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. binbagsss

    I Integration - chain rule / functional

    I have ## \int_{t = 0}^{t = 1} \frac{1}{x} \frac{dx}{dt} dt = \int_{t = 0}^{t = 1} (1-y) dt ## [1] The LHS evaluates to ## ln \frac{(x(t_0+T))}{x(t_0)} ##, where ##t_{1}=t_{0}+T## My issue is that, asked to write out the intermediatary step, I could not. I am unsure how you do this when the...
  2. S

    Partial derivatives and chain rule

    Homework Statement a. Given u=F(x,y,z) and z=f(x,y) find { f }_{ xx } in terms of the partial derivatives of of F. b. Given { z }^{ 3 }+xyz=8 find { f }_{ x }(0,1)\quad { f }_{ y }(0,1)\quad { f }_{ xx }(0,1) Homework Equations Implicit function theorem, chain rule diagrams, Clairaut's...
  3. A

    I Force Applied to Chain at Max Length

    I want to know. If you have a car that weighs 4 thousand pounds, and it is moving at 60 mph with a chain on it. What would be the amount of force applied to the chain when the chain reaches its maximum length. This is ignoring the fact that the car would rip apart when the chain stops. This is...
  4. F

    Construct a Markov Chain: How to Generate Xn's Using the Sequence U0, U1, ...?

    Homework Statement Describe the construction of a Markov chain X0, X1, ... on Ω ∈ (0, 1) with state space S = {1, 2, ..., s} and S X S PTM P and initial state X0 ~ ν (probabilities distributed like vector ν). Use the sequence U0, U1, ... to generate the Xn's Homework Equations U0, U1 is a...
  5. S

    I Energy spectrum of a chain of quantum oscillators

    I am trying to derive the energy spectrum of a 1D chain of identical quantum oscillators from its Hamiltonian by Fourier transforming the position and momentum operator. I came across this: https://en.wikipedia.org/wiki/Phonon#Quantum_treatment However, I am unsure of the mathematics...
  6. A

    A Canonical perturbation for infinite chain

    I've been Dealing with a problem of perturbation of the movement of an infinite chain of harmonic oscillator and I tried to apply the von Zeippel-Poincare formalism of canonical perturbation theory just to see what I get. This was too naive since I quickly stumbled into the problem of defining...
  7. Turbodog66

    Using Chain Rule to Find Partial Derivatives of a Multivariable Function

    Homework Statement Suppose $$z=x^2 sin(y), x=5t^2-5s^2, y=4st$$ Use the chain rule to find $$\frac{\partial z}{\partial s} \text{ and } \frac{\partial z}{\partial t}$$ Homework Equations $$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} +...
  8. ManicPIxie

    Fundamental Theorem of Calc Problem using Chain Rule

    Homework Statement F(x) = (integral from 1 to x^3) (t^2 - 10)/(t + 1) dt Evaluate F'(x) Homework Equations Using the chain rule The Attempt at a Solution Let u = x^3 Then: [((x^3)^2 - 10) / (x^3 + 1)] ⋅ 3x^2 *step cancelling powers of x from fraction* = (x^3 - 10)(3x^2) = 3x^5 - 30x^2 I am...
  9. D

    Can you continue the French Names Chain?

    Lets make a chain that would cover French Names. I will start with a French name and next person will answer the name which will start with the last letter of the name. I'll start with Abban.
  10. I

    I Understanding the Chain Rule in Derivatives: An Analysis of MIT Lecture Video

    While solving an equation, the lecturer was using substitution in this video: x=au was subbed in for Psi at timestamp 39:27 d/dx = (1/a)(d/du). I get that. But then the second derivative is stated as being d2/dx2 = (1/a2)(d2/du2) How is it (1/a2) if we do not know if there is an "a" in the...
  11. K

    Free-body diagrams and Newton's laws with a suspended chain

    Homework Statement The chain comprising three rings (each of mass ##0.25kg##) is suspended from a massless rope, and a pulling force ##\left(F=9N\right)## is exerted upwards on the rope. Picture: http://i.imgur.com/xeaiBsc.jpg?1. I need to find the values of all the unknowns. Homework...
  12. karush

    MHB What is the Chain Rule for Integration?

