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. B

    Nonlinear DE similar to a Bernoulli equation

    Hi all, I've got a nonlinear differential equation of the general form y' + f(x)y + g(x) = h(x)(y^n) to solve. For g(x) = 0 this is your standard Bernoulli equation. I've been trying to think of a way to solve it but haven't managed so far. Any ideas would be appreciated. Many...
  2. J

    Bernoulli/ Continuity Eq'n problem

    First of all, thanks to all the PF mentors out here, especially TSny and pgardn, who have made physics doable and are helping me accomplish my dreams! Even when putting in the work its not easy to get all this stuff! Homework Statement A large water tank has an inlet pipe and an outlet...
  3. C

    MHB Solving the General Bernoulli / Riccati Equation

    Hello, I have an equation of the following form: y'(t) + Ay(t) + By(t)^m + c = 0 1/ With c=0, the equation is of the Bernoulli form, and might be integrated. 2/ With m = 2, it is a Riccati equation, which might be turn back to a Bernoulli one and then integrated 3/ My question is about the...
  4. C

    Estimating p with Bernoulli Sample

    A Bernoulli random sample : X=1 , Pr(X=1)=p; X=0 , Pr(X=0)=1-p; taken X1, ..., Xn and if it is known that 0=< p =<0.5 find a method of moments estimator of p If I just take estimator of P =(sum Xi )/n than the estimate p may be bigger than 0.5 as in extreme case all Xi =1 ...
  5. phion

    Reducing the Bernoulli Equation

    I'm practicing first-order linear differential equations, and have come across something I find interesting - being able to reduce nonlinear equations to linear equation with appropriate substitutions. I'll start with the well-known Bernoulli equation, and if there are other ways to do this...
  6. M

    Fluid mechanics, open water channel velocity - bernoulli & conservation of flow

    Homework Statement If in an open water channel, the approach speed of an incompressible ideal fluid, of constant depth h1, is not too large, a small bump in height H (H<< h1) in the base of the channel will cause a dip Δh (Δh << H) in the water level directly above it, such that h1 = Δh + H...
  7. A

    Bernoulli principle or due to Viscosity?

    Why the speed of water in the middle of the smooth flowing stream is high than its speed on the sides? Is it due to Adhesion or cohesion forces or viscosity or Bernoulli principle holds here? Kindly tell me exact reason .
  8. M

    Does Bernoulli's Equation Apply to Ideal Gases in a Piston-Driven System?

    consider a hypodermic needle like structure with a piston in the larger tube and a piston in the smaller tube as well. let the cavity be filled with an ideal incompressible fluid. if i understand the Bernoulli theorem correctly, if a pressure is applied to the larger piston, causing the fluid...
  9. L

    Reaaranging a Bernoulli Equation.

    Homework Statement How can I rearrange this equation to graphically show that hole diametre is inversely proportional to sink time? So my experiment results show a nice diametre=1/t , and I would now like to take the equation I've come up with to show this relationship within the equation...
  10. L

    Bernoulli Process Statistics

    Homework Statement Consider Bernoulli single-server queuing process with an arrival rate of 2 jobs per minute, a service rate of 4 jobs per minute, frames of 0.2 minutes, and a capacity limited by 2 jobs. Compute the steady-state distribution of the number of jobs in the system. Homework...
  11. JJBladester

    Hydrostatic pressure in the Bernoulli Equation

    Homework Statement The hydrostatic pressure term in the Bernoulli equation (ρgz) decreases with fluid depth. Why? Homework Equations Bernoulli Equation (multiplied by density ρ to give us pressure units): P+\rho\frac{V^2}{2}+\rho gz=constant The Attempt at a Solution In the hydrostatics...
  12. JJBladester

    Bernoulli Equation - Units Question

    Homework Statement My book says that each term in the Bernoulli equation (when divided by ρ, has pressure units). I don't see how. Homework Equations The Bernoulli equation for steady, incompressible flow is: \frac{P}{\rho}+\frac{V^{2}}{2}+gz=constant The Attempt at a Solution...
  13. A

    Help with bernoulli differential equation

    I need help with solving equation y'=(x^2-y^2)/xy as bernoulli differential equation. I also tried to do it differently by plugging in y=xv, so I got f(v)=1/v-v. Finally I got answer y=(+/-) [(x^4+c)^(1/2)]/[(x*2^(1/2)] but I still have no idea how to do it as bernoulli equation
  14. S

