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

    How Do You Apply the Multidimensional Chain Rule in Variable Transformations?

    hey pf! suppose i have a function ##f( x , y)##. i make a change of variables such that ##z(x,y)## in such a way that now ##f( z , y)##. how do i find $$\frac{\partial f}{\partial y}$$ $$\frac{\partial f}{\partial x}$$ $$\frac{\partial^2 f}{\partial y^2}$$ $$\frac{\partial^2 f}{\partial x}$$...
  2. P

    What is the correct expression for tension in a chain falling through a hole?

    Homework Statement Pls help me with the (d) option of the question asked in the link https://www.physicsforums.com/showthread.php?t=724332&page=1 Correct expression for tension is ρgx/6 (as given in the answer sheet) Homework Equations The Attempt at a Solution...
  3. N

    Astrophysics (Stars - PP chain)

    Homework Statement Estimate the lifetime of a proton against fusion to 4He in the center of a Zero-Age-Main-Sequence solar mass star. First calculate the energy generation, εpp in the center of the star from the p-p chain. Then convert this to the number of fusions (conversion of 4 protons...
  4. S

    Partial derivatives chain rule

    Suppose we have a function V(x,y)=x^2 + axy + y^2 how do we write \frac{dV}{dt} For instance if V(x,y)=x^2 + y^2, then \frac{dV}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} So, is the solution \frac{dV}{dt} = 2x \frac{dx}{dt} + ay\frac{dx}{dt} + ax\frac{dy}{dt} + 2y \frac{dy}{dt}
  5. M

    Observation of a single polymer chain

    Hi, I have a question regarding polymers. You know that polymers, basically, consist of chains of polymers, each chain including number of repeating units (monomer). These chains can be in amorphous or crystalline states. Experimentally, is it possible to fabricate and see a single chain...
  6. K

    Proving C is a Differentiable Function: Inverse Function Theorem & Chain Rule

    Homework Statement Problem: Given C is the graph of the equation 2radical3 * sinpi(x)/3 =y^5+5y-3 Homework Equations (1) Prove that as a set C= {(x,y) Exists at all Real Numbers Squared | 2radical3 * sinpi(x)/3 =y^5+5y-3 is the graph of a function differentiable on all real...
  7. Radarithm

    Kleppner and Kolenkow 5.10 - Falling Chain

    Homework Statement A chain of mass M and length ##\ell## 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 of chain, ##x##, has fallen? (Neglect the size of individual links.)...
  8. B

    Chain Rule of a functional to an exponential

    Homework Statement Suppose f is differentiable on \mathbb R and \alpha is a real number. Let G(x) = [f(x)]^a Find the expression for G'(x) Homework Equations I'm pretty sure that I got this one right, but I really want to double check and make sure. The Attempt at a Solution...
  9. D

    Radius of hellical chain (radius of path of electron)?

    Homework Statement here's the question, is my concept correct? the ans is 9.86 cm, but my ans is 13 cm, can anyone tell me which part is wrong? Homework Equations The Attempt at a Solution
  10. 9

    Partial derivative with chain rule: check work

    Homework Statement If possible, please check my work for any large errors. y = 10kl - √k - √l k = (t/5) + 5 l = 5e^t/10 Evaluate at t = 0 using chain rule. Homework Equations y = 10kl - √k - √l k = (t/5) + 5 l = 5e^t/10 The Attempt at a Solution = ∂y/∂k * dk/dt + ∂y/∂l * dl/dt = (10l -...
  11. 9

    Chain rule: y = f(t, t^2, t^3) and y = g(t, h(t), k(t^2))

    Homework Statement I am confused because for each problem there is no equation and for one no intermediate variables. Compute dy/dt when a) y = f(t, t^2, t^3) b) y = g(t, h(t), k(t^2)) Homework Equations a) y = f(t, t^2, t^3) b) y = g(t, h(t), k(t^2)) The Attempt at a...
  12. Y

    MHB Solving a Chain Rule Problem with F(x,y)

    Hello all, I need some help with this chain rule problem. \[F(x,y)=f\left (\frac{x-y}{x+y} \right )\] It is known that: f'(1)=20,f'(2)=30, f'(3)=40 and \[f''(1)=5,f''(2)=6,f''(3))=7\]Find \frac{\partial F}{\partial x}(2,-1) and \[\frac{\partial^2 F}{\partial x\partial y}\]The final...
  13. S

    Adding Power of Two Chain Drives Without Slipping

    is there any way to add power of two independent chain drives having different rpm , such that slipping doesn't occur and driven shaft move with the added resultant power
  14. D

    Energy Question - Chain sliding off frictionless table

    Energy Question -- Chain sliding off frictionless table Homework Statement Here is the problem that's confusing me: A frictionless chain of length 2.00m is held with 20.0% of its length hanging over the edge of a table. The chain is then released. Determine its speed the moment the entire...
  15. B

    Speed of chain with few variables

    I have no idea how to solve this. All i can think of is that you need mass to solve the problem. Homework Statement A chain is lying flat on a table. It has no friction against the flat surface. You pull the chain to the edge of the table so that it from a resting state starts to slide down...
  16. S

    MHB How can I apply the chain rule correctly to functions?