    $\tiny\text{Whitman 8.7.18 chain rule} $ $$\displaystyle I=\int { \left({t}^{3/2}+47\right)^3 \sqrt{t} } \ d{t} ={ \left({t}^{3/2}+{47}^{}\right)^4/6 } + C$$ $$\begin{align} \displaystyle u& = {t}^{3/2}+47& du&=\frac{3}{2}{t}^{1/2} \ d{t}& \\ \end{align}$$...
  13. H

    I Derive the formula for gradient using chain rule

    Consider a surface defined by the equation ##g(x, y, z)=0##. The intersection between this surface and the plane ##z=c## produces a curve that can be plotted on an x-y plane. Find the gradient of this curve. By chain rule, ##\frac{\partial y}{\partial x}=\frac{\partial y}{\partial...
  14. binbagsss

    Chain rule / Taylor expansion / functional derivative

    Homework Statement To show that ##\rho(p',s)>\rho(p',s') => (\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0## where ##p=p(z)##, ##p'=p(z+dz)##, ##s'=s(z+dz)##, ##s=s(z)## Homework Equations I have no idea how to approach this. I'm thinking functional derivatives, taylor expansions...
  15. H

    I Is it necessary to use a different function name in the chain rule?

    Is the chain rule below wrong? What I propose is as follows: Given that ##x_i=x_i(u_1, u_2, ..., u_m)##. If we define the function ##g## such that ##g(u_1, u_2, ..., u_m)=f(x_1, x_2, ..., x_n)##, then ##\frac{\partial g}{\partial u_j}=\sum_{i=1}^n\frac{\partial f}{\partial x_i}\frac{\partial...
  16. karush

    MHB Can the Chain Rule Help Me Integrate This Tricky Function?

    $$\tiny\text{Whitman 8.7.15 Chain Rule} $$ $$\displaystyle I=\int \frac{\sec^2\left({t}\right)}{\left(1+\tan\left({t}\right)\right)^2}\ d{t} =\frac{-1}{2\left(1+\tan\left({t}\right)\right)} + C$$ $\begin{align}\displaystyle u& = \tan\left({t}\right)& du&= \sec^2 \left({t}\right)\ d{t} \\...
  17. F

    B How do forces propagate through the chain?

    If I have a chain suspended from a hook, such that gravity is pulling it downward, how does that gravitational force propagate through the chain? What would happen if the gravitational source suddenly disappeared? What would happen if the hook suddenly disappeared? What I'm interested in, is...
  18. H

    I Chain on a slope remains still if its end points are of equal height?

    Let ##\vec{a}## be the gravitational acceleration along the curve. Then ##|\vec{a}|=a=-g\sin\theta## and ##\vec{a}=a_x\,\vec{i}+a_y\,\vec{j}=-g\sin\theta\cos\theta\,\vec{i}-g\sin^2\theta\,\vec{j}## My question is why does the solution below ignore the direction of ##\vec{a}## and calculate...
  19. S

    Three/four sprockets in horizontal line can be run by chain?

    Hello people, I have a serious design problem for designing a sprocket-chain system. I want to know that can three sprockets as in picture (in attachment) can be rolled by a chain? The main concern is for middle sprocket it have few teeth above and few teeth in bottom connected to chain. The...
  20. wrobel

    A Finding Equilibriums on a Cone with a Chain Loop

    Imagine a right circular cone with smooth surface. The cone is stated such that its axis is parallel to the standard gravitational field g. And you have a piece a thin homogeneous chain. Then you connect the tips of the chain to obtain a loop. You put this loop on the cone: It is clear...
  21. C

    Music About music, sounds and how they interact with the body

    First of all, Thank you very much for you time if you're reading my question. I just want to know how things work (i guess all of us), and some guidance will be more than welcomed. One of the questions that i have in my mind is: Could the sound waves (e.g:a song or a sound) make some...
  22. A

    I Chain rule in a multi-variable function

    Suppose you have a parameterized muli-varied function of the from ##F[x(t),y(t),\dot{x}(t),\dot{y}(t)]## and asked to find ##\frac{dF}{dt}##, is this the correct expression according to chain rule? I am confused because of the derivative terms involved. ##\frac{dF}{dt}=\frac{\partial...
  23. r_prieto5

    Sprocket diameter, chain link pressure, transverse vibration

    Homework Statement Power=P, rotation speed n1, rotation speed n2, chain center distance c, life = Lh All I need for this one is the formula for sprocket diameter. I have found calculators (https://www.rbracing-rsr.com/calcsprocketdiam.html) but no reference to the formula. Chain pitch and...
  24. kdrdgn07

    I Fission Chain Reaction: Is it Possible?

