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. O

    Understanding the Chain Rule for Derivatives

    Homework Statement Homework Equations The Attempt at a Solution My dought is about if ,dw/dy*dy/dt = (-2)t^(-3)*dy/dt
  2. F

    What is the speed of the chain at this instant?

    Homework Statement A chain of metal links with total mass M = 6 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 = 69 N. Eventually the chain straightens out to its full length L = 1.0 m, and you keep pulling until...
  3. F

    Determining Speed of a Chain: Calculating for a Point Particle System

    speed of chain?? Homework Statement A chain of metal links with total mass M = 6 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 = 69 N. Eventually the chain straightens out to its full length L = 1.0 m, and you...
  4. H

    Solving Tension in Chain Homework

    Homework Statement You have a new job designing rides for an amusement park. In one ride, the rider's chair is attached by a 9.0-m-long chain to the top of a tall rotating tower. The tower spins the chair and rider around at the rate of 1 rev every 4.0 s. In your design, you've assumed that...
  5. M

    How does the chain rule apply to acceleration in the context of mechanics?

    Hi, I'm new to these forums so not exactly sure where to place this question, although calculus seems a good bet, so here goes: I'm currently taking a mechanics course at my university (current subject is work/energy), and I'll just post this snippit from our textbook (Physics for Scientists...
  6. B

    Differentiation Question - Chain Rule

    Homework Statement Differentiate y = \left(\frac{x+2}{\sqrt[3]{x}}\right)3 Homework Equations -Chain Rule -Quotient Rule -Power Rule -Product Rule? The Attempt at a Solution First I got rid of the fraction by taking the negative of x^3, and then used the chain rule to differentiate...
  7. K

    Partial derivatives with Gradient and the chain rule

    Homework Statement First problem: Let f(x,y) = x-y and u = vi+wj. In which direction does the function decrease and increase the most? And what u (all of them) satisfies Duf = 0 Second problem: Let z = f(x,y), where x = 2s+3t and y = 3s-2t. Determine \partial{z^2}/\partial{s^2}...
  8. W

    Boulder being pulled up by a chain. Find max acceleration

    Homework Statement A 800 kg boulder is raised from a quarry 150 m deep by a long uniform chain having a mass of 580 kg. This chain is of uniform strength, but at any point it can support a maximum tension no greater than 2.90 times its weight without breaking. What is the maximum acceleration...
  9. B

    Solve Chain Rule Annoyance with e^-t^2 and y' = e^-t^2(y'-2ty)

    Homework Statement It is given that, \left(e^{-t^2}y\right)'=e^{-t^2}\left(y'-2ty\right), which I am trying to work out. Homework Equations f'(t)=h'(g(t))g'(t) (u\cdot v)'=u'v+uv'The Attempt at a Solution f(t)=e^{-t^2}y=h(g(t)) \text{let}\;g(t)=u=t^2\;\text{and}\;h(u)=e^{-u}y...
  10. Д

    Understanding the Chain Rule Proof: Explained with Examples

    Hello! I got one question for you. How come that (f \circ g)'(x) = f'(g(x)) g'(x) ? Since (f \circ g)'(x)=f(g(x))' , f'(g(x))=f'(g(x)) g'(x). And now we can rewrite the equation like 1=g'(x) I don't understand that part. Also I don't understand why the flawed proof of the chain rule...
  11. C

    Calculating the Force on a Falling Chain: Solving a Momentum Problem

    Homework Statement A chain of length L and total mass M is released from rest with its lower end just touching the top of a table. Find the force exerted by the table on the chain after the chain has fallen through a distance x. (Assume each link comes to rest the instant it reaches the...
  12. N

    Chain rule: partial derivatives transformation

    Hello. Let g(x,y) be a function that has second order partial derivatives. Transform the differential equation \frac{\delta ^{2}g}{\delta x^{2}}-\frac{\delta ^{2}g}{\delta y^{2}}=xyg by chaning to the new variables u=x^2-y^2 and v=xy The equation doesn't have to be solved. Okay, so this is...
  13. K

    Rubber Band Model for 1D Chain of N Links

    In the text, there is a rubber band model for 1D chain of N links. Assume each link of the rubber band is align horizontally either to the right or to the left. In the text, define N_+ as the number of links directed to the right and N_- represents the number of links directed to the left. I...
  14. G

    Selecting a Chain for 11kW Electric Motor Gearbox

    Hi I want't your help to learn about chain selection I have an electric motor of 11 kw and a gearbox I want to find a suitable chain can you help me?
  15. S

    Help with determining the transition matrix for a markov chain

    Homework Statement well i have my algebra exam coming up and my teacher told us that there is going to be a markov chain problem. the only problem i have is that i don't know how to get the initial transition matrix, which is crucial in getting full marks. can someone help me in determining how...
  16. S

    Chain rule for several variables: Implicit diff.

