Torricelli's Principle: Outflow Velocity and Liquid Drainage in a Tank

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In summary: It shows that the change in height of the liquid in the tank is related to the outflow velocity and the contraction coefficient. This equation can be used to solve for h(t), the height of the liquid at any given time. Part (c) uses the given information to find the specific values needed to solve the equation from part (b). By plugging in the values for A(h), a, g, and α, we can solve for h(t). Once we have h(t), we can find the time it takes to drain the tank by setting h(t) equal to the height of the outlet and solving for t. This gives us the time it takes to drain the tank down to the level of the outlet. In summary
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Suppose that a tank containing a certain liquid has an outlet near the bottom. Let be the height of the liquid surface above the outlet at time t. Torricelli's principle states that the outflow velocity v at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height h.

(a)

Show that v=[tex]\sqrt{2gh}[/tex], where g is the acceleration due to gravity.

(b)

By equating the rate of outflow to the rate of change of liquid in the tank, show that h(t) satisfies the equation

A(h)(dh/dt)=-α*a*[tex]\sqrt{2gh}[/tex]

where A(h) is the area of the cross section of the tank at height h and a is the area of the outlet. The constant α is a contraction coefficient that accounts for the observed fact that the cross section of the (smooth) outflow stream is smaller than a. The value of α for water is about 0.6.

(c)

Consider a water tank in the form of a right circular cylinder that is 3 m high above the outlet. The radius of the tank is 1 m and the radius of the circular outlet is 0.1 m. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet.


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I know that for part (a) you can use conservation of energy, mgh=.5mv^2 or bernoulli's principle.

part (b) i just get lost. I'm pretty bad at related rates.
 
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Part (b) is just the continuity equation.
 

What is Torricelli's principle?

Torricelli's principle, also known as the "law of Torricelli" or the "Torricelli's theorem", is a physics principle that describes the motion of fluids in a container. It states that the speed of a fluid flowing out of an opening at the bottom of a container is equal to the speed that an object would have if it were dropped from the same height as the fluid surface.

Who is Torricelli?

Evangelista Torricelli was an Italian physicist and mathematician who first discovered and described this principle in the 17th century. He is also known for inventing the barometer and making important contributions to the field of optics.

What is the significance of Torricelli's principle?

Torricelli's principle is important in understanding the behavior of fluids in a container, such as water in a water tank or air in a balloon. It also has practical applications in engineering and design, such as in the construction of fountains and sprinkler systems.

How is Torricelli's principle related to Bernoulli's principle?

Torricelli's principle is a specific case of Bernoulli's principle, which describes the relationship between fluid speed and pressure in a moving fluid. Torricelli's principle only applies to fluids that are not compressible, such as water, while Bernoulli's principle applies to all fluids, including gases.

What is an example of Torricelli's principle in everyday life?

An example of Torricelli's principle in everyday life is when you fill a bathtub with water and then open the drain. The water flows out of the drain at the same speed as an object would fall if it were dropped from the surface of the water at the same height as the drain. This is also why water can shoot out of a faucet at a high speed when the water pressure is high.

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