    I'm randomly having trouble applying the chain rule to functions (well, 1 function in particular), I was hoping someone could quickly walk me through this simple problem as I don't know where I've gone wrong. I've tried U substitution, chain/product rule, factoring answer...but I just can't see...
  17. anthonyk2013

    Applying the Chain Rule to y=(2x2+4x)5

    Differentiate the following by rule y=(2x2+4x)5 Is the chain rule the right rule to use? dy/dx=dy/du*du/dx Let U=2x2+4x du/dx=4x+4 y=(u)5 → dy/du=5(u)4 dy/dx=5(u)4*4x+4 dy/dx=5(2x2+4x)4*4x+4 dy/dx= 30(2x2+4x)44x dy/dx= 30(2x216x)4 I'm wondering if I am on the right...
  18. C

    Curvature of flexible chain

    Dear all, I was wondering how the radii of curvature can be calculated of a flexible chain (polymer chain). I have the x,y and z values of the polymer chain. For a 2D chain, I can calculate the curvature radii (http://www.intmath.com/applications-differentiation/8-radius-curvature.php). I am...
  19. Radarithm

    Solve Falling Chain (3.11) Length & Speed

    Homework Statement A chain with length ℓ is held stretched out on a frictionless horizontal table, with a length y0 hanging down through a hole in the table. The chain is released. As a function of time, find the length that hangs down through the hole (don't bother with t after the chain...
  20. Rugile

    Chain and spinning disk problem

    Homework Statement A chain is wrapped around a disk of radius R. The tension of chain is T. What is the coefficient of friction, if when the disk is spinning at angular velocity ω, the chain slips down? See image attached. Homework Equations II Newton law a_{centripetal} =...
  21. Julio1

    MHB Chain Rule in 2D Function Transformation

    Let $z:\mathbb{R}^2\to \mathbb{R}$ an function of kind $C^2(\mathbb{R}^2)$. What transforms the equation $2\dfrac{\partial^2 z}{\partial x^2}+\dfrac{\partial^2 z}{\partial x\partial y}-\dfrac{\partial^2 z}{\partial y^2}+\dfrac{\partial z}{\partial x}+\dfrac{\partial z}{\partial y}=0$ under the...
  22. Y

    MHB Proofing the Relation of u, v, x, y, a

    Hello, I need to do this proof here: I tried but didn't get what I wanted, so I was re-thinking the whole thing. If I say u=y+ax and v=y-ax, should I do something like (dz/df)*(df/du)*(du/dx)+...? Because I tried just with u and v (without f and g), and I got almost what I wanted, with a...
  23. J

    Question about determining whether to use the chain rule or not?

    For example, if you differentiate 6*sqrt(x^5), would you use the chain rule? If not, why? Thank you!
  24. Y

    MHB Second derivative with chain rule

    Hello all, I have a problem with second derivatives and chain rule. I am working on the question attached (sorry, my Latex editor wasn't working...) I need to find F'(1) and F''(1). I managed to solve F'(1), but I can't figure out F''(1). In the second image attached, you can see the solution...
  25. Saitama

    Breaking the chain of charges

    Homework Statement Homework Equations The Attempt at a Solution I am not sure how to proceed with the given problem. I can write down the net force on any of the charges but what should be the condition that the chain breaks? :confused: Any help is appreciated. Thanks!
  26. B

    How can we choose y and k to express f(g(x+h)) in the desired form?

    Hello everyone, I am reading a proof of the chain rule given in this link: http://kruel.co/math/chainrule.pdf Here is the portion I am troubled with: "We know use these equations to rewrite f(g(x+h)). In particular, use the first equation to obtain f(g(x+h)) = f(g(x) + [g'(x) + v]h)...
  27. M

    Finding dE/dx: Using the Chain Rule

    Hi fi you look at quesiotn 16b in the following link they try to find dE/dx. they use the chain rule. the chain rule says dF/dt=dx/dt*dF/dx+dy/dt*dF/dy if F=f(x,y) and x=f(t) and y=f(t). But in 16b they're trying to find dE/dx and as part of the use of the chain rule they try to find...
  28. M

    Use of chain rule in showing invariance.