    Find the speed of fluid using bernoulli

    Homework Statement Find v Homework Equations Bernoulli The Attempt at a Solution Here's what I've done I took the surface of water as first reference and the right-end of the pipe as second reference, so: P1 + 1/2 ρV12 + ρgh1 = P2 + 1/2 ρV22 + ρgh2 P1 = P2 = Patm and I...
  15. T

    Bernoulli Differential Equations

    Suppose I have a Bernoulli differential equation; that is, an equation of the form: y' + p(x)y = g(x) y^n. Supposing that I let n=1, the equation is linear. Can I solve it by constructing an integrating factor? That is, can I observe: y' + p(x)y = g(x) y → y' + y[(p + g)(x)] = 0. I would then...
  16. F

    Compressed Air - Bernoulli Equation

    I'm trying to ascertain the dynamics of a compressed air system in terms of flow rate, air consumption, pressure drop etc. To keep it simple my system consists of a 5.5 bar supply (550000Pa) supply on 12mm OD Nylon Tubing and is reduced to 6mm OD Nylon through a valve (pressure drop assumed...
  17. S

    MLE of Bernoulli trials

    Homework Statement Two independent bernoulli trials resulted in one failure and one success. What is the MLE of the probability of success θ is it is know that θ is at most 1/4 Homework Equations f(x,θ) = θx (1-θ)1-x The Attempt at a Solution Now, I know how to find the likelihood...
  18. 1

    Frustrating Bernoulli Equation

    I've been unable to fully solve this: \frac{dy}{dx} + y = xy^4 The U: u = y^{-3}, so y = u^\frac{-1}{3}, and \frac{dy}{dx} = \frac{-1}{3}u^\frac{-4}{3}\frac{du}{dx} The substitution: \frac{-1}{3}u^\frac{-4}{3}\frac{du}{dx} + u^\frac{-1}{3} = xu^\frac{-4}{3} Simplified: \frac{du}{dx} -...
  19. B

    Bernoulli Differential Equation

    Homework Statement solve the given differential equation: xdy/dx + y = y^-2 Homework Equations The Attempt at a Solution I don't understand how to do these substitutions...i got n=-2 then u=y^3, du/dx=3y^2 from there i don't know where to place them
  20. D

    Understanding Bernoulli Loss in Fluid Dynamics

    "Bernoulli Loss" Hi, I am reading a fluid dynamics paper about dissipation due to turbulence in the fluid causing "Bernoulli loss". What exactly is meant by this? The Bernoulli principle concerns energy conservation. So, does Bernoulli loss simply refer to a loss of energy within the fluid...
  21. C

    Solving Bernoulli Problem with Water & Air Density

    hi i want to check if i have done this the right way as their are two density values given. i used bernoulli's equation for incompressable fluid flow. V^2 = 2(p1-p2)/air density = 0.166 meters/s is it correct to use the air density in this way even though the manamoter is filled with...
  22. W

    Hydraulics, Flow rate from Bernoulli Equation

    Homework Statement Flow rate in Venturi meter of Petrol (SG 0.85). Diam1 - 0.2m Diam2 - 0.15m Height Diff in manometer of Mercury (SG 13.6) 0.012m, so Δp= 13600 x 9.81 x 0.012 = 1601? Cd = 0.98 Homework Equations Bernoulli, p/ρg + U2/2g + Z U=Q/A The Attempt at a...
  23. S

    If this is not a Bernoulli spiral, what is it then?

    I met this old nerd wearing a tie with a certain spiral, which does not look like a Bernoulli spiral to me. He actually told me, what it was, but I seem to have lost the note and time made me forget it. I wouldn´t be able to describe properly, so I have a photograph of the tie on my flickr...
  24. M

    Using Bernoulli to find volume of water to pass through hole

    Homework Statement Consider the fresh water damn shown in the picture (attached below). The water behind the dam has depth D = 15 m. A horizontal pipe 4 cm in diameter passes through the dam at depth d = 6 m. A plug secures the opening. The plug is removed. Water water volume exits the pipe in...
  25. L

    Paradox of Bernoulli's Theorem and Moving Frames

    Here is a paradox that came to my mind during my fluid mechanics course last term. I don't know the solution to it: We have this experiment in which we hold two sheets of paper parallel to each other and blow between them. They are brought closer to each other...
  26. T

