These are some questions about deriving Bernoulli\'s Principle. The first pictur

Question

These are some questions about deriving Bernoulli's Principle. The first picture has all of the equations. Please provide a step by step solution. Thank you so much for your assistance. Theory: The figure shows a tank of cross-section 4 filled with water to a height h above a small hole of cross-section a. You will show (see 03a) that the speed vh at which the water level drops depends on the height according to eqn la below. You'll also show (see Q3b) this can be recast as eqn lb (la) & (lb) al You'll also explain (see Q2a) why we can ignore the (a/A) term in the denominator of the radical, arriving at: We also note that v (speed that the water level descends in the tank) is simply the rate at which the water level drops- except we note dhidt will be negative (h decreases) while vh (a speed) is positive. We just need an extra minus to make them equate (eqn 3a below). Then we sub into it eqn 2 from above, transforming it into eqn 3b: dh Un-at (a) & (3b) hence dt=-AV2gh

Explanation / Answer

a.

from bernoulli's equations, two points 1 and 2, haing pressure P1, P2, flow velocities, V1, V2 and height from reference h1 and h2, then these quantitiels can be related as under

rho*gh1 + 0.5*rho*v1^2 + P1 = rho*gh2 + 0.5*rho*v2^2 + P2

now, from the figure

point 1 is a point on the free surface of the water, point 2 is a point on the hole

so P1 = P2 = Patm

rho -> density of the liquid

h1 - h2 = h

so, rhog(h1 - h2) + 0.5*rho*(v1^2 - v2^2) = 0

2*g*h = v2^2 - v1^2

now, volume flow rate at top of container = volume flow rate through the hole

area of cross section of top = A1

area of cross section of hole = A2

so, A1v1 = A2v2

v1 = A2v2/A1

2gh = v2^2 - A2^2 v2^2 / A1^2 = v2^2 [ 1 - (A2/A1)^2]

so v2 = sqroot(2gh/(1 - (A2/A1)^2))

b. v2 = sqroot(2gh/(1 - (A2/A1)^2))

v2 = sqroot(2gh/(A2/A1)^2((A1/A2)^2 - 1) = sqroot(2gh/((A1/A2)^2 - 1) / (A2/A1) = (A1/A2) * sqroot(2gh/((A1/A2)^2 - 1)

c. now, if A1 > > A2

then A2/A1 = 0

then v2 = sqroot(2gh/(1 - (A2/A1)^2)) = (A1/A2) * sqroot(2gh)

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