    Homework Statement Z is defined implicitly as a function of x,y by equation (z^2)x + 3xy^2 + e^((y^2)z) = 4. Find dz/dx Homework Equations dz/dx = -Fx/FzThe Attempt at a Solution Fx= z^2 + 3y^2 Fz=2zx+(y^2)e^((y^2)z) dz/dx= (z^2+3y^2)/[2zx+(y^2)e^((y^2)z)] I'm not sure if I used the partial...
  17. P

    Do neutron sources play a crucial role in the function of nuclear bombs?

    Do Uranium-235 nuclei ever undergo fission spontaneously? If not how does a nuclear bomb actually work? I understand that two pieces of Uranium (which are subcritical) are driven togther by a chemical explosion and this initiates the chain reaction; however, if nuclei can not undergo fission...
  18. S

    1D chain with loads of disorder

    Hi all, I'm wanting to write a small program simulating a 1D lattice with some motion. I have the equation: m_{n}\frac{d^{2}u_{n}}{dt^{2}}= k_{n,n+1}(u_{n+1}-u_{n})+k_{n-1,n}(u_{n-1}-u_{n}) Then using a simple trial plane wave (u_{n}=Ae^{-i\omega t}). It boils down to: - \omega^{2}...
  19. malawi_glenn

    Chain rule and Analytical Mechanics

    This is stuff I do in order to understand analytical mechanics better, I encounter the followin thing: \frac{\partial L}{\partial \dot{\phi}} = \text{?} Where \dot{\phi} = \frac{\partial \phi}{\partial q} \frac{dq}{dt} = \frac{\partial \phi}{\partial q} \dot{q} I should know this! It is...
  20. A

    Motorcycle vs Bicycle chain tension

    Trek recently released a belt driven bicycle (instead of chain drive) which led to a discussion among some cycling friends. Which led to discussion of torque, power output, etc. Anyway, I'm trying to compare the chain tension of a motorcycle vs the chain tension of a bicycle. I'm better with...
  21. S

    Solving Markov Chain Problem for Proportions in Areas A, B & C

    Homework Statement i have a scenario which i have to find the proportion of time spent in each area by a person using markov chains. i was given a word problem, which i have put into a matrix and the question asks what the proportion of time is spent in each area A, B and C. Homework...
  22. S

    What's stronger then Gear OR Chain?

    Sorry if the title is a bit blunt, but it's basically like this. There's this machine I'm working with. It's kinda using Gear + Chain to drive the output. Initially, they were all driven by gears but the gear is spoil every 2 months. So then it was changed to chain instead but now the...
  23. L

    How to Derive Using the Chain Rule for 2x^2+5xy-y^2=1?

    Homework Statement 2x^2+5xy-y^2=1 Homework Equations d/dx(f(u)x))=df/du * du/dx The Attempt at a Solution i got (2y-4x)/5x but I'm almost certain that its wrong...can anyone help me?
  24. P

    Chain sliding off a table, find the function a(t)

    Homework Statement A chain lies on a frictionless table at rest, half off the edge, and half on. As soon as it is let go, it begins accelerating due to gravity only. Determine the acceleration of the chain as a function of time. The mass is m, gravity is g, and the length of the chain is L...
  25. M

    Solving for f '(x) using the chain rule

    Homework Statement f(x)= x^2(x-2)^4 solve for f '(x) Homework Equations f(x) = x^2(x-2)^4 The Attempt at a Solution 4x^2(x-2)^3 The answer is given in the book as 2x(x-2)^3(3x-2) i'm not following any progression that gets me to that solution regardless of how many times I...
  26. J

    Why Does My Spin Chain Energy Calculation Differ from the Textbook?

    Hi all, I am supposed to calculate the energy of chain of spins where the magnetic field H = 0. For the first chain the spins are all aligned in the same direction - up - hence the energy U = -NJ where N is the total number of spins. Next, the half the chain is spin up and the other...
  27. S

    What is the derivative of y=e^square root of 1+tan(sinx)?

