Speed of Efflux Torricelli`s Law Derivation Class 11

The application of Bernoulli`s principle$$p_1 + rho gy + frac{1}{2}rho v_1^2 = p_2 + rho gY$$Here $displaystyle{frac{1}{2}rho v_2^2}$ is zero because $v_2$ appears to us zero. So then-$$frac{1}{2}rho v_1^2 = (p_2 – p_1) + rho g(Y -y)$$And we know that, $Y – y = h$So, it goes-begin{align*}frac{1}{2}rho v_1^2& = (p_2 -p_1) + rho ghv_1^2&= 2gh + frac{2(p_2 – p_1)}{rho}end{align*}Then the discharge speed is -$$v_1 = sqrt{2gh + frac{2(p2 – p1)}{rho}}qquad (p_1 = p(text{ atm})$$ This formula applies when the tank is closed. When the tank is opened, $p_$2 becomes p(atm). And the formula was-$$v_1 = sqrt{2gh}$$ [p_2-p_1 = 0], because both are atmospheric pressure. In kinematics, this formula is the speed of objects in free fall. According to Torricelli`s law for Newtonian fluids, the density of the flow of a liquid flowing through a sharp-edged hole at the bottom of a reservoir filled with liquid at a depth of h is the same velocity that a free-falling body would reach if it fell from a height h. If the liquid flows like a jet, it should be a horizontal distance called the zone. Then, this area can be calculated simply by multiplying the velocity of the flow and the time it takes to touch the ground. If y is the height of the opening and Y is the height of the liquid column, then the time it takes to reach the ground is given as follows: If we apply the second equation of motion, we get -begin{align*}y& = ut + frac{1}{2}gt^2y& = frac{1}{2}gt^2qquad [ ut = 0]\text{Then}; t& = sqrt{frac{2y}{g}}end{align*}Then the range-begin{align*}R& = v_1times t = sqrt{2gh}times sqrt{frac{y}{g}}R& = sqrt{4yh} = 2sqrt{yh}end{align*}Then the maximum range is given as follows:$$R = 2sqrt{yh} = 2sqrt{y(Y – y)}$$Differentiate Range w.r.t to y, we get-$$R`(y) = 2 left[frac{1}{{2sqrt{y(Y – y)}}} times(Y-2y)right]$$Equating R`(y) is equal at 0$$frac{(Y -2y)}{sqrt{y(Y-y)}} = 0$$We get $Y – 2y =0$Then double differentiation from Y-2y w.r.t to y, then we get (-2), which is less than 0, so it is maxima. Coordinates $displaystyle{y = frac{Y}{2}}$ is the maxima pointIf you put the value $displaystyle{y = frac{Y}{2}}$ in the range formula, then we get-begin{align*}R& = 2sqrt{frac{Yleft[Y -left(frac{Y}{2}right)right]}{2}}R& = 2sqrt{frac{Y^2}{4}} = Yend{align*}This means that its maximum reach is equal to the height of the liquid column in the tank. Watch this video for more information. Since the water level H − h {displaystyle H-h} is above the aperture, the horizontal output velocity v = 2 g ( H − h ) , {displaystyle v={sqrt {2g(H-h)}},} according to Torricelli`s law. So we have two equations, where t {displaystyle t} is the time it takes for the beam particle to fall from the aperture to the ground.

If the horizontal output velocity v is {displaystyle v}, then the horizontal distance traveled by the beam particle during period t {displaystyle t} is Evangelista Torricelli`s original derivation can be found in the second book `De motu aquarum` of his `Opera Geometrica` (see [4]): He starts a tube filled with water AB (Figure (a)) to plane A. Then a narrow opening is drilled at height B and connected to a second vertical tube BC. Due to the hydrostatic principle of the communicating tanks, the water in both pipes rises to the same AC level (Figure (b)). Finally, when the BC pipe is removed (Figure (c)), the water must rise to this height, called AD in Figure (c). The reason for this behavior is the fact that the rate of fall of a droplet from altitude A to B is equal to the muzzle velocity required to lift a fall from B to A. When we talk about Torricelli`s law, Torricelli`s law only tells us the exit rate. And says that if a reservoir is filled with the liquid up to the height h, above the opening, then the velocity of the flow is equal to the rate of free fall of a drop of liquid from the free surface of the liquid at the opening. This flow rate can be found by equating the kinetic energy gained with the potential energy lost during the free fall of a drop of liquid. We will now derive the formula of the rate of discharge. In this article, we will discuss the velocity of flow, its calculation, Torricelli`s law, its derivative, venturimeter, operating principle and working equation.

Going further in the article, we must first understand the meaning of efflux. So what is Efflux? If we talk about efflux, then the efflux is nothing, but it is a simple flow of liquid, just like a jet from a very small hole, that is, an opening, and it is necessary that the cross-section of the opening is very small compared to the cross-section of the tank. [latexpage] A klepsydra is a clock that measures time through the flow of water. It consists of a pot with a small hole at the bottom through which water can escape. The amount of water escaping indicates the measurement of time. As Torricelli`s law gives, the velocity of flow through the hole depends on the water level; And when the water level drops, the outflow is not uniform. A simple solution is to keep the water level constant. This can be achieved by a constant flow of water flowing into the container, the overflow of which can escape from above another hole. With a constant height, the water flowing from the bottom can be collected in another cylindrical container with a uniform graduation to measure time. It is a tributary of the Klepsydra. “V” is considered “0” because the surface of the liquid falls slowly relative to the rate at which the liquid leaves the tank. at any two points of the flowing liquid.

Here, v {displaystyle v} is the velocity of the fluid, g {displaystyle g} is the acceleration due to gravity, y {displaystyle y} is the height above a reference point, p {displaystyle p} is the pressure, and ρ {displaystyle rho } is the density. So today we are going to talk about the concept of Torricelli`s law and the velocity of EFFLUX and we are also going to derive the expression of the efflux velocity. This concept is the application of Bernoulli`s principle. This is a very interesting phenomenon of hydrodynamics, this law is given in 1643 by the Italian scientist Evangelista Torricelli. It calculates the velocity of liquids flowing like a jet from a hole in a tank. Torricelli`s law, also known as Torricelli`s theorem, is a theorem in fluid dynamics that relates the flow velocity of the fluid of an aperture to the height of the fluid above the aperture. The law states that the velocity v of the flow of a liquid through a sharp-edged hole at the bottom of the container, filled to a depth h, is the same velocity that a body (in this case, a drop of water) would reach if it fell freely from a height h, that is.

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