    Hi, I’m a bit confused. I am familiar with the chain rule: if y=f(g(t,x),h(t,x)) then dy/dt=dy/dg*dg/dt+dy/dh*dh/dt To show that an equation is invariant under a galiliean transform, it’s partially necessary to show that the equation takes the same form both for x and for x’=x-v(T). So if you...
  29. M

    MHB Markov Chain - Is state 2 periodic?

    Hey! :o Given the Markov chain $\{X_n, n \geq 1\}$ and the following probability transition matrix: $\begin{pmatrix} 0 & 1/3 & 2/3\\ 1/4 & 3/4 & 0\\ 2/5 & 0 & 3/5 \end{pmatrix}$ All states communicate, so the chain is irreducible, isn't? Could you tell me if the state $2$ is periodic?
  30. N

    Calculating df/dt using the chain rule

    Hello everyone, first post here. Homework Statement Let f(x,y)=x2y+y2x , where x=sin2t and y=cos2t. Use the chain rule to compute df/dt Homework Equations f(x,y)=x2y+y2x x=sin2t y=cos2t The Attempt at a Solution This is pretty much the exact wording of the question...
  31. H

    Calculus 1 Differentiation Problem: Chain Rule with Binomial Theorem Application

    I'm not entirely sure if this belongs in homework or elsewhere -- I'm self-teaching working through a basic calculus text, so it's not homework per se. In any case it's a simple differentiation problem wherein I am supposed to differentiate: f(x) = x(3x-9)^3 f'(x) = 3x(3)(3x-9)^2 Applying...
  32. G

    How do I compute the following differentiation by chain rule?

    How do I compute the following differentiation by chain rule? \frac{d}{d\lambda}(\lambda^{-1}\phi(\lambda^{-1}x)) It is not a homework, but I can't figure out the exact way of getting the answer -\phi(x)-x^{s}\partial_{s}\phi(x)
  33. AJKing

    Force to lift a chain: Conservative or not?

    Homework Statement You are lifting a chain straight up at a constant velocity v_0. The chain has a linear mass density λ. What is the force required to lift the chain as a function of height? The Attempt at a Solution U = mgh = λygh The height in the potential energy is the same as...
  34. Saitama

    What Is the Incorrect Statement About the Dynamics of a Falling Chain?

    Homework Statement One end of the chain falls through a hole in its support and pulls the remaining links after it in a steady flow. If the links which are initially at rest, acquire the velocity of the chain suddenly and without frictional resistance or interference from the support or from...
  35. S

    What is the effect of an elastic collision between two blocks and a chain?

    Homework Statement We have a chain of 10 blocks, all of them joined by a thin rope and placed in a straight line. Suddenly, other two blocks collide with v speed at one end with the chain of the 10 blocks. It is assumed that the table is frictionless and the collision is elastic. The main...
  36. Y

    Prove the Second Derivative of a Multivariate Function Using the Chain Rule

    Homework Statement Let h(u,v) = f(a(u,v), b(u,v)), where a_u = b_v and a_v = -b_u. Show that h_{uu} + h_{vv} = (f_{xx} + f_{yy}) (a^2_u + a^2_v). Homework Equations The Attempt at a Solution I suppose my first question is where the x's and y's come from. (I thought at first it...
  37. A

    Multivariable Calculus Chain Rule Problem: Wave equation

    Homework Statement Show that any function of the form ##z = f(x + at) + g(x - at)## is a solution to the wave equation ##\frac {\partial^2 z} {\partial t^2} = a^2 \frac {\partial^2 z} {\partial x^2}## [Hint: Let u = x + at, v = x - at] 2. The attempt at a solution My problem with this is...
  38. D

    MHB Unraveling the Chain Rule in Differentials

    Take \(U(\eta) = u(x - ct)\) and the wave equation \(u_{tt} - u_{xx} = \sin(u)\). Then making the transformation, we have \[ (1 - c^2)U_{\eta\eta} = \sin(u). \] My question is the chain rule on the differential. \[ U_{\eta} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial\eta} +...
  39. P

    How do you find the second derivative of a function using the chain rule?

