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. F

    How Much Work Is Required to Lift a 25-ft Chain Up a Building?

    How much work is done lifting a 25-ft chain over the side of a building? Assume that the chain has a density of 4 lb/ft. Hint: Break up the chain into N segments, estimate the work performed on a segment, and compute the limit as N approaches infinity as an integral. (The small segment of...
  2. T

    Calculate work when winding chain (GRE Problem 66 - Exam 9277)

    This problem is in a GRE exam which I think might have big mistake in forming the answers: A thin uniform steel chain is 10 meters long with a mass density of 2 kg per meter. One end of the chain is attached to a horizontal axle having a radius that is small compared to the length of the...
  3. A

    Free Body Diagrams & Newton's Law Pairs For a 4-Link Chain

    Homework Statement I don't have a picture of the diagram, but I'll do my best to describe it. A chain consisting of 4 identical links is hanging by a rope. The question wants to know what the free body diagrams would be for each link. It also wants all of the Newton's third law pairs to...
  4. S

    Solving g(t): Seeking Help with Chain Rule

    Homework Statement g(t) = 4ln(5ln(4t)) Homework Equations What is the derivative of g(t)? The Attempt at a Solution I have tried to use the chain rule in many different ways, and still can't come up with the correct answer. Anybody care to walk me through this? Is the...
  5. M

    Atomic Chain in BCC - # Atoms/Unit Length & Area

    i'd like to know what is the number of atoms per unit length for an atomic chain in the [1 1 1] direction of a BCC structure. i can figure out the unit length, however I'm not sure how many atoms to use. Is it 1/8+1/8+1? That means 1/8s for the two corner atoms and 1 atom for the middle? or...
  6. E

    Chain rule for second derivatives

    Homework Statement I cannot believe I do not know this! dx/dt = dx/dt' dt'/dt is the chain rule for first derivatives d^2x/dt^2 = ? is the chain rule for second derivatives if it is complicated could you link me to a source that explains it please Homework Equations The Attempt at a...
  7. A

    Solving Chain Homework w/ Newton's Laws: Acceleration & Force

    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 chian is connected to a string and then suspended vertically, witht the student holding the upper end of the string and pulling upward. Because of the...
  8. C

    Functional derivative: chain rule

    Hmm, I've been working with functional derivatives lately, and some things aren't particularly clear. I took the definition Wikipedia gives, but since I know little of distribution theory I don't fully get it all (I just read the bracket thing as a function inner product :)). Anyway, I tried...
  9. W

    Does the fusion process cause a chain reaction?

    does the fusion process cause a chain reaction? or in a fusion reactor are lots of hydrogen atoms released to produce energy. does a fusion reactor heat water and produce steam to generate electricity? if you know anything then please let me know. thanks
  10. L

    Radioactive Decay chain and Masses of Products

    Hello, Based on a starting isotope, isotope mass, and duration of time, what calculations would be used to find the mass remaining of each individual nuclide in the decay chain after the time period? I think that the rate at which the final product is formed depends most on the nuclide with...
  11. F

    Explain why autotrophs occupy the lowest level of the food chain.

    Homework Statement Explain why autotrophs occupy the lowest level of the food chain. The Attempt at a Solution I not sure but i think its because they are producers and have the highest energy level (10 000 J) so they the primary source of energy for other consumers.
  12. A

    How can I integrate (Sin^4(x)*Cos^4(x)) without a calculator?

    Homework Statement Find the integral of (Sin^4(x)*Cos^4(x)) in respect to x without using a calculator. Homework Equations Sin^2(x) + Cos^2(x) = 1 Sin^2(x) = (1-Sin(2X))/2 Cos^2(x) = (1+Sin(2X))/2 Sin(2x) = 2Sin(x)Cos(x) Cos(2x) = Cos^2(x) - Sin^2(x) The Attempt at a Solution...
  13. C

    Solve Markov Chain Problem: Find Initial Probability Vector p(1)

    [SOLVED] Markov chain problem Hi, all! This is not a homework thing or something like that. I'm just curious. Suppose a Markov chain p(n+1)=Q*p(n), where Q=[q11, q12; q21, q22] is the transition probability matrix and p(n)=[p1(n);p2(n)] is the probability vector. Given Q and p(1), we can...
  14. E

    How do I apply the chain rule in division rule for this calculus problem?

