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. T

    Multivariable chain rule and differentiability

    Homework Statement Hi I'm currently trying to revise for a Calculus exam, and have very little idea of how to do the following: Let f be defined by f(x,y) = (y+e^x, sin(x+y)) Let g be of class C2 (twice differentiable with continuous second derivatives) with grad(g)(1,0) = (1,-1) and Hg(1,0)...
  2. S

    Exploring the Multivariable Chain Rule Proof

    I'm looking at the proof of the multivariable chain rule & just a little bit curious about something. In the single variable chain rule proof the way I know it is that you take the derivative: f'(x) \ = \ \lim_{ \Delta x \to \infty} \frac{ \Delta y}{ \Delta x} and manipulate it as follows...
  3. S

    What Is the Total Force Exerted by a Falling Chain on the Ground?

    A chain of length L,mass M kept vertical so it just touches ground.A height x of it from bottom falls down.Find the total force exerted by it on ground. My attempt: Force due to its weight=Mg Force due to falling and change in momentum: consider small element dy at height y Change in...
  4. X

    What is the solution to the Hanging Chain Problem?

    Homework Statement A uniform chain of total length 'a' has a portion 0<b<a hanging over the edge of a smooth table AB. Prove that the time taken for the chain to slide off the table if it starts from rest is (a/g)1/2*ln(a+((a2-b2)/b)1/2)
  5. S

    Higher order differential equations and the chain rule (2 variables)

    Homework Statement The function F is defined by F (r, θ) = f (x(r, θ), y(r, θ)), where f is twice continuously differentiable and x(r, θ) = r cos θ, y(r, θ) = r sin θ. Use the chain rule to find d2F/dθ2Homework Equations The Attempt at a Solution I know that dF/dθ = (df/dx)(dx/dθ) +...
  6. D

    What is the role of kinetic energy in proton-proton fusion?

    Ive studied how chain reaction of proton-proton powers the stars, I also know that proton-proton fusion will convert firstly one proton to neutron then fuse it with the other proton...my question is... Is the main reason for them not being fused is that they both have +ve charge?
  7. D

    Environmental disaster chain of events (for story)

    I've written a scifi short story for a contest on the theme of environmental disaster, and I want to check the plausibility of the chain of events that occurs in the story. 1. It opens with a spill of massive quantities of methyl and phenyl isocyanate into a bay area on a planet in the...
  8. S

    Solving Tension in a Chain of 5 Links

    Homework Statement A chain consisting of five links, each of mass 0.10 kg, is lifted vertically with a constant acceleration of 2.5m/s^2. a.) Find the forces acting between adjacent links. b.) Find the force F exerted in the top link. Homework Equations F=ma W=mg The Attempt at...
  9. T

    How does the chain reaction start in a BWR reactor?

    Greetings, As someone who is interested in the history of the nuclear age, I have been unable to find an answer for this: Once the fuel rods are loaded into the fuel assembly and the assembly is loaded into the core, how does the chain reaction start which creates the heat? Do they...
  10. C

    Equilibrium Distance for Spring & Chain Homework

    Homework Statement Two carts can slide along a horizontal rail without friction. The carts are connected: (a) by an elastic spring of spring constant k and unstretched length l; (b) by a chain of length l and linear density p. The spring is going along the rail, the chain hangs in the...
  11. T

    Markov Chain Conditional Expectation

    Hello, in relation to Markov chains, could you please clarify the following equations: In particular, could you please expand on why the first line is equal. Surely from , along with the first equation, this implies that: I just don't see why they are all equal. Please could you...
  12. C

    Equilibrium Distance Between Carts with Spring and Chain

    Homework Statement Two carts can slide along a horizontal rail without friction. The carts are connected: (a) by an elastic spring of spring constant k and unstretched length l; (b) by a chain of length l and linear density p. The spring is going along the rail, the chain hangs in the...
  13. Z

    Proof of the chain rule

    Homework Statement I'm looking for a proof for the chain rule that is relatively easy to understand. Can someone show / link me one? Thanks. Homework Equations The Attempt at a Solution
  14. L

    Using the chain rule on a fraction?

    Homework Statement Find dy/dx at x = 2. y = (1 + s)/(1 - s); s = t - 1/t; t = sqrt(x) Homework Equations I know if f = f(g(x)), then f1(x) = f1(g(x)) * g1(x) The Attempt at a Solution I think I may need to combine the chain rule and quotient rule, but all of the separate...
  15. J

    Multivariate Calculus Chain Rule.

