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# Fluids: Pressure of fluid column, Burnoulli equation

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1. Examine the schematic of the tower. Water flows in at the top to balance the water flowing out through the holes, so the height of the water in the tower stays fixed. Take state1 at the top of the water in the tower and state 2 in the water just as it flows out of the hole.
a. What is the pressure at states 1 and 2? Explain
b. Write down Bernoulli's equation
c. If the holes are at heights of 3.0 cm, 13 cm and 23 cm above the surface of the water in the lower basin and the top hole is 3.0 cm below the top of the water in the tower, predict which one you believe will go farthest (top, middle, or bottom). Explain your reasoning.

a) The pressure in state 1 and 2 are different. The pressure at state 2 will be more then state 1 because of the weight of water above it.

The pressure is the force per unit area normal to the area.
Let the area of the water column is A. then if we consider a column of height h then its volume of water in the column of height h will be
V = Ah
Its mass m = V =  Ah ( is the density)
And weight W - mg =  A hg
This extra weight is balanced by the increase in pressure. Let the pressure of the liquid increases from P1 at the top to P2 at the bottom of the column of height h then balancing the force in vertical direction we get
P1A + W - P2A = 0
Or P2A - P1A =  A hg
Or P2 - P1 = h  g
Thence the pressure difference of the column of liquid of height h is
P=h g
And hence the pressure at state 2 will be more than that in state 1.
b)
Bernoulli's theorem can be stated in three different ways
1. if we consider energy of a mass m of the fluid, having volume V, v is the flow velocity at height h, then we can write

PV + m g h + (1/2) mv2 = constant (for all points).
2. If the energy per unit volume is considered then the same equation is written as

P +  g h + (1/2) v2 = constant;
(Dividing equation 1. by V) where  is the density of the fluid, and

3. If the pressure and velocity is measured in terms of height (heads) then
P/g) + h + v2/2gh = constant
(Dividing equation 2. by g)

According to the given data we have to choose the equation.

c)
The horizontal distance reached by the water from a hole depends on the velocity of the water in horizontal direction and the time taken to reach the bottom.
As we have seen above the velocity of the water coming out of the whole increases with the depth means the velocity of water at the top will be the minimum and greatest at the bottom hole.
The time required to reach the floor will depend on the height from which water is falling and hence it increases from top to bottom hole.
Thus we can predict that the water from the middle hole should go farthest on the bottom.
Because for the water from the top the velocity of water is too low and the horizontal distance covered will be less and for the water from the lowest hole the time to reach too less and hence the horizontal distance covered will be less. The velocity and time both for water from the middle hole are reasonable and hence will go the farthest.

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