    Homework Statement Find the second derivative of $$9x^2+y^2=9$$ Homework Equations Chain rule The Attempt at a Solution I find the first derivative first. $$18x+2y\frac{dy}{dx}=0$$ $$\frac{dy}{dx}=-9\frac{x}{y}$$ I then find the second derivative...
  40. P

    Derivatives of Trigonometric Functions with the Chain Rule

    Homework Statement Find the derivative of $$y=cos(\frac{1-e^{2x}}{1+e^{2x}})$$ Homework Equations Chain rule The Attempt at a Solution $$y=cosu$$ $$\frac{dy}{du}=-sinu$$ $$u=\frac{1-e^{2x}}{1+e^{2x}}$$ $$ \frac{du}{dx}=(1-e^{2x})(-(1+e^{2x})^{-2})+(1+e^{2x})^{-1}(-2e^{2x})$$...
  41. P

    Derivative of y=cos(a3+x3)What is the derivative of y=cos(a3+x3)?

    Homework Statement Find the derivative of y=cos(a3+x3) Homework Equations Chain rule The Attempt at a Solution y=cosu \frac{dy}{du} = -sinu u=a3+x3 \frac{du}{dx} = 3a2+3x2 \frac{dy}{dx} = -3sin(a3+x3)(a2+x2). The answer is supposed to be -3x2sin(a3+x3). What did...
  42. P

    Finding the Derivative of y=xe-kx using the Chain Rule

    Homework Statement Find the derivative of y=xe-kx Homework Equations Chain rule The Attempt at a Solution y = xeu \frac{dy}{du} = xeu+eu u = -kx \frac{du}{dx} = -k \frac{dy}{dx} = (xe-kx+e-kx)(-k) = e-kx(x+1)(-k) = e-kx(-kx-k) The answer is e-kx(-kx+1)...
  43. P

    Derivatives, chain rule

    1. Find the derivative of y=e\sqrt{x} Homework Equations Chain rule The Attempt at a Solution y=eu \frac{dy}{du}= ueu-1 u=\sqrt{x} \frac{du}{dx}= \frac{1}{2}x-1/2 \frac{dy}{dx}= \sqrt{x}e\sqrt{x}-1 × \frac{1}{2}x-1/2 = \sqrt{x} \frac{e^\sqrt{x}}{e} ×...
  44. A

    Chain Rule with Partials & Evaluation Question

    Hi, I have a test prep question regarding Chain Rule, please see the problem and my attempt below. I believe part A is okay but part B, I'm just confused, seems like there is a part missing from the question, or at least how I'm use to doing it. Homework Statement A. Let f(x, y) =...
  45. Y

    Check whether I am correct in Chain Rule

    If ##r## is a function of ## x,y##, then \delta r= \frac{\partial r}{\partial x}\delta x + \frac{\partial r}{\partial y}\delta y Means Small change of r = ##\left[\frac{\partial r}{\partial x}\right]_{y=k}## X (Small change of x) + ##\left[\frac{\partial r}{\partial y}\right]_{x=k}## X...
  46. W

    Second Order differential equation involving chain rule

    Homework Statement Solve d^2x/dt^2 = (3x^3)/2 when dx/dt = -8 and x = 4 when t = 0 2. The attempt at a solution v = dx/dt dv/dx = d^2/dx^2 d^2x/dt^2 = v(dv/dx) = (3x^3)/2 v dv = (3x^3)/2 dx integrating and using limits and you get : v^2/2 -32 = (3x^4)/8 - 96 ...
  47. H

    Chain Link Problem? [Tension Forces; Newton's Third Law]

    Homework Statement A chain of three links, each with a mass 0.2 kg, is being pulled up by a person lifting the top link with 8.88 N of force and the chain accelerates upward. Calculate three forces that are acting on the middle link while the chain is accelerating. Homework Equations ƩF = ma...
  48. W

    Second Order differential equation involving chain rule

    Homework Statement Solve (d^2x)/(dt^2) = 2x(9 + x^2) given that dx/dt = 9 when x = 0 and x = 3 when t = 0 Homework Equations The Attempt at a Solution v = dx/dt ...... dv/dx = d^2x/dt^2 dv/dx = v(dv/dx) v(dv/dx) = 18x +2x^3 integrating and evaluating using...
  49. E

    Checking work on an unforced vibrations / hanging chain problem

    Homework Statement We are deriving the PDE that models a hanging chain of Length L. The x-axis is placed vertically. Positive direction points upwards. The fixed end of the chain is at x=L. Let u(x,t) denote the deflection of the chain. We assume the deflection is in the x,u plane. Let...
  50. W

    Linear chain with m th nearest neighbor interactions

    [b]1. Ashcroft and Mermin 22.1 Reexamine the theory of the linear chain without making the assumption that only nearest neighbors interact, using the harmonic potential energy of the form: U^harm=∑_n▒∑_(m>0)▒1/2 K_m [u(na)-u([n+m]a) ]^(1/2) Show that the dispersion relation must be...
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