    Homework Statement y = (2x-3)/(x^2+4)^2 Homework Equations The Attempt at a Solution I am trying to relearn the calculus that I forgot from many moons ago. I am struggling with the chain rule in the above example. I tried to set it up as follows: This is what I know...
  15. P

    Is my proof for the Chain Rule invalid?

    All the proofs I have found for the Chain Rule involve limits and the fundamental theorem of Algebra... So I came up with a PROOF, not a derivation. But my teacher claims that my proof is invalid. Is it? If so, why? let: u=z(x) y=f(u)=f(z(x)) therefore: dy/du = f ' (u) therefore...
  16. D

    Lattice Waves: Atom Chain Fixed at Ends or Free is Space?

    My textbook treats lattice waves as if they are waves propagating through a mass-spring chain. Kind of like this: M -/\/\/- M -/\/\/- M -/\/\/- M The book talks about the motion of the nth atom. It says that the motion of the nth atom has been simplified by only considering the motion...
  17. S

    The Dynamics of a Heavy Chain on a Smooth Peg

    Would anyone kindly discuss the motion of a heavy chain ? A typical problem I am stuck with is as follows: A uniform chain of length 2a is hung over a smooth peg, with the lengths (a+b) and (a-b) of two sides. Motion is allowed to ensue - show that the time the chain would take to leave the...
  18. D

    Trouble using the chain rule product rule and quotient rule

    im having a lot of trouble using the chain rule product rule and quotient rule..i can do them fine seperatly but when they're put together i can't get them like if you have (x^2-1)^4 (2-3x) i would start with 4(x^2-1)^3(2x)(2-3x)+(x^2-1)^4(-3) have i done something wrong here because i never...
  19. S

    What is the Power Spectrum of a Markov Chain?

    I'm reading the wikipedia article on them and I can't really get an understanding of what they are. I'm writing a paper for psychology, and I keep coming across articles that say 'learning can be modeled with markov chains' what does that mean?
  20. A

    Mastering the Chain Rule: A Quick Guide for Calculus Students

    Homework Statement y = 2x / (1+x^2)^2 Find dy/dx Homework Equations Chain rule The Attempt at a Solution I completely forgot how to apply the chain rule.. I mean, I can always apply the quotient rule, but I'm sure this is 1000 times easier if you can apply the chain rule. Do you...
  21. L

    Understanding the Chain Rule: When to Use and Why

    When do you know to use the chain rule?? I really don't understand what this means: dr/dt = dr/ds * ds/dt Like in one example in the book they have: r(s) = cos2si + sin2sj + e^-3sk where s = t^4 Then they find dr/dt using the chain rule. My question is why can't you just...
  22. R

    Troubleshooting Product and Chain Rule for h(t) Derivatives

    i am having trouble with this one problem. maybe you can tell me where i am going wrong. find h'(t) if h(t)=(t^6-1)^5(t^5+1)^6 so i am using product rule and to find the derivatives of each expression i am using chain rule... so i get h'(t)=30t^4(t^6-1)^4(t^5+1)^6+30^4(t^5+1)^5(t^6-1)^5...
  23. S

    Neutron to start the chain reaction

    Hello, I can't find :confused: where do we get the starting neutron(s) for the chain reaction in the reactor. I'm aware that then the splitted atom emits enough(in fact even too much?) neutrons to selfsustain the reaction, which are then slowed down to be able to split uranium and so on...
  24. P

    What is the Relationship between Tension and Force in a Pulley System?