    Chain rule difficulties, due tomorrow! Homework Statement Find the derivative of y=e^square root of 1+tan(sinx) Homework Equations chain rule: F'(x)=f'(g(x)) * g'(x)The Attempt at a Solution I thought I had it and then while I was looking at other chain rules and started doubting my...
  28. C

    Is the Chain Strong Enough?

    Homework Statement You have a new job designing rides for an amusement park. In one ride, the rider's chair is attached by a 9.0-m-long chain to the top of a tall rotating tower. The tower spins the chair and rider around at the rate of 1 rev every 4.0 s. In your design, you've assumed that...
  29. I

    How much work is required to lift a 10m chain with 80kg from one end to 6ft?

    Homework Statement You have a chain of length 10m with 80kg, how much work does it take to lift this chain from one end to 6ft? Homework Equations \delta = \frac{10}{80} = .125 The Attempt at a Solution W = \int{F(x)}\,dx = \int^{6}_{0}{\delta lg}\,dl = \frac{\delta l^2g}{2}...
  30. D

    Calculating Work Required to Lift a Hanging Chain

    Homework Statement A chain hangs verticaly from a building. The chain is 30 ft long and is 5 lb/ft3, how much work is needed ot lift the bottom of the chain to the top.Homework Equations If you put the axis where the chain is hanging your limits would be 0 and -30The Attempt at a Solution So I...
  31. C

    Coefficient of static friction of chain

    Hi! I'm studying for an exam on Friday, and I'm stuck on this problem: A uniform chain of length 8.00m initially lies stretched out on a horizontal table. A. Assuming the coefficient of static friction between chain and table is 0.600, show that the chain will begin to slide off the table if...
  32. T

    Applying the Chain Rule to Derivatives with Square Roots

    Chain Rule Question is Find the derivative of F(x)= 3 sq rt of x^3-1 First step I did was changing the Sq RT to (x^3-1)^3/2 Then I solved it by 3/2(X^3-1)^1/2*3X^2 Another problem very similar F(X)= 3 SQ RT of X^4+3x+2 Step 1 (X^4+3x+2)^3/2 Then 3/2(X^4+3x+2)*4x^3+3 I know how...
  33. B

    Derivative of ( (X^3-1)/(X^3+1) )^1/3

    Homework Statement Find the derivative: ( (X^3-1)/(X^3+1) )^1/3 Homework Equations d/dx f(g(x)) = f'(g(x)) * g'(x) quotient rule x/a x'a-xa'/a^2 The Attempt at a Solution first i used the chain rule and quotient rule to get 1/3 ((x^3-1)/(x^3+1))^-2/3 * ((3x^2(x^3+1) -...
  34. A

    Learn the Chain Rule for Finding the Derivative of e^sec(x) | Homework Question

    Homework Statement derivative of esec(x) The Attempt at a Solutionu = sec(x) y = eu du/dx = tan(x)sec(x) dy/du = eu dy/dx = dy/du * du/dx = esec(x)tan(x)sec(x)
  35. Somefantastik

    Understanding the Chain Rule for Partial Derivatives in Multivariable Calculus

    u^{*}(r^{*},\theta^{*},\phi^{*}) = \frac{a}{r^{*}}u(\frac{a^{2}}{r^{*}},\theta^{*},\phi^{*}) \frac{\partial u^{*}}{\partial r^{*}}= \frac{a}{r^{*}}u_{r^{*}} \left( \frac{a^{2}}{r^{*}},\theta^{*},\phi^{*}\right) \left( -\frac{a^{2}}{r^{2*}} \right) - \frac{a}{r^{*2}} u \left(...
  36. J

    Markov Chain - Rat w/4 Rooms

    I've attached the diagram of 4 rooms, which a rat must move through. Each period he changes his room (his state). As you can see if you click on the image, the rat cannot access room 2 from 3, vice versa. If I assume the rat begins in room 1, how do I calculate the probability he will be in...
  37. F

    Q. f(x)=ln (12x-5/9x-2)So by using the chain rule, i can

    Q. f(x)=ln (12x-5/9x-2) So by using the chain rule, i can get: (-4/3)((9x-2)2/(12x-5)2) and by using the quotient rule, i can get the final answer, which is: (2(-36x-8)(-36x-15)2-2(-36x-15)(-36x-8)2) ------------------------------------------------------------------...
  38. R

    Chain Rule Practice: d/dx (cos2x*sinx) = cos3x-2sin2x*cosx?

