where ##y = y(x)##, ##\eta = \eta (x)##, and...
  34. U

    How Is the Polyacetylene Chain Structured in a 1D Lattice Model?

    Homework Statement The polyacetylene chain is a 1D chain of Carbon atoms with single bonds and double bonds in succession. Spacing for single bond is ##a_s = 0.144~nm## and spacing for double bond is ##a_d = 0.136~nm##. Describe the structure using a "lattice" and a "basis". Sketch the...
  35. S

    Solid State Physics: Atomic chain oscillations

    Homework Statement Ok, don't get angry with me. The original problem is from Solid State Physics but my problem is very well in "Introductory physics". Here is the problem: The chain consists of molecules, which has three atoms, each with mass ##M##. Spring constant between the atoms inside...
  36. W

    Dispersion relation for diatomic linear chain.

    Hi. Here's the dispersion relation for a diatomic linear chain, where the distance is a/2 between each atom. My issue here is that if you set m_1=m_2=m, i.e. set both atoms equal to each other, it doesn't automatically reduce to the old acoustic dispersion relation as the ± term doesn't...
  37. George Zucas

    Drive Chain Connection: Single or Double Roller?

    Hello all, I have two rollers that will be rotated which will in turn rotate another tube on top of them. The connection will be made with a drive chain between the motor sprocket and the roller sprocket(s). My question is, would it be better to connect the chain to one roller, or to both? I'd...
  38. gfd43tg

    Maximize crude oil chain weekly profit

    Homework Statement Homework EquationsThe Attempt at a Solution I sort of want to make this into a linear program problem, but I think that it should be solvable without it, since I never learned about it in this particular course. I will just work with Fuel Process 1 as an example, and I can...
  39. C

    Can Calculus Problems be Solved by Factoring?

    Homework Statement [/B] hi could some body please help me factorise this please ? any chance of a few stages would be much appreciated Homework EquationsThe Attempt at a Solution my attempt , but my solutions say otherwise ? [/B]
  40. R

    Change of Variables Question with chain rule

    Homework Statement Consider the function of two variables: u(x,y) = f(x-y) + g(x+(1/3)y) where f(s) and g(t) are each arbitrary functions of a single variable. Using the change of variables: s = x-y t = x-(1/3)y use the chain rule to determine the first and second derivatives of u with...
  41. Drakkith

    Mastering the Chain Rule: Tips and Tricks for Calculus Students

    I'm in Calc 1 and the Chain Rule is giving me one hell of a rough time. I've spent about 10-12 hours over the last few days just on the homework problems in this one section (only getting about 15-20 problems done) and still feel like I barely understand it. Does anyone have any tips, tricks...
  42. binbagsss

    Algebra /derivatives/ chain rule/

    Homework Statement ##J=r^{2}\dot{\phi}## [1] ##\dot{r^{2}}=E^{2}-1-\frac{J^{2}}{r^{2}}+\frac{2MJ^{2}}{r^{3}}+\frac{2M}{r}##. [2] (the context is geodesic equation GR, but I'm pretty sure this is irrelevant). where ##u=r^{-1}## Question: From these two equations to derive...
  43. goonking

    Calculating Force Needed to Pull Chain Onto Table

    Homework Statement Homework Equations W = F d F = ma The Attempt at a Solution so in order to get the whole chain on the table, we need to pull the chain 0.65 meters onto the table. since 0.65 meters is hanging off the table, the gravity is acting on it, therefore F=ma where m is half the...
  44. M

    Chain Rule with Partial Derivative?

    Homework Statement Given that the surface x^7y^2+y^4z^6+z^8x^8+9xyz=12 has the equation z=f(xy) in a neighbourhod of the point (1,1,1) with f(x,y) differentiable, find the derivatives. df/dx (1,1) = ? d^2f/dx^2 (1,1) = ? Homework EquationsThe Attempt at a Solution df/dx (1,1) I got -24/23 or...
  45. powerof

    Symmetry in second order partial derivatives and chain rule

    When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...
  46. P

    MHB Some more Brownian motions and Birth-Death processes in Markov Chain

    Hi, I have a deadline tomorrow, and I need some urgent help now. I don't have a background in Markov chain, so I would be very very very very thankful if you can help with solutions step-by-step. Thanks a lot in advance. Below come the questions: 2) Give an example of birthrates λ(i) > 0 such...
  47. P

    Finite T transverse magnetization of transverse Ising chain

    Homework Statement Consider the transverse field Ising model, with $$H=-J\sum_i\left(\sigma^x_i\sigma^x_{i+1}+g\sigma^z_i\right)$$ I have to calculate the magnetization $$\langle\sigma_z\rangle$$ at finite temperature. Homework EquationsThe Attempt at a Solution I have to say, I'm a bit lost.
  48. P

    MHB Some questions about Brownian Motion and Birth-Death in Markov chain

    Hi, I need urgent answers. Basically, I don't have background in Markov and I don't need to learn it now actually. But I have to solve the questions below somehow. If somebody can give detailed answers to the questions below (From beginning to the final solution with explanations), then I will...
  49. ElijahRockers

    What is the proportion of time the runner runs barefooted using a Markov Chain?

    Homework Statement A certain laid-back runner owns three pairs of shoes, which he keeps by either the front or back doors of his house. Each morning he is equally likely to leave through the front or back door, and then after his run, he is equally likely to enter through the front or back...
  50. R

    Solve Chain Rule Equation: y=(tan^-1(6x))^2

    Homework Statement y = (tan^-1(6x))^2 Homework Equations Chain Rule, power rule? The Attempt at a Solution Okay, so I did power rule to bring it to 2(tan^-1(6x)) Then, I know to use the chain rule... I get 2(tan^-1(6x)*(1/1+x^2)... I know u = 6x so I play 6 into x^2 and I get 6x^2... I see...
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