    Distribution of Bernoulli random variable

    Homework Statement a) Let X1, X2, ...XN be a collection of independent Bernoulli random variables. What is the distribution of Y = \sumNi = 1 Xi b) Show E(Y) = np Homework Equations Bernoulli equations f(x) = px(1-p)1-x The Attempt at a Solution a)X1 + X2 + ... + XN = p...
  27. Z

    Bernoulli equation in terms of energy per unit volume

    dear PF.. what is the Bernoulli equation in terms of energy per unit volume the equation include the 1) static pressure 2) hydrostatic pressure 3) dynamic pressure 4) stagnation pressure
  28. H

    Bernoulli? Difference in water pressure between floors

    Homework Statement A water line enters a house 2.0 m below the ground. A smaller diameter pipe carries water to a faucet 5.0 m above the ground, on the second floor. Water flows at 2.0 m/s in the main line and at 7.0 m/s on the second floor. Take the density of water to be 1.0 x 103 kg/m3...
  29. D

    Bernoulli Principle of fluid question about velocity

    Homework Statement A dairy farmer notices that a circular water trough near the barn has become rusty and now has a hole near the base. The hole is 0.11 m below the level of the water that is in the tank. If the top of the trough is open to the atmosphere, what is the speed of the water as it...
  30. P

    Pressure and Bernoulli + Continuity

    Homework Statement Water flows through a horizontal tapered pipe. At the wide end its speed is 4.0 m/s and at the narrow end it is 5.0 m/s. The pressure in the wide pipe is 2.5 x 10^5 Pa. What is the pressure in the narrow pipe? a. 2.5 x 10^2 Pa b. 3.4 x 10^3 Pa c. 4.5 x 10^3 Pa d. 2.3 x...
  31. J

    Maximizing b(n,k) in Bernoulli Trials Homework

    Homework Statement Suppose 0 < p < 1. Define b(n,k) = \binom{n}{k}p^k(1-p)^k. For what value of k is b(n,k) a maximum? Homework Equations N/A The Attempt at a Solution Is there any way to get a nice closed form solution to this problem? I've already proved that it has a maximum so there...
  32. E

    Bernoulli - Find the general solution

    Bernoulli -- Find the general solution Homework Statement Find the general solution of: y'+xy=xy^3 Homework Equations Bernoulli's Equation The Attempt at a Solution y'+xy=xy^3 (y^-3)y'+x(y^-2)=x Let v=(y^-2), thus v'=((-2y^-3)y' Then, -v'/2+xv=x Multiply...
  33. A

    Find Expected Number of Cakes per Person: Bernoulli to Gamma

    Solve this?? There are w cakes and n people. The cakes are distributed to people randomly. Find the expected number of cakes a 'i' person can receive. The distribution is: a) Bernoulli distribution b) Binomial distribution c) Poisson distribution d) Uniform distribution e)...
  34. Jonnyb42

    ODE, bernoulli equation -> leads to crazy integral

    ODE, bernoulli equation --> leads to crazy integral ! Homework Statement An Initial Value Problem, ODE (Bernoulli equation) ODE: [x^2]*y' + 2*[x^3]*y = [y^2]*(1+2*[x^2]) IV: y(1) = 1/2 Homework Equations general form of Bernoulli's equation: y' + a(x)y = b(x)*[y^n] First...
  35. P

    Bernoulli propulsion-the inducement of water over a hydrofoil

    I am trying to find any references to research being done on the process of inducement being used as a form of propulsion in water. I would appreciate any help on finding sources on any general information about the inducement of fluid flow however I am specifically trying to find information...
  36. G

    Bernoulli Equation VS. V1A1 = V2A2

    Hi! I came across a problem involving a spigot. In order to find the velocity in the spigot which is located h meters in the tank, one uses the reduced V= sqrt (2gh) equation... how is this different from using V1A1 = V2A2? When would you use this equation? For instance, say you have a...
  37. S

    Bernoulli Principle with Bullets

    Hi I was reading about long range sniper bullets been affected by gyroscopic affect with the spin of the bullet. I was wondering if Bernoulli's principle could be applied to this, as the pressure at one side of the bullet is greater than the other (depending on the rotation). Is this true and...
  38. A

    Does Decreasing the Volume of a Pipe Increase Fluid Flow Pressure According to Bernoulli's Equation?

