What is Bernoulli: Definition and 343 Discussions

Jacob Bernoulli (also known as James or Jacques; 6 January 1655 [O.S. 27 December 1654] – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy. He is known for his numerous contributions to calculus, and along with his brother Johann, was one of the founders of the calculus of variations. He also discovered the fundamental mathematical constant e. However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work Ars Conjectandi.

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  1. F

    Fluid dynamics problem- bernoulli?

    Homework Statement A very large open water tank has a 50 cm diameter output pipe connected 10 m above the water main which is 6 cm in diameter. If the gauge pressure in the water main reads 9.8 E4 Pa what is the speed at whic the water is supplied? Homework Equations Bernoulli's...
  2. S

    Bernoulli and flow in a flooding river

    Does the top of a flowing river curve convex because of Bernoulli's Principle? I was looking at a picture of a flooding river on Google Earth, and noticed that most of the debris was concentrated in a line down the middle of the river, corresponding to the area of most rapid flow. I know...
  3. R

    Check my proof on showing two Bernoulli RV's are independent

    I've got a pretty good answer to this one already, yet I'd like to see how solid it is. I'll list the question first in quotes. Here's my work below. I credit http://arxiv.org/PS_cache/arxiv/pdf/0909/0909.1685v4.pdf" for the answer. X and Y are independent if and only if...
  4. E

    Bernoulli for incompressible inviscid flow

    [b]1. Investigate and apply equations of stae; speed of sound; continuity; bernoulli for incompressible inviscid flow; shear forces and stresses; laminar flow; flow in boundary layer link to shear forces and viscosity- continuity equation [b]2. [b]3. I have answered 90% of the...
  5. P

    Uncovering the Bernoulli Equation: Solving y'+P(x)y=Q(x)y^n

    Homework Statement I think this is a bernoulli equation in disguise (y^7-6x)y'+y=0 Homework Equations I'm having trouble putting it in the form of y'+P(x)y=Q(x)y^n The Attempt at a Solution I've tried to divide out (y^7-6x) as well as some other alebraic manipulation but i keep...
  6. S

    Understanding the Bernoulli Differential Equation

    Homework Statement This diff.eqn here. \frac{dy}{dx} = \frac{y}{x} - y^2 which in my mind can be re-written to \frac{dy}{dx} -\frac{1}{x}y = -y^2 does this mean that the above is a socalled bernoulli diff.eqn? And should solved as such? I may be dumb, but that the only way I...
  7. Saladsamurai

    Solving Bernoulli Equation: Confirm Results and Identify Error

    I have solved the following Bernoulli equation by letting Z = y1 - 2: xy' - 2y = x3y2. I obtained the solution y = 5/(5c1 - x5) which Wolfram Alpha has confirmed. From this result, I have obtained y' to be y' = 25x4/(5c1 - x5)2 The problem is when I go to check the solution by plugging...
  8. L

    Bernoulli Differential Equation

    \frac{dy}{dx} - \frac{1}{x}*y + \frac{1}{y^2}*x = 0 This is a bernoulli differential equation which follows the following procedure http://en.wikipedia.org/wiki/Bernoulli_differential_equation Can someone provide the full steps because somewhere I am mistaken?
  9. G

    Flow through a pipe, pressure, velocity, Bernoulli, etc

    Hello everyone, Thank you to all who offer their help. Homework Statement If the volumetric output of the pump were increased, keeping the pipe diameter constant, would the pressures at A, B, and C increase or decrease? Would the heights within each column at each location rise or fall...
  10. G

    Bernoulli confidence intervals

    Confidence intervals [b]1. Homework Statement [/ Use CLT to construct approximate symmetric 100(1-alpha)% confidence interval [L,R] for p then show that [L,1] is then an approximate 100(1-alpha/2)% confidence interval for p The Attempt at a Solution When [L,1] then we have a one...
  11. G

    Bernoulli Variance: Calculating Var[Xi] as a Function of p

    Homework Statement Show how Var(Xi) depends on p writing it as a function \sigma^2(p) The Attempt at a Solution Var[Xi] = E[Xi^2] - E^2[Xi] = p-p^2 = p(1-p) not sure where to go from here to get it in the form \sigma^2(p) ?
  12. F

    Bernoulli and darcy equation in split flow system

    Refering to the image uploaded can we say by applying bernouli equation and darcy equation the following : Pa/(guamma) + Va^2/2.g + Za - (head loss in pipe ab using darcy equation )-(head lossin pipe bc using darcy equation) = Pc/(guamma) + Vc^2/2.g + zc . ?? and the same thing between...
  13. Z