    Homework Statement Apply the two cases of th change rule. For example: The voltage V in an electrical circuit is slowly decreasing as a battery wears out and the resistance R is slowly increasing as the resistor heats up Use Ohm's law V=IR to find how the current is changing (with respect to...
  16. B

    Problem with proof of Chain rule for f:R->R

    Problem with proof of Chain rule for f:R-->R Hi, Analysts: I am going over problems in Rosenlicht's Intro. Analysis book. In this problem , he asks one to find the flaw in this argument to the effect that (f(g(x))'=f'(g(x))g'(x). Unfortunately, author does not clearly state the...
  17. B

    Solving the Chain Rule Equation with Differential Calculus

    Homework Statement Homework Equations The Attempt at a Solution a) (∂z/∂x)=-(∂f/∂x)*(∂z/∂f) i used that (AxB)=-(BxA) so i get (∂z/∂x)=-[-(∂z/∂f)(∂f/∂x)] =(∂z/∂x) is this correct if not can someone give me hints pls thanks
  18. S

    Changing V(x) to V(t): Chain Rule Application?

    I have a function for velocity, V in terms of position, x. The equation is of the form V(x) = a*x2+b*x+c. Initial conditions are x=0, t=0. How do I change from V(x) to V(t)? It seems this would be an application of the chain rule, dy/dx = dy/du * du/dx, but I'm struggling to adapt it to...
  19. S

    How Does Pulling a Chain Affect Its Thermal Energy?

    Homework Statement A chain of metal links with total mass m = 7 kg is coiled up in a tight ball on a low-friction table. You pull on a link at one end of the chain with a constant force F = 52 N. Eventually the chain straightens out to its full length L = 0.8 m, and you keep pulling until you...
  20. S

    Partial Derivates using Chain Rule

    Homework Statement Find: ∂f/∂x f(r,θ)=rsin^2(θ), x=rcosθ, y=rsinθ The Attempt at a Solution ∂f/∂r=sin^2θ ∂r/∂x=-cosθ/x ∂f/∂θ=2*r*cosθ*sinθ ∂θ/∂x=-1/sqrt(1-(x^2)/(r^2) ∂f/∂x = -sin^2θcosθ/x^2 + -2*r*cosθ*sinθ/sqrt(1-(x^2)/(r^2) ∂f/∂x = -y-sqrt(x^2+y^2) /...
  21. U

    Derivatives of Square Root Functions: Understanding the Chain Rule

    I'm kind of confused about how to approach a function with the chain rule. For example in the equation ƒ(x) = sqrt(1-sin(x)) I know i simplify it to ƒ(x) = 1-sin(x)^(1/2) but I'm lost from there.
  22. S

    Cannon Ball Chain Shot Impulse Problem

    Homework Statement Image you have one of the old fashion chain shots from the 1800’s where they had two cannon balls connected by a chain. Now let’s say you wish to try to fire the chain shot from two cannons. The synchronized cannon firing nearly works, but one cannon ball receives a...
  23. C

    Solving Chain Rule A(r,t) Problem

    Problem Use the chain rule to proof \dot{A}=\partial_t A+v_j\partial_jA_i Attempt at Solution \dot{A}=\frac{dA_i}{dt} = \partial_t A_i+\frac{dr_i}{dt}\frac{\partial A_i}{\partial r_i} Obviously v_j = \frac{dr_j}{dt} I'm puzzled where the v_j and partial d_j come in
  24. J

    Reducing the Order of a Cauchy-Euler Equation

    Homework Statement Reduce the order of a Cauchy-Euler Equation Homework Equations x = e^t \mbox{ and } \ln x = t The Attempt at a Solution \displaystyle \frac{d y}{d x} = \displaystyle \frac{d y}{d t} \displaystyle \frac{d t}{d x} = \displaystyle \frac{d y}{d t} \cdot...
  25. S

    Stationary distribution of countable Markov chain

    How do I find the stationary distribution of the Markov chain with the countable state space {0, 1, 2, ..., n, ...}, where each point, including 0, can either a. return to 0 with probability 1/2, or b. move to the right n -> n+1 with probability 1/2? Thanks.
  26. S

