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. M

    Computing Hamiltonian matrix for a 1-D spin chain.

    I'm not going to "follow the template provided" in the strictest sense - but I'm going to include all the same information expected - statement of the problem and a showing of how I've tried to do it, in the intended "spirit" of the template, since these different components are kind of "mixed"...
  2. F

    What is the correct method for calculating the speed of a bicycle chain?

    Homework Statement A bicycle has wheels 67.3 cm in diameter and pedal cranks 17.5 cm long. The cyclist pedals at a steady angular rate of 71.5 rev/min. The chain engages with a front sprocket 15.2 cm in diameter and a rear sprocket 6.45 cm in diameter. Homework Equations v=rw The...
  3. B

    What is the Work Done Lifting a Chain?

    Homework Statement A chain 40 feet in length and weighing 3 pounds per foot is hanging fully extended from a winch. Find the work done by the winch in winding up 30 feet of the chain. Homework Equations Typical segment of the chain: \Delta y_{i} Weight of typical segment of the...
  4. P

    Maximum Tension in the Bicycle Chain

    Figure 1 below shows drawings of a typical bicycle chain. Consider two components of the chain: (i) the pin, and (ii) the link. The chain is under tension caused by a load applied by a person pushing on one of the pedals. Calculate the maximum tension in the chain. Calculate the stresses in...
  5. N

    Solving Stress in Bicycle Chain for NDO

    Homework Statement See attachment for the problem. or visit the link http://i908.photobucket.com/albums/ac286/NDO_02/Capture.jpg Basically calculate the stress in the pin and the link in a bicycle chain. Homework Equations the average shearing stress = P/A = F/A i think the chain...
  6. S

    What is meant by the Pka value of a side chain in an amino acid?

    I know how to get the answer for this question using hesselback equation but I don't understand the question. I have some questions about the questiuon. 1. The PKa value of the sulphydryl (-SH) group of cysteine is 8.33. Calculate the fraction of anion to free sulfhydryl group at PH 7.O...
  7. M

    Can a Sprocket and Chain System Be Compared to a Rope on a Pulley?

    Hi everyone, I am just looking at some calculations i did a while ago, where i have a hydraulic ram with a rotating sprocket on top, with chain passing over it attached to a load at the the other end. It is a similar set up to a forklift. My query is: Can the sprocket and chain set up be...
  8. C

    How Do You Differentiate e^(b*t*ln(t)) + ln(x) with Respect to t?

    D e^(b*t*ln(t)) + ln(x) respect to t my answer: t^(b*t)*(ln(t)*b)+b + 1/x
  9. Q

    Why Multiply by the Derivative of the Inner Function in the Chain Rule?

    Chain Rule - intuitive "Proof" Suppose y = f(u), and u = g(x), then dy/dx = dy/du * du/dx. In an intuitive "proof" of the chain rule, it has this step: dy/dx = \lim_{\Delta x \to 0} \frac {\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac {\Delta y}{\Delta u} * \frac {\Delta u}{\Delta x}...
  10. mg0stisha

    Chain Rule for Derivatives: Differentiating a Product with Chain Rule

    Homework Statement Differentiate f(x)=(3x^{2}+4)^{3}(5-3x)^{4} Homework Equations N/A The Attempt at a Solution I can see that this derivative is a product, yet also involves using chain rule. With this being said, am i just supposed to evaluate these separately using chain...
  11. R

    How do I expand the chain rule to second partial derivatives?

    We have f(x(y,z),t(y,z)). This is more of a study question. I don't know how to expand out d^{2}f/dz^2 I know df/dz = df/dx*dx/dz + df/dt*dt/dz, but I don't know how to expand this to the 2nd derivative. I think the product rule comes into play? Not really sure. Thanks for your help.
  12. W

    Chain Rule with partial derivatives

    Homework Statement Let T= g(x,y) be the temperature at the point (x,y) on the ellipse x=2sqrt2 cos(t) and y= sqrt2 sin(t), t is from 0 to 2pi. suppose that partial derivative of T with respect to x is equal to y and partial derivative of T with respect to y is equal to x. Locate the max and...
  13. S