    Bernoulli polynomials evaluated at 1/4

    Homework Statement The problem is to prove the identity B_k(1/4) = 2^{-k} B_k(1/2) for even k. Homework Equations The Bernoulli polynomials B_k(y) are defined by the generating function relation \frac{xe^{xy}}{e^x-1} = \sum_{k=0}^{\infty} \frac{B_k(y) x^k}{k!}. The Attempt at a Solution...
  14. S

    Bernoulli random variable problem

    Homework Statement Suppose we want to estimate a binomial proportion, p. We take a sample of size n and count X successes. Consider a Bernoulli random variable, Y that is 1 with probability p and 0 otherwise. Show that the mean and variance of Y are p and p(1-p), respectively...
  15. J

    How Does Bernoulli's Equation Apply to a Water Fountain System?

    Homework Statement An open tank of water is sited on top of a hill and has a pipe leading from the bottom leading down to a vertical water fountain. The base of the tank is 15m above the fountain nozzle and the depth of water in the tank is 1.0m. 1. Draw a ruled sketch of the system, showing...
  16. S

    Skewness and Kurtosis of Bernoulli Distributions

    Suppose you have multiple independent Bernoulli random variables, X_1,X_2,...,X_n, with respective probabilities of success p_1,p_2,...,p_n. So E(X_i)=p_i, and E(X_i+X_j)=E(X_i)+E(X_j). Also, \text{var}(X_i)=p\cdot (1-p), and \text{var}(X_i+X_j)=\text{var}(X_i)+\text{var}(X_j). (Though...
  17. J

    Bernoulli Differential Equation?

    I keep getting stuck on one part of the equation. can somebody help? dy/dx = y(xy3 - 1) dy/dx = xy4 - y dy/dx + y = xy4 u = y1-n = y-3 y = u-3 dy/dx = dy/du * du/dx dy/dx = -3u-4 du/dx Substituting terms into the original equation -3u-4 du/dx + u-3 = x(u-3)4 Here is where I get stuck...
  18. S

    Comparing A & B in Bernoulli Trials w/ X & P

    Let X be the set X=\left\{x_1,x_2,...,x_n\right\}, where each x_i is a Bernoulli random variable and P be the set P=\left\{p_1,p_2,...,p_n\right\}, where p_i is the probability that x_i=1. Now, suppose there are two other sets, A=\left\{a_1,a_2,...,a_n\right\} and...
  19. S

    Quick question about Bernoulli principle and resistance?

    Ok I studied bernoulli and I can see how fluid would travel faster in narrowed regions but in most examples I see when something is narrowed the flow is reduced. Is this because of resistance? Also when there is resistance speed in the narrowed section is reduced right? I like to know what do...
  20. F

    Bernoulli appliance to garden hose

    I know the topic has been dealt in the past, I read it but still couldn't extrapolate a satisfying answer. I would like to know how Berboulli equation applies to a hose without restrictions in which fluid (water) exits at atmospheric pressure, the typical garden situation without any nozzle. I...
  21. S

    I think I understood Bernoulli principle but I need to clarify one thing quickly

    Ok after hours of contemplating I think I finally understood the concept(hopefully ). However I still need 1 quick clarification. When water flows suddenly from a larger to a small dimater Bernoulli says 1. Pressure decreases 2. Velocity of fluid increases I'll put numbers just for...
  22. K

    Bernoulli equation (self made diagram)

    An object is accidentally dropped into a water pipe. Pressure applied to pipe is 10 atm. The depth of lake is 10m. What will be speed of object when it exits the pipe and first enters the lake? Assume that the object is carried along with the surrounding water and does not affect flow of...
  23. J

    Difference between Magnus effect and Bernoulli

    Hi, I need some help as I’m a bit confused, subsequently, Magnus has stated that due to the velocity of the air around the spinning cylinder there will be a difference of pressure and therefore lift, whereas, Bernoulli said that there will be a lower pressure where the fluid will travel...
  24. S

    Fluid mechanics - Bernoulli Ball

    sorry for the "bad" title, I couldn't think of anything else. We all have seen the experiment where you take a hair dryer and a ping-pong ball and the hair dryer blows the ball into the air and you can move and tilt the ball around... now, that is a very simplified version of my final...
  25. P

    What is the Bernoulli Equation?