    Chain rule derivative applied to an ice cube

    Homework Statement A cubical block of ice is melting in such a way that each edge decreases steadily by 9.8 cm every hour. At what rate is its volume decreasing when each edge is 10 meters long? Homework Equations V(t) = (l(t))^3 m^3 l'(t) = 0.098 m/h The Attempt at a Solution...
  27. M

    Simple word problem: Chain rule

    Homework Statement One side of a triangle is increasing at a rate of 3cm/s and a second side is decreasing at a rate of 2cm/s. If the area of the triangle remain constant, at what rate does the angle between the sides change when the first side is 20cm long and the second side is 30cm, and...
  28. J

    What Determines the Velocity of a Falling Chain?

    Homework Statement uploaded Homework Equations rocket equation The Attempt at a Solution i can calculate the force acting on the chain by the ground using rocket equation but i cannot show that the velocity is that.
  29. M

    Prove the equality : Multivariable chain rule problem

    Homework Statement Prove that (\frac{\partial u}{\partial x})^{2} + (\frac{\partial u}{\partial t})^{2} = e^{-2s}[(\frac{\partial u}{\partial s})^{2} + (\frac{\partial u}{\partial t})^{2}].Homework Equations u = f(x,y) x = e^{s}cost y = e^{s}sint The Attempt at a Solution I started out by...
  30. E

    Discrete time Markov chain

    Homework Statement Let \left( X_n \right)_{n \geq 0} be a Markov chain on {0,1,...} with transition probabilities given by: p_{01} = 1, p_{i,i+1} + p_{i,i-1} = 1, p_{i,i+1} = \left(\frac{i+1}{i} \right)^2 p_{i,i-1} Show that if X_0 = 0 then the probability that X_n \geq 1 for all n \geq 1 is...
  31. pellman

    Chain rule for functions of operators?

    This is strictly a math question but I figured that since it is something which would show up in QM, the quantum folks might be already familiar with it. Suppose we have an operator valued function A(x) of a real parameter x and another function f, both of which have well defined derivatives...
  32. R

    Apparently easy Chain Rule Problem

    Homework Statement F(s) = ( s - \frac{1}{s^2})3 I have to calculate the derivative of this using chain rule everytime i try i get something way different than in the back of the book... my first move is 3( s - \frac{1}{s^2})2 X ( 1 + \frac{2}{s^3}) is this correct? then expand...
  33. S

    Equation of motion for a chain sliding down an edge

    I studied physics a long time ago and somebody just asked me this question. After trying for a while I couldn't work it out. The situation is this: there's a chain of length $l$ on a table, of which a portion, of length $x_0$, is hanging out (enough so that when you stop holding it down, the...
  34. M

    Chain of mass M with length L (SPhO 2009)

    Homework Statement A chain of mass M and length L is suspended vertically with its lower end touching a scale. The chain is released and falls onto the scale. What is the reading of the scale when a length of the chain x has fallen? You may neglect the size of the individual links. [10]...
  35. L

    Multivariable calculus. The chain rule.

    Homework Statement Let x=x^2ysin(u)tan(v), where x(u,v) and y(u,v) are smooth functions that, when evaluated at u=1 and v=-3 satisfy x=2.112, y=4.797, \partialx/\partialu = -3.491, \partialx/\partialv = -2.230 , \partialy/\partialu = 1.787 , \partialy/\partialv = 1.554. Then the...
  36. Y

    What Am I Missing in the Chain Rule Calculation?

    \hbox { Let }\; u(x,y)=v(x^2-y^2,2xy) \;\hbox { and let }\; t=x^2-y^2,\;s=2xy u_x = 2xv_t \;+\; 2yv_s u_{xx} = 2v_t + 4x^2 v_{tt} + 8xyv_{ts} + 4y^2 v_{ss} The u_{yy} can be done the same way and is not shown here. According to Chain Rule: u_x = \frac{\partial v}{\partial x}...
  37. M

    Learning Calculus: Chain Rule and Derivatives

    I am currently learning calculus and just had my lecture on the chain rule. I noticed that we haven't learned how to take the derivative of a function like 2^2+x or 3^4+x. Any example works.. Is this something I will learn later as I progress through calculus or what?
  38. Telemachus

    Composition and the chain rule

    Homework Statement I have a problem with the next exercise: Given de function f(x,y)=\begin{Bmatrix} \displaystyle\frac{xy^2}{x^2+y^2} & \mbox{ if }& (x,y)\neq{(0,0)}\\0 & \mbox{if}& (x,y)=(0,0)\end{matrix} with \vec{g}(t)=\begin{Bmatrix} x=at \\y=bt \end{matrix},t\in{\mathbb{R}} a) Find...
  39. T

    How Does the Falling Chain Problem Illustrate Variable Mass Dynamics?