    Homework Statement A worker lifts a weight w by pulling down a rope with a force F. The upper pulley is attached to a ceiling by a chain, and the lower pulley is attached to the weight by another chain. The weight is lifted at constant speed. Assume the chain, rope, and pulley all have...
  25. P

    Motion of a Uniform Chain falling I desperatly

    Motion of a Uniform Chain falling...I desperatly need help Homework Statement A uniform heavy chain of length "a" hangs initially with a part of length, "b", hanging over the edge of a table. The remaining part of length a-b is coiled up at the edge of the table. If the chain is released show...
  26. A

    Optimum values and w/ chain rule

    A boat leaves dock at 2:00 PM, heading west at 15 km/h. Another boat heads south at 12 km/h and reaches the same dock at 3:00 PM. When were the boats closest to each other? I've solved other problems similar to this one, but I can't seem to figure this one out. I'm not sure what I have to do...
  27. S

    Energy problem - Chain on low friction surface.

    Homework Statement Consider a chain of total mass M = 5kg that is coiled into a tight ball on a low friction floor(like an ideal ice rink). You pull on a link at one end with a constant horizontal force F=20 N, just until it reaches its full length L = 3 m. What is the final velocity of the...
  28. L

    IUPAC name of a branched chain hydrocarbon

    Hi, I'm just starting high-school organic chemistry and as is customary I've begun with nomenclature of organic compounds. Till now I've covered aliphatic hydrocarbons and will be starting on nomenclature of functional groups. The following problem from K.L. Chugh's Senior Chemistry Vol. II...
  29. T

    How to Explain the Chain Rule: A Visual, Verbal, and Mathematical Approach

    Homework Statement The question is: How would you explain the chain rule to a classmate? My teacher wants: 1.a graph, 2. in words, 3. using math Homework Equations The Attempt at a Solution I said: for the function y=f[g(x)], the chain rule is d/dx[g(x)]. Whats the graph like?
  30. T

    What is the Chain Rule for Derivatives?

    Given: h=f\circ g , g(2)=5, g\prime(2)=3, and f\prime(5)=-2 Determine h\prime(2)
  31. C

    Games: Gravity is a chain reaction within an equilibrium of chaos& entro

    [SOLVED] Games: Gravity is a chain reaction within an equilibrium of chaos& entro Roulette is a 50/50 game if you are playing black vs red, except for the one green slip of the wheel, but the odds of hitting that are very remote. So here is an experiment you can test out at home. Get 21...
  32. T

    Using the Quotient Rule with the Chain Rule

    Hi, I'm struggling to understand how to find the derivative of something like this... [(2x - 1)^2] / [(x - 2)^3] The answer in my book says it is supposed to be [-(2x - 1)(2x + 5)] / [(x - 2)^4] How do I use the chain rule with the quotient rule at the same time? I even multiplied...
  33. J

    Exploring the Second Partials Chain Rule in Multivariable Calculus

    This is a stupid question but... The regular multivariable chain rule is: u_x = u_v v_x + u_w w_x and u_y = u_v v_y + u_w w_y where u(v(x, y), w(x, y)) Now, are there formulae for the second partials u_{xx}, u_{xy}, u_{yy} I just want to check myself (this isn't a homework problem, though...
  34. F

    Conditionally independent with a Random Chain.

    I found this post somewhere on the net. I am also getting a similar problem. Can you all please help? +++++++++++++++ I have a problem, I try to solve but still get stuck. Can you please read my solution and point out what my wrong points are? Xn is a Markov Chain . n=1...M. with...
  35. H

    Differentiating f(t) with Chain Rule

    f(t) = (1+tan t)^(1/3) differentiate using chain rule. u = 1 + tan t y = u^(1/3) dy/dt = dy/du x du/dt u=1+tan t 1/3 u^(-2/3) when u = 1 + tan t x sec^(2)t = = sec^(2)t/3(1+tan t)^(2/3) Did I do this correct??
  36. C

    A chain sliding down an incline

    My first post :D I have an incline of angle (θ) to the horizontal, and a chain of length (l) of a uniform mass (m), I place the chain so that one end is at the bottom of the incline, the entire system is frictionless. I need to find what the velocity is as the last end leaves the incline, and...
  37. L

    Master the Chain Rule to Finding Derivatives of Complex Functions

    Using the "Chain Rule" Alright I've been working on some more practice problems and doing fairly well in finding the correct derivative, until I came to this question: y = [PLAIN]http://upload.wikimedia.org/math/1/6/0/160a5b4ac79375bd4c5e13c0f3a95f73.pngcos(sin^2x)[/URL] (everything is...
  38. M

    How Does Tension in a Spinning Chain Relate to EM Wave Propagation Speed?