    One form of bernoullis equation (fluid flow) incompressible is the change in pressure plus [the change in (velocity squared)*density/2] = 0 when the height of the fluid remains the same. This along with the continuity of mass A1v1=A2v2 allows you to infer that decreasing the volume of...
  39. U

    Euler Bernoulli to second order ode

    hello, I have read in a number of papers that if we have a cantilever beam and are only interested in the movement of the tip when the base is being excited at the frequency of the first eigen mode , then the whole beam can be replaced by a spring mass system. Can anyone tell me of a...
  40. N

    Work and Bernoulli Eqn in Rotating Frame with Constant Angular Speed

    Q 1:Consider a disk of radius R. This disk is rotating around its center with a constarnt angular speed of w. Find the necessary work to move a body of mass m radially with respect to the disk from r=a to r=b. Q 2:The Bernoulli equation for a unit mass can be written as gdz+1/2VdV+vdP=0...
  41. P

    Differential Equation - Bernoulli Equation

    Homework Statement solve the differential equation y'(t)=-4y+6y^3 Homework Equations The Attempt at a Solution I'm pretty sure (not positive) that this is a Bernoulli Equation. I've been following this wiki in an attempt to solve...
  42. U

    Damping term in Euler Bernoulli equation

    hello, I have made an FEM simulation of a cantilever beam in Matlab. I have included the damping using damping matrix C= alpha x M + beta x K. Problem is that I want to compare my result with this paper http://flyingv.ucsd.edu/rvazquez/Journal/nano.pdf (See eq.1) where the...
  43. B

    Bernoulli Differential Equation

    Homework Statement Solve: 3dy/dx+y=(1-2x)y^4 Homework Equations None, really. The Attempt at a Solution y'+y/3=((1-2x)/3)y^4 y'/y^4+1/3y^3=(1-2x)/3 w=1/y^3 w'=(-3/y^4)y' w'/-3+w/3=(1-2x)/3 w'-w=2x-1 e^int(-dx)=e^-x e^(-x)*w=-e^-x(2x-1)+c w=1-2x+c*e^x...
  44. C

    Solve Bernoulli ODE with Integrating Factor x^-4 and n=3: Am I Missing a Step?

    So I have the following Bernoulli ODE: x^2*y' + 2xy = 5y^3 I use an integrating factor of x^-4, my n value is 3. I am okay until I get to the very end, where I get y^-2 = (2+Cx^5)/x; the book shows y^2 = x/(2+Cx^5). Am I must missing an algebra step or did I make a mistake somewhere within?
  45. G

    Differential Equation (Bernoulli?)

    Homework Statement Solve the differential equation: √(y)*(3*y'+y)=x Homework Equations Bernoulli equations: dy/dx+P(x)y=Q(x)yn dv/dx+(1-n)P(x)v=(1-n)Q(x) Integration by Parts: ∫u*dv = u*v - ∫v*du The Attempt at a Solution Since it's not separable, failed the homogeneous test, and failed...
  46. L

    Solving Bernoulli Equation with y'+3y=e^(-3x)*y^4

    Homework Statement y'+3y=e^(-3x)*y^4 , IC: y(1) = (12/4e^-3)^(-1/3) Homework Equations Bernoulli Method The Attempt at a Solution So n=4, i can substitue u=y^-3 u'+(-3)(3)u=(-3)e^(-3x) determine an integrating factor of e^-9x, then integrate both sides...
  47. C

    Solving Bernoulli ODE | Step-by-Step Guide | Example with n=-2

    I am working on the following Bernoulli ODE: 3xy^2 y' = 3x^4 + y^3. I come up with n = -2, so v = y^3 and y' = (1/3)v^(-2/3) v'. My integrating factor was x^-1. I end up with y^3 = X^2 + Cx yet the book has the same thing except X^4 instead of X^2. That makes me think I'm going wrong with...
  48. M

    Bernoulli s Principal would a fan matter?

    Bernoulli"s Principal would a fan matter? If you have air moving through a 3' tube connected to a 1' tube with a taperd connection. Than the pressure in the 3' tube would be greater than the 1' tube due to the speed in the 1' tube being higher than the 3' tube, correct? Then if you put a...
  49. I

    Bernoulli equation with pressure tank

    Homework Statement Water flows at a rate of 30ml/s from an opening in the bottom of a 4m high pressure tank (a tank with a plunger type lid). Calculate the flow rate when an extra 50 kPa of pressure is applied. Homework Equations Bernoulli's equation and \frac{V}{t} = AvThe Attempt at a...
  50. 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 which the water is supplied? Homework Equations I know you need...
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