    Homework Statement This is from Serway's book Prob 9.71...(busying preparing for GRE) A chain of length L and total mass M is released from rest with its lower end just touching the top of a table, as in figure (a). Find the force exerted by the table on the chain after the chain has...
  40. A

    Solving G(y) with the Chain Rule: Where Do I Start?

    I need some help with the chain rule...Thank you for helping me:) Question: G(y)=((x-1)^4)/(((x^2)+2*x))^7 I have no idea where to start.
  41. S

    Designing Chain Lifting Device for 400 Metric Tonnes

    I'm looking for some input on the design of a lifting device. The design is required to accurately lower lower a load of 400 metric tonnes through a height of 50m, return to the top of it's travel and lower the next load and so on. I'm currently thinking along the lines of a two-pronged tower...
  42. S

    How to Obtain the Transition Probability Matrix in a Birth Death Markov Chain?

    Hi I am trying to model the behaviour of 2 independent ON-OFF sources. My state diagram is as follows state 0 = both sources are OFF state 1 = 1 of the sources are ON state 2 = both sources are ON The transition rates are given as BIRTH RATE = lamda(i) = (N-I)*lamda DEATH RATE =...
  43. B

    Chain rule with second derivative

    Homework Statement I trying to find the second derivative of xe^x Homework Equations chain rule The Attempt at a Solution Two find the first derivative I use the chain rule. f'(y)g(y)+f(y)g'(y) so I get e^x+xe^x is the second derivative e^x+f'(y)g(y)+f(y)g'(y)...
  44. M

    Chain with distance-dependent mass problem

    1. A particle of mass m is tied on one end of a very long chain which has a linear density μ (kg/m) and lies on a surface with the chain wound next to it. The particle is thrown upwards with an initial velocity V. Find the maximum height the particle is going to reach. My question is not what...
  45. R

    Understanding Tension in Chain Slings: A Statics Problem

    [PLAIN]http://img245.imageshack.us/img245/7903/staticsproblem.jpg I don't know where to start with this because I'm unsure as to how the force is distributed in the chain.
  46. C

    Derivation of Decay Chain Formulae

    I'm afraid I'm suffering from a bit of brain block in try to get from the simple statement of change in the number of daughter nuclei arising from the decay of parent nuclei. The basic statement is straight forward... \frac {dN_d}{dt} = \lambda_pN_p - \lambda_dN_d Subscripts d and p denote...
  47. S

    Chain rule with leibniz notation

    Homework Statement If y=f((x2+9)0.5) and f'(5)=-2, find dy/dx when x=4 Homework Equations chain rule: dy/dx=(dy/du)(du/dx) The Attempt at a Solution In my opinion giving f'(5)=-2 is unnecessary as: y=f(u)=u, u=(x2+9)0.5 dy/dx= (dy/du)(du/dx) (dy/du)= 1 (du/dx)= x/((x2+9))0.5 dy/dx =...
  48. jegues

    How does the chain rule apply to partial derivatives?

    Homework Statement See figure. Homework Equations The Attempt at a Solution Here's what I got, \frac{ \partial z}{\partial x} = \left( \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial x} \right) + \left( \frac{\partial z}{\partial v} \cdot \frac{\partial...
  49. D

    What is the Chain Rule for Differentiating tan^3(x) + tan(x^3)?

    Homework Statement tan^3(x) + tan(x^3) Homework Equations The Attempt at a Solution tan^3(x) + sec^2(x^3) + 3x^2 Im not sure how to do the tan^3(x) and not even sure I did the tan(x^3) right
  50. D

    Chain Rule: Solving y' for y=cot^7(x^5)

    Homework Statement y=cot^7(x^5) Homework Equations f(x)=f(g(x)) The Attempt at a Solution u=(x^5) y'=7(-csc^2)^6(x^5) * 5x^4
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