    I need to find the tension of a spinning chain with mass per unit length u and angular speed w. I need to show that em waves on the chain travel at the same speed as the chains linear speed. please help if you can,
  39. mbrmbrg

    Simple integration based on chain rule

    I think I've got this, but I'm not quite sure, especially about multiplying to get du to be what I want. Can someone please tell me if this is correct? (The first line is the problem.) Thanks! \int\frac{3xdx}{\sqrt[3]{2x^2+3}} I set u=2x^2+3, du=4x so the problem becomes \int...
  40. L

    Solving a Logarithmic Chain Rule Problem with u(r) Differentiation

    Problem: u(x) = u_1(x)u_2(x)\cdot\cdot\cdot u_n(x) Show that u'(x) = u(x)\left[\frac{u'_1(x)}{u_1(x)} + \frac{u'_2(x)}{u_2(x)} + ... + \frac{u'_n(x)}{u_n(x)}\right] Hint: Take the logarithm of u(x) first. I solved this by using the chain rule. Regarding the hint, am I missing...
  41. E

    Calculating Force of Link 2 on Link 1 in Chain of 5 Links

    umm this question seems simple enough but I don't really know what I'm doing... A chain consisting of 5 links, each of mass .100 kg is lifted vertically upwards with a constant acceleration of 2.50 m/s^2. Calculate the force that link 2 exerts on link 1. so what I did was add the masses...
  42. T

    Chain rule with table of values

    ok firstly i understand how the chain rule works however I am given this table x f '(x) g(x) g'(x) h'(x) 0 7 2 ___ 32 2 8 0 -3 ___ im told to fill in the blanks where h(x)=f(g(x^2-x)) this problem has become an annoyance at first i assumed to solve i would take...
  43. C

    Solve Chain Rule Problem: Finding dy/dx for y=2t+3 and x=t^2-t

    I need help solving this problem. It is in my textbook but no answer is provided in the appendix. If y=2t+3 and x=t^{2}-t, find \frac{dy}{dx} In theory this should be fairly straight forward! Simply find \frac{dy}{dt} and \frac{dt}{dx} and multiply both derivatives together to find...
  44. C

    Need help with combination of product and chain rule

    I need help differentiating y=\sqrt{x-2}\sqrt{x+1} I am using a mix of the chain rule and product rule which my textbook for school wants me to use for this. So suggestions for different ways of approaching it won't help :P Anyways, thanks in advance for looking over it. It's my first...
  45. B

    Long chain of c=c carbon double for room temp superconductivity

    could you have a long chain of c=c carbon double bonded to carbon each carbon would be sp hybridzed, with two pi orbals bonded to adjacent carbons c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c going on for millions or even billions of repeating elements which would be in resonance c-c...
  46. P

    Find derivative with chain rule

    hey how would i find the derivative of y= -18 \sin 80 t? thanks pavadrin
  47. M

    Chain Rule/Product Rule Question

    I am suppose to derive the following and then find x if f´(x)= 0 f(x) = (6-x2) \bullet e2x Which one should be used? I guess I am confused with the fact that it is e2x Product rule gives: (6-x2) \bullet e2x + e2x \bullet (-2x) e2x(6-x2-2x) (x1 = 0) -x2-2x+6 = 0 gives decimal values that...
  48. wolram

    Designing a Custom Chain Case for Bike Rebuild

    The rebuild of my bike is well under way but i have hit a snag, i decided to convert the primary drive from chain to belt drive, all the parts are under construction, the problem is the origonal housing is 3/8 to narrow, and these housings are notorious for leaking oil any way, the oil comes...
  49. J

    Where Can I Find Reliable Information on Polymerase Chain Reaction?

    Does anyone know of a website or two that would have some good information on Polymerase Chain Reaction?
  50. S

    Is the Chain Rule Needed to Differentiate 6e^{0.013t}?

    f(t) = 6e^{0.013t} How do I find the derivative of this? I'm confused. Do I have to use the chain rule here or the product rule, or both?
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