    I'm wondering that is it possible? I mean, certain radioactive matter can undergo fission? Sure I know half-life and radioactive decay. This question is asked roughly. I just learn is it possible? Thank you for answers
  25. S

    MHB Differentiation with fractions, radicands, and the power chain rule

    Differentiate the following two problems. 1. x divided by the square root of x squared+ 1 2. The square root of x + 2 divided by the square root of x - 1 Thank you.
  26. C

    A thermodynamics polymer chain problem

    A one-dimensional polymer molecule (rubber) is chain of N links of the same length a, the links can go either forward or backward but always stay parallel to the x axis. If one denotes the coordinates of the joints are ${x_0, x_1, . . . , x_N}$ , then $|x_n − x_{n+1}| = a$. The energy of the...
  27. K

    I Solve Chain Rule Confusion with Diff. Eq. | Help

    while solving differential equations, I got a bit confused with chain rule problem. The solution says below yprime = z then y double prime = z (dz/dy) = z prime but I don't understand why the differentiation of z is in that form. Please help...
  28. Amrator

    Partial Derivatives Using Chain Rule

    Homework Statement Suppose ω = g(u,v) is a differentiable function of u = x/y and v = z/y. Using the chain rule evaluate $$x \frac{\partial ω}{\partial x} + y \frac {\partial ω}{\partial y} + z \frac {\partial ω}{\partial z}$$ Homework EquationsThe Attempt at a Solution u = f(x,y) v = h(y,z)...
  29. K

    Oscillations of a free hanging chain

    Homework Statement I am trying to find an equation for a free hanging chain of mass m and length L. The chain is hanging vertically downwards where x is measured vertically upwards from the free end of the chain and y is measured horizontally. Homework Equations [/B] I derived this...
  30. C

    Can the Markov chain be used in qubit manipulation?

    Can a Markov chain be used to generate a set of qubits given their initial state?
  31. C

    Translate change of length to force in hanging chain problem

    I'm trying to write a simple script in blender python in order to show load deflection in cloth simulation. My question is: Is it possible to translate the change of length (distance between two nodes), into a force? (Newton between those nodes)? In addition you will find a minimal example. The...
  32. J

    Understanding the Chain Rule: (df/dx) + (df/dy)* (dy/dx)

    (df/dx) + (df/dy)* (dy/dx) = df(x,y)/dx My book mentions the chain rule to obtain the right side of the equation, but I don't see how. The chain rule has no mention of addition. The furthest I got was applying the chain rule to the right operant resulting in: df/dx + df/dx = 2(df/dx)
  33. B3NR4Y

    Using the mean value theorem to prove the chain rule

    Homework Statement I and J are open subsets of the real line. The function f takes I to J, and the function g take J to R. The functions are in C1. Use the mean value theorem to prove the chain rule. Homework Equations (g o f)' (x) = g'(f (x)) f'(x) MVT The Attempt at a Solution [/B] I know...
  34. Math Amateur

    MHB Real Valued Functions on R^3 - Chain Rule ....?

    I am reading Barrett O'Neil's book: Elementary Differential Geometry ... I need help to get started on Exercise 4(a) of Section 1.1 Euclidean Space ... Exercise 4 of Section 1.1 reads as follows:Can anyone help me to get started on Exercise 4(a) ... I would guess that we need the chain rule...
  35. M

    Chain Rule W/ Composite Functions

    Homework Statement If d/dx(f(x)) = g(x) and d/dx(g(x)) = f(x2), then d2/dx2(f(x3)) = a) f(x6) b) g(x3) c) 3x2*g(x3) d) 9x4*f(x6) + 6x*g(x3) e) f(x6) + g(x3) Homework EquationsThe Attempt at a Solution The answer is D. Since d/dx(f(x)) = g(x), I said that d/dx(f(x3)) should equal 3x2*g(x3), then...
  36. J

    Solving dy/dt = 0: Chain Rule & y(t)

    Not sure if this is the correct place to post this. dy/dt = 0, find y(t) My professor told me that the chain rule is used to determine that (dy/dt)*dt = dy, but I just don't see it. Multiply both sides by dt. (dy/dt)*dt = 0dt (dy/dt)*dt = 0 dy = 0, then integrating both sides: y = C dy/dt is...
  37. terryds

    What Force Does the Top Link Exert on the Middle Link in a Suspended Chain?

    Homework Statement A student tries to raise a chain consisting of three identical links. Each link has a mass of 200 g. The three-piece chain is connected to light string and then suspended vertically, with the student holding the upper end of the string and pulling upward. Because of the...
  38. M

    What is the acceleration and velocity of a falling chain on a cylinder?

