• Kaxa2000
In summary, the problem involves an object dropped into a water pipe with a pressure of 10 atm and a depth of 10m in a lake. The speed of the object when it exits the pipe and enters the lake needs to be determined, assuming the object is carried along with the surrounding water and friction is neglected. Bernoulli's equation is needed to solve the problem, but unlike most examples, the areas at each end are the same. The height of the lake and pipe may also affect the pressure. The equation (.5)(density)(vf)^2 + density(g)(y height) + 2atm can be used to solve for vf.
Kaxa2000
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 water in any way. Neglect friction.

Diagram is in attachment

I know you have to use Bernoullis equation for this problem. The examples I've looked at usually have a different area at each end. This problem has apparently the same area at each end. Also I don't understand how to factor in the height included in the problem.

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.5*density *(0)2 + 0 + 10atm(convert) = constant

(.5)(density)(vf)2 + density(g)(y height) + 2atm

Would I set these equal and solve for vf??

What about the height of the lake and the height of the pipe?? Do those add more pressure??

I would first start by breaking down the Bernoulli equation and understanding each component. The Bernoulli equation states that the total energy of a fluid remains constant along a streamline. In this case, the fluid is water and the streamline is the pipe.

The first component of the equation is the static pressure, which is the pressure applied to the pipe. In this case, it is given as 10 atm.

The second component is the dynamic pressure, which is the kinetic energy of the fluid. In this case, the object is dropped into the pipe, so the fluid is moving and therefore has kinetic energy.

The third component is the potential energy, which is determined by the elevation or height of the fluid. In this case, the pipe is connected to a lake, which is 10m deep. This means that the potential energy of the fluid is affected by the depth of the lake.

Now, to solve for the speed of the object when it exits the pipe and first enters the lake, we need to use the Bernoulli equation in the form of:

P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2

Where P1 is the initial pressure (10 atm), v1 is the initial velocity (unknown), h1 is the initial height (unknown), P2 is the final pressure (atmospheric pressure), v2 is the final velocity (unknown), and h2 is the final height (10m).

Since we are neglecting friction, we can also assume that the pressure remains constant throughout the pipe, so P1 = P2.

Plugging in the known values and solving for v2, we get:

10 atm + 0 + ρgh1 = 1 atm + 1/2ρv2^2 + ρgh2

Since the object is dropped into the pipe, we can assume that its initial height (h1) is 0. Also, since the object is carried along with the surrounding water, we can assume that its final height (h2) is also 0. This simplifies the equation to:

10 atm = 1 atm + 1/2ρv2^2

Solving for v2, we get:

v2 = √(2(10 atm - 1 atm)/ρ)

To determine the density (ρ) of

1. How is the Bernoulli equation derived?

The Bernoulli equation is derived from the principle of conservation of energy, which states that the total energy of a closed system remains constant. It is based on the concepts of kinetic energy, potential energy, and pressure energy.

2. What are the assumptions made in the Bernoulli equation?

The Bernoulli equation assumes that the fluid is incompressible, non-viscous, and flowing along a streamline. It also assumes that the flow is steady and the forces acting on the fluid are conservative.

3. What is the significance of the Bernoulli equation in fluid mechanics?

The Bernoulli equation is a fundamental equation in fluid mechanics and is used to describe the behavior of fluids in various applications such as aerodynamics, hydraulics, and thermodynamics. It helps in understanding the relationship between velocity, pressure, and height in a fluid flow.

4. How is the Bernoulli equation applied in real-life situations?

The Bernoulli equation is applied in various real-life situations such as calculating the lift force on an airplane wing, determining the pressure difference in a water pipe, and designing water fountains. It is also used in medical devices such as ventilators and blood flow meters.

5. What are the limitations of the Bernoulli equation?

The Bernoulli equation is only applicable to ideal fluids and cannot accurately predict the behavior of real fluids with high viscosity or compressibility. It also does not take into account factors such as turbulence, friction, and heat transfer. Additionally, it only applies to steady flow along a streamline and cannot be used for unsteady or non-uniform flows.

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