    Derivatives Chain Rule question

    Homework Statement http://img21.imageshack.us/img21/6784/probi.jpg Homework Equations The Attempt at a Solution i have no idea where to begin and my textbook doesn't have any examples that look like this question.. can someone give me hints? whats the equation that l=10m after...
  14. S

    Markov Chain of Stochastic Processes

    I would like to construct a model using a markov chain that has different stochastic processes for each state in the chain. Is there a term for such a thing, or anything similar to it? Thanks
  15. S

    Using chain rule to differentiate x^x

    Homework Statement Use the chain rule to find (d/dx)(xx) by using the function f(y,z)=yz. Homework Equations Chain rule: \frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt} The Attempt at a Solution I honestly have no clue on how to use...
  16. Y

    How do i show that a markov chain is irreducible?

    how do i show that a markov chain is irreducible?
  17. N

    Exponential of (Markov Chain) Transition matrix

    Hi, I have a (markov chain) transition matrix X which I understand. In particular each row of this matrix sums to 1. I have used this transition matrix to construct it's generator, Y. I.e. Y is the continuously compounded transition matrix, X = exp(Y) X*X = exp(2Y), etc both X and Y...
  18. L

    Do complementary genes move closer on the DNA chain?

    1. I read that each gene has a 50% chance to be transferred to a single gamete. Does the distance between two genes on the same chromosome influence the conditional probability for the second gene to be transferred to the same gamete as the first one? 2. Can the process of reproduction...
  19. K

    Partial Derivatives and The Chain Rule

    Homework Statement The length l, width w, and height h of a box change with time. At a certain instant the dimensions are l = 7 m and w = h = 9 m, and l and w are increasing at a rate of 6 m/s while h is decreasing at a rate of 3 m/s. At that instant find the rates at which the following...
  20. F

    Markov chain and state diagram

    Homework Statement I have the following queuing system: http://img39.imageshack.us/img39/8264/immaginetd.jpg that models voice traffic that come up with \alpha e \beta parameters, on both queue 1 and 2. When a source of voice is active causes traffic with exponential inter-arrival time which...
  21. D

    Max Tension of a Chain on Vertical Poles: Find T as a Function of L

    A chain is attached to the top of a vertical pole of height H located at x=0. The other end of the chain is attached to another pole of height H at x=a. Find the maximum tension T of the chain as a function of the length of the chain L. Assume that the weight density of the chain is $ (lambda)
  22. E

    Solving a Chain Rule Problem: Find Derivative of y

    Homework Statement I need to find the derivative of: y=\left(4x+3\right)^{4}\cdot\left(x+1\right)^{-3} Homework Equations Chain Rule Quotient or Product Rule The Attempt at a Solution So I tried to use quotient rule because...
  23. B

    Chain rule in partial derivative

    There is a theorem in partial derivative If x= x(t) , y= y(t), z= z(t) are differentiable at t_{0}, and if w= f(x,y,z) is differentiable at the point (x,y,z)=(x(t),y(t),z(t)),then w=f(x(t),y(t),z(t)) is differentiable at t and \frac{dw}{dt}=\frac{\partial w}{\partial x}\frac{dx}{dt} +...
  24. V

    Intermediate variable chain rule question.

    Homework Statement Suppose that w=f(x,y), x=r*cos(θ), y=r*sin(θ). Show that: (\frac{\partial w}{\partial x})^2 + (\frac{\partial w}{\partial y})^2 =(\frac{\partial w}{\partial r})^2 + \frac{1}{r^2} (\frac{\partial w}{\partial \theta})^2 Homework Equations the multivariable chain rule...
  25. M

    Would a knot in a falling chain make a difference?

    While I was looking up about free falling chain problem(the question about a chain falling on to a table), I wondered how it would be if the chain is knotted. What I'm thinking is that if the chain has a knot, would it make much difference? Two questions; 1) Once the knot hits the table...
  26. S

    How Does a Falling Chain Affect Scale Readings?

    Homework Statement Hi guys, I have a question on problem 4.11 in Kleppner and Kolenkow's mechanics book. A chain of mass M and length L is suspended vertically with its lowest end touching a scale. The chain is released and falls onto the scale. What is the reading of the scale when a length...
  27. J

    Help understanding derivatives of time; chain rule.