    Homework Statement I recently was given an interesting problem I thought I'd share with the community here. It's a fun exercise in mathematics. Given the equation: Po/P = [1+((k-1)/2)*Ma^2]^(k/(k-1)) From this stagnation ratio, derive the Bernoulli Equation: Po = P + 1/2 * rho * V^2...
  26. H

    Calculating Variance of Two Bernoulli Experiments

    say you perform one bernoulli experiment where outcome X = 0 and X=1 are failure and success, respectively. then you have P(X=0) = 1-p and P(X=1) = p. what if you expand this to two bernoulli experiments? only 1/4 cases would be successful, so you would have: P(X=1) = p^2 E(X) = p^2...
  27. Saladsamurai

    Deriving the Bernoulli EquationFluids

    I am trying to follow along with the text's derivation of the Bernoulli equation and I am a little confused with a particular line. The images show an elemental streamtube control volume (CV) of variable area A(s) and length ds. We assume that though the properties may change with time and...
  28. D

    Bernoulli Number's conjeture?

    Is this a known bernoulli number conjeture/theorem?: The denominators of B_n (when expressed as an irreducible fraction), doesn't contain as a factor powers of prime numbers (ex. isn't divided by 5^2) Example: B_2 = 1/6 6 = 2*3 B_4 = -1/30 30 = 2*3*5 B_{24} = -236364091/2730 2730...
  29. H

    Differential Equation, Bernoulli

    Homework Statement 3y^2*y'+y^3= e^-x The Attempt at a Solution I am using Bernoulli's equation to substitute V in and I keep coming out with Y^3= e^x(-e^-x+c) my V = y^3 Y = v^(1/3) and dy/dv=(1/3)v^(-2/3) I peeked to see if I was correct, the right answer is supposed to...
  30. D

    How can I solve this differential equation using Bernoulli's equation?

    Ugh Bernoulli Equations! Homework Statement dy/dx = y(xy^3-1) I tried to set it up and use the bernoulli equation method as a substitution but it didn't work. Any tips? The Attempt at a Solution I set u=y^(-3) and had it set up like this dy/dx + y = xy^4 Ok HELP...
  31. S

    Which Ball Wins the Race According to Bernoulli's Principle?

    http://up2.up-images.com/up//uploads2/images/hosting-036ee42f15.jpg in the figure, which ball will win in the race? note: there is no energy loss
  32. D

    Linear and Bernoulli First Order

    1. The problem statement. 1. http://img194.imageshack.us/img194/3815/linear.png 2. http://img25.imageshack.us/img25/1543/bernoulli1.jpg 3. http://img23.imageshack.us/img23/7991/bernoulli2.jpg 2.Relevant equations Problem 1 is Linear Differential Equation and Problems 2 and 3...
  33. J

    Bernoulli Effect & Friction: Lift Away Slower Interfaces?

    Is nothing of the Bernoulli effect due to friction? Passing fluid trying to lift away a slower interface?
  34. R

    Find the general solution of Bernoulli Eq

    Problem:y' + y(1/x) = 3 x^2 y^2 Solution:p(x)= 1/x q(x) = 3x^2 <--- These are countinuous functions on the interval (0,+inf) y^-2[y' + y(1/x) = 3 x^2 y^2] A => y'y^-2 + y^-1(1/x) = 3x^2 v = y^(-n+1) = y^-1 v' = -y^-2 y' Plugin to problem A v' + v/x = 3x^2 B p(x) = 1/x...
  35. C

    A question on window draughts, hot days and bernoulli

    Hello there! It's been about a grillion degrees up here in my home town recently and last night this led me to a sort of conundrum. If I open the window in my room but leave my door shut, hardly any air comes it at all. If I open my door, there's a strong puff of wind that comes into the room...
  36. C

    A question on window breezes and bernoulli effect

    Hello there! It's been about a grillion degrees up here in my home town recently and last night this led me to a sort of conundrum. If I open the window in my room but leave my door shut, hardly any air comes it at all. If I open my door, there's a strong puff of wind that comes into the room...
  37. WCOLtd

    A Bernoulli Bet (experts please)

    I have a bet with one of my friends, it has to do with the bernoulli equation that says the higher the velocity of a gas, the lower the pressure. I came up with an invention idea that involves a gas cylinder, the gas cylinder has a maximum pressure of 4000 PSI and the gas inside exerts...
  38. E

    Amazing 16-Year-Old Finds Bernoulli Numbers Relationship

    "Mathematics holds a vital place in a wide variety of fields, including the sciences and finance. Number calculations have proved the viability of countless scientific phenomenon. Money lenders generally need statistical data to evaluate a consumer’s financial situation before granting them a...
  39. Saladsamurai

    What is the role of Q1 in finding V2 using Bernoulli's equation?