    Homework Statement d/dx (cos2x*sinx) The Attempt at a Solution Does this equal cos3x-2sin2x*cosx ?
  39. J

    Power needed to accelerate a chain.

    I'm not a student, but this seemed like the correct place to put a question. I need to know how much power it takes to accelerate a motorcycle chain weighing 1 pound in 1 second to a speed that would equal 20mph at the wheel from a dead stop. The wheel has a diameter of 20 inches. The...
  40. G

    Partial Derivatives and using the chain rule

    Homework Statement If V=x^{3}f(y/x) show that x^{2}Vxx + 2xyVxy + y^{2}Vyy = 6VThe Attempt at a Solution i would normally just use the chain rule to differenciate this with respect to x and then so on but the f(y/x) is throwing me. Do i just treat the f like a constant or is it a whole new...
  41. I

    Markov chain calibration to a set of cumulated frequencies.

    Homework Statement Hi! I have been given such a task: A population of firms can assume three states: good-bad-bankrupt (default) The cumulated frequencies of default (DP) from year 1 to 10 are given. Find an appropriate transition matrix (TM) I'm given a matrix of historical cumulated...
  42. F

    Confusing chain differentiation rule

    If I have a function f from RxR to R, and a function g from RxR to RxR. What are the partial derivatives of the composition f(g)? I end up multiplying the derivative of f with g, but g is a vector? The partial derivative should have its image in R.
  43. R

    Chain sliding from a sphere.

    Homework Statement Refer to diagram for this question: A uniform flexible chain of length 1.50 m rests on a fixed smooth sphere of radius R=\frac{2}{pi}such that one end A of the chain is at the top of the sphere while the other end B is hanging freely. Chain is held stationary by a...
  44. P

    Wave on a string and the chain rule Argh

    Wave on a string and the chain rule...Argh So, I am working through the wave equation for a review before my friend and I go off to grad school. It has been a couple of years since we both graduated with our BS in Physics. So, here is the question: Suppose I want to solve the wave...
  45. P

    Wave on a string, and the chain rule argh

    So, I am working through the wave equation for a review before my friend and I go off to grad school. It has been a couple of years since we both graduated with our BS in Physics. So, here is the question: Suppose I want to solve the wave equation using a change of variables. Let's use...
  46. K

    Understanding Chain Rule: Differentiating P(x, y) and S(x, y)

    If P: R2 -> R is defined by p(x,y) = x . y, then Dp(a,b)(x,y) = bx + ay. Please tell me in words how to read Dp(a,b)(x,y). Is this a product? a composition of functions? Is this the differential of p(x,y) at (a,b)? If that's the case, why does the text also state: If s: R2 -> R...
  47. K

    Proving a known function of position via Chain Rule

    Homework Statement Use the Chain Rule to prove that for rectilinear motion, when the acceleration is a known function of position, you can find the velocity as a function of position via the integral \frac{v^{2}-v_{0}^{2}}{2} = \int^{s}_{s_{0}}a(s)ds Homework Equations...
  48. J

    How to Use the Chain Rule for Derivatives with sqrt(tan(sin^2 x))?

    Homework Statement y= squareroot tan(sin^2 x) Homework Equations chain rule The Attempt at a Solution f(x)= sqaureroot tan x g(x)= (sinx)^2 f'(x)=1/2 sec^2x ^1/2 g'(x)= 2 * sinx * cosx I don't know if my f'(x) is right if it is then do i just do the chain rule?
  49. P

    C. What is the average energy released per fission event in this chain reaction?

    The fission reaction n + 235U → 236U* → 141Ba + 92Kr + 3n produced 170 MeV of kinetic energy. A. How many of these fission events are needed to produce energy of 1 kilowatt- hour (kWh), that is, the energy it takes to run your blow dryer for an hour? B. How many neutrons are produced...
  50. J

    Chain rule with functional derivative

    Given that F = \int{f[h(s),s]ds} does \frac{\partial}{\partial h}ln(F)=\frac{1}{F}\frac{\delta F}{\delta h}=\frac{1}{F}\frac{\partial f}{\partial h} ?
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