    I'm having a bit of a hiccup understanding the differentiation that I am doing... I'd like to be clear on the concept rather than just knowing 'apply chain rule'. So I have a particle with equation: y=a(1+cos\theta) now the derivative with respect to time (the velocity in y) is...
  28. M

    Ball and chain- (kidding, no chain)

    If one ball is dropped and another is thrown down from the same height, which will arrive at the ground first? Won't they get there at the same time? Thanks MB
  29. H

    Markov Chain Homework: Expressing Transition Matrix P in Terms of G

    Please Help! Markov Chain Homework Statement Let X0 be a random variable with values in a countable set I. Let Y1, Y2, ... be a sequence of independent random variables, uniformly distributed on [0, 1]. Suppose we are given a function G : I x [0, 1] -> I and define inductively for n >=...
  30. Z

    The Chain Rule - Simple but Complicated Problem

    Find the value of (f o g)' at the given value of x. f(u) = u5 + 1 u = g(x) = sqrt(x) x = 1 Ok so the section is based on the chain rule and came right out of my calculus book. I seem to be doing the problem right, i check my attempt over a few times and cannot seem to find the problem (the...
  31. M

    Chain Rule and Partial Derivatives for Differentiable Functions

    Homework Statement Prove that if z(x,y)=e^y f(ye^{\frac{x^2}{2y^2}}) is differentiable, then (x^2-y^2) \frac{\partial z}{\partial x} + xy\frac{\partial z}{\partial y} = xyz Homework Equations Chain Rule. The Attempt at a Solution A similar question is solved like this: Have this: z(x,y) =...
  32. T

    Mastering the Chain Rule with Fractions for Calculus Students

    Okay, I know how to differentiate regular functions. But when it comes to fractions, I'm hopeless. This may be an extremely simple one to some, here is the function; "1/4x-7" I have to differentiate that using the chain rule. I think that u=4x-7, but I am not sure. As i said, I am horrible...
  33. 5

    A multivariable chain rule problem

    Hello all, I am stuck on what seems like a rather simple problem: Let f:\mathbb{R}^3 \rightarrow \mathbb{R} and g:\mathbb{R}^2\rightarrow \mathbb{R} be differentiable. Let F:\mathbb{R}^2 \rightarrow \mathbb{R} be defined by the equation F(x,y)=f(x,y,g(x,y)). Find DF in terms of the...
  34. G

    Deriving y= [x+(x+(sin(x)2))5]3

    Find the derivative of y = [x + (x + (sin(x)2))5]3 I know that power and chain rule combined uses the equation n[g(x)]n-1 * g'(x) I don't even really know where to start with so many layers in the equation. I can only find examples with only one power. with my attempt I got...
  35. J

    Does a Recruit Have to Follow All Orders?

    Random question: suppose in the military that a recruit is given a direct order to do something by his captain, which obviously goes against the captain's own orders. Is the recruit exempt from the punishment because he was just following his own orders? If he is not exempt from the...
  36. R

    Proving Horizontal Component of Tension in Stationary Chain

    I'd just like some confirmation on my answers, and I'd appreciate it if someone could take the time to explain why what I did is right. I solved it with a lot of hand-waving, so I'm very unsure of how I reached the answers. A uniform chain of weight W is strung between two vertical walls. The...
  37. B

    Critical exponents for the Heisenberg AFM spin-1/2 chain

    Hi everybody! I'm looking for the critical exponent ν (i.e. the one of the correlation length) of the Heisenberg (i.e. equal coupling in all directions) antiferromagnetic spin-1/2 model in 1D... Furthermore, do you know to which universality class it belongs? Is it true that it's the...
  38. D

    Euler Problem #14: Find Longest Chain <1M

    im trying to do the Euler problem #14 - to determine which starting value under one million produces the longest chain and here's my code in Python Count=[] Count2=[] List=[] def seq(n): if n%2==0: return n/2 else: return 3*n+1 m=1 for i in range(1,1000000): j=i List.append(j) while...
  39. Saladsamurai