    Homework Statement Homework Equations Bernoulli eq and Conservation of Mass The Attempt at a Solution Treating the ENTIRE system as my control volume, from conservation of mass we have: \frac{d}{dt}\int\int\int_{cv}\rho\, dV +\dot{m_2}-\dot{m_1}=0 \Rightarrow \frac{d}{dt}\int_h\rho A_o\...
  40. Saladsamurai

    Bernoulli Equation: Finding t_fall Expression

    Homework Statement Homework Equations X=Vo*tfall Bernoulli Eq: \frac {p}{\rho}+\frac{V^2}{2}+gz=const I used B.E. to find an expression for the velocity of the exit jet: V_o=\sqrt{2g(H-h)} For some reason I am confused as to how to find an expression for t_fall ? I know it should be...
  41. R

    Bernoulli : temperature lower in a restriction?

    Hello I’ d like to have your thoughts about any temperature variation when a fluid passes through a narrower part of a tube, a subject quite rarely discussed.. Can we start with « perfect Bernoulli conditions »and an incompressible fluid having only translational energy,. flowing theough a...
  42. M

    General Solution for a Bernoulli Equation with Trigonometric Functions

    Homework Statement Find the general solution of the following differential equation. y'+4xy=10xy2cos(x2) Homework Equations The usual Bernoulli equation ones: y'+p(x)y = g(x)ya u(x)=[y(x)]1-a u'+(1-a)pu = (1-a)g The Attempt at a Solution I got up until the general solution part.. I'll just...
  43. R

    Bernoulli Equation: Compressible vs Incompressible Flows

    Hi, urgent question. Bernoulli's equation seems to be conservation of energy. I read that it's only for incompressible flows; but isn't the term involving pressure the energy due to the work done on a mass of air in compressing it? Thanks
  44. H

    Finding Summation of n^p with Bernoulli Numbers

    Hey everyone, I need some help trying to figure out how to find the summation of n \sum_{}^{\6}i^p i=0 I was looking on the web and found on Wikipedia this formula off the http://en.wikipedia.org/wiki/Summation" page. It looks like this assuming I copied it right (ignore the periods)...
  45. T

    Bernoulli differential equation

    Homework Statement Solve the Bernoulli equation, y'(x) - 4y(x) = 2e^(x) * sqrt(y(x)) Homework Equations y' + P(x)y = Q(x)y^n - Bernoulli Eqn e^(∫P(x) dx) - Integrating Factor The Attempt at a Solution y' - 4y = 2e^(x) * y^(1/2) Divided both sides by y^(1/2) y'/y^(1/2) - 4y/y^(1/2) =...
  46. H

    Is Bernoulli principle the cause of suction truck?

    Is Bernoulli principle the cause of suction truck?? Hello! I’d like to know this question: When a truck walks to a high speed we see how the road tree leaves are dragged behind it. I would like to know the real cause of this problem. I think the air viscosity makes the truck drag a thin...
  47. K

    Bernoulli Numbers, Euler-Maclauring Formula - Math Methods class

    Homework Statement The Bloch-Gruneissen approximation for the resistance on a monovalent metal is \rho=C(T^{5}/\Theta^{6})\int^{\Theta/T}_{0}\frac{x^{5}dx}{(e^{x}-1)(1-e^{-x})} (a)For T->\infty, show that \rho=(C/4)(T/\Theta^{2}) (b)For T->0, show that...
  48. 5

    Fluid Mechanics - Split Pipe Flow Question (momentum and Bernoulli eqns)

    Hi, I am stuck on this question as I haven't encountered anything similar to it. Normally I have only dealt with pipe with one inlet/outlet. Homework Statement I am given a diagram of the pipe shown on the attachment (Sorry for the crude diagram). Hosted it while the attachment is "reviewed"...
  49. R

    Volume of solid formed by revolution of one loop of Lemniscate of bernoulli

    Hello ppl. I have a problem in finding out the volume of solid formed by the revolution of one loop of lemniscate of bernoulli ( r²=a²cos2θ) about the initial line θ =0 Using the relevant forumula for the volume of the solid generated by the revolution of one loop of the polar curve about the...
  50. A

    Static Pressure in Flowing Fluids: Meaning and Causes

    well,i ve some problems with bernouuli equation.first,i'd like to ask abt meaning of static pressure in flowing fluid because i really ve read abt that in references and many websites but still can't get exactly what's meaningt of static pressure in flowing fluid. i ve read on website that...
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