    Hi! I am struggling with this problem the last two days and I cannot decide which solution is correct. Homework Statement A cylinder of radius R is fixed horizontally on the floor. A uniform chain of mass M and length L (L<πR/2) is placed on the cylinder in such a way that one end of the...
  39. J

    Limit of a continuous time Markov chain

    Homework Statement Calculate the limit $$lim_{s,t→∞} R_X(s, s+t) = lim_{s,t→∞}E(X(s)X(s+t))$$ for a continuous time Markov chain $$(X(t) ; t ≥ 0)$$ with state space S and generator G given by $$S = (0, 1)$$ $$ G= \begin{pmatrix} -\alpha & \alpha \\ \beta & -\beta\...
  40. F

    Do the Krebs cycle and electron transport chain require ATP?

    I cannot find a website that answers this question, and all diagrams I see do not show that ATP is used. Does this mean that these processes are intrinsically spontaneous? I can see how the electron transport chain is spontaneous, as the oxidation of oxygen to water is favorable (E 1/2=0.7V).
  41. A

    Can QM be described by Markov chain theory?

    Can we describe the intensity of spectral lines using Markov theory? No matter what is the initial state vector of the system, the final state will be reduced to a stationary vector whose elements represent the intensity of the spectral lines.
  42. S

    Chain rule for product of functions

    Here is a simple question : let f(g(x)) = h(x)*g(x). I want to calculate df/dx. If I use the product rule, I get g(x)h'(x) + h(x)g'x). Now if I use the composition/chain rule, I get df/dx = df/dg * dg/dx = h(x) * g'(x) which is different. I guess my df/dg = h is wrong, but I can't see what...
  43. O

    Calculating Forces in a Concrete Slab Supported by a Chain

    Homework Statement A 500-kg concrete slab is supported by a chain and sling attached to the bucket of the front-end loader shown. The action of the bucket is controlled by two identical mechanisms, only one of which is shown. Knowing that the mechanism shown supports half of the 500-kg slab...
  44. J

    Markov Chain - Time Reversibility proof

    Homework Statement Let X = {Xn : n ≥ 0} be an irreducible, aperiodic Markov chain with finite state space S, transition matrix P, and stationary distribution π. For x,y ∈ R|S|, define the inner product ⟨x,y⟩ = ∑i∈S xiyiπi, and let L2(π) = {x ∈ R|S| : ⟨x,x⟩ < ∞}. Show that X is time-reversible...
  45. throneoo

    Equation of motion of magnetic dipole chain

    Homework Statement Find the equation of motion of a chain of atoms in 1D with alternating magnetic dipoles At stationary equilibrium the atoms of mass m are separated by d , all displacements are small compared to d Homework Equations U=μBx=2μ2(μ0/4π)(1/x^3) F(x)=-dU/dx The Attempt at a...
  46. E

    Calculating Density of States and Occupied States in 1D Chain of Atoms

    Homework Statement The system is a chain of atoms in 1D length L and number of atoms N. and \epsilon_k=\hbar c_s k a) What is the density of states? b)The number of states that can be occupied (use boundary conditions) c) Determine w_d(I think this is the debye frequency) in terms of N,L,k...
  47. gonadas91

    Is it possible to have a spin chain with both bosons and fermions interacting?

    Hello, just as a thought, we are used to describe spin systems of either fermions or bosons (for example, the Ising or Heisenber models, where spins are considered to be all of the same type, in both cases fermions). However, I was wondering if it makes some sense to have a system where, for...
  48. R

    Solving the Mass of a Chain Hanging from a Ceiling

    1. https://gyazo.com/a298396e34c32669755a5caffe82290e https://gyazo.com/a298396e34c32669755a5caffe82290e A small ball of mass m = 30 g hangs from the ceiling on a l = 50 cm long light rope. From the side a chain is attached, as seen on the picture. Whats the mass of the chain? Heres the part I...
  49. dwdoyle8854

    Classical Dynamics -- Falling chain and energy conservation

    Homework Statement The statement of the question is:A chain of uniform linear mass density ##\rho##, length ##b## and mass ##M## hands as shown in the figure below. At time t=0, the ends A and B are adjacent, but end B is released. Find the tension in the chain at point A after end B has...
  50. M

    Partial derivatives and chain rule?

    F(r,s,t,v) = r^2 + sv + t^3, where: r = x^2 +y^2+z^2 /// s = xyz /// v = xe^y /// t = yz^2 find Fxx i have 2 solutions for this and i am not sure what is the right one the first solution finds Fx then uses formula : Fxx = Fxr.Rx + Fxs.Sx + Fxv.Vx+Fxt.Tx the 2nd solution find Fx then uses the...
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