    Partial Differentiation Help with Chain Rule

    In fluid mechanics velocity is given in the form \textbf{V}=u\textbf{i}+v\textbf{j}+w\textbf{k} Homework Statement A two-dimensional velocity field is given by \textbf{V}=(x^2-y^2+x)\textbf{i}+(-2xy-y)\textbf{j} At (x_o,y_o) compute the accelerations a_x\text{ and }a_y I am...
  40. P

    Proof of Chain Rule: Understanding Delta(u) & Delta(x)

    Hello everyone, I was looking at the proof of chain rule as posted here: http://web.mit.edu/wwmath/calculus/differentiation/chain-proof.html" I am having trouble understanding why delta(u) tends to 0 as delta(x) tends to 0. Can someone point out to me why that is so? Many thanks, Luca
  41. B

    How Do You Model Radioactive Chain Decay in Differential Equations?

    Homework Statement I'm looking at a problem from MIT's Open Courseware on radioactive chain decay, i.e. one element decays into another decays into another, finding the quantity at time t.Homework Equations The standard linear differential equation governing exponential decay.The Attempt at a...
  42. S

    What is the Chain Rule and How Do You Apply It?

    Homework Statement x2+y2=1 I want to differentiate this equation. I know that the answer is 2x+2y*y'=0. Homework Equations The chain rule. The Attempt at a Solution I don't understand how you get 2y*y' from y2. Shouldn't it just be 2x+2y=0?
  43. MathematicalPhysicist

    Stationary probabilities.(Markov chain).

    We are given two states 1,2 in an irreducible and positive recurrent Markov chain, and their stationary probabilities \pi_1 and \pi_2 respectively, try to characterise in general the probability (distribution) of the number of visits in state 2 after two consecutive visits in state 1. Any hints?
  44. N

    Solve Chain Torque Problem: Find Distance from Left Chain

    Homework Statement A horizontal uniform board of weight 125N and length 4m is supported by vertical chains at the ends. A person weighing 500N is sitting on the board. The tension in the right chain is 250N. How far from the left chain is the person sitting? Homework Equations...
  45. A

    Rat and Cat Markov Chain

    Homework Statement Rat and Cat move between room 1 and 2 using different paths. Their motions are governed by their respective transition matrices: [0.9, 0.1 ; 0.2, 0.8] [0.6, 0.4 ; 0.3, 0.7] (semi colon is a new line in the matrix, like in matlab) If they are ever in the same room...
  46. D

    Solving Mass-on-Scale Problem with Falling Chain

    Homework Statement A chain of mass M and length L is suspended vertically with the lower end touching a scale. the chain is released and falls onto the scale. what is the reading of the chain when a length x is fallen? neglect the size of individual links Homework Equations dp = IMPULSE=F*dt...
  47. F

    Calculating Chain Velocity Through Point Particle System

    chain velocity?? Homework Statement A chain of metal links with total mass M = 6 kg is coiled up in a tight ball on a low-rriction table. you pull on a link at one end of the chain with a constant force F = 67 N. Evntually the chain straightens out to its full length L = 0.9 m, and you keep...
  48. C

    How Does Energy Transformation Occur in Solar-Powered vs Electric Synthesizers?

    Homework Statement When the synthesizer Iimaginary device is used on solar power, describe, in simplest terms, the chain of energy transformations required. As well describe the chain for the original synthesizer running on electric power. Homework Equations I'm not to familiar with...
  49. N

    Finding Time-Derivative of V(x,y) with Chain Rule

    Homework Statement Hi all. I have an expression given by V(x,y) = ay+x2y2, where a is a constant. I wish to find the time-derivative of V(x,y), and this is what I have done: \frac{dV}{dt} = a\dot y + \frac{d}{dt}x^2y^2, where the dot over y represents differentiation w.r.t. time. My...
  50. E

    Solving Chain on Pulley: Find Velocity

    [SOLVED] Chain On Pulley Homework Statement Given: g = 9.8m/s^2 . A uniform flexible chain whose mass is 7 kg and length is 6 m. Given: A small frictionless pulley whose circumference is negligible compared to the length of the chain. Initially the chain is hung over the pulley with nearly...
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