Physics, 05.05.2020 22:32 Shubbs

Two circular holes, one larger than the other, are cut in the side of a large water tank whose top is open to the atmosphere. Hole 1 is the larger hole, and hole 2 is the smaller hole. The center of one of these holes is located 6 times as far beneath the surface of the water as the other. The volume flow rate of the water coming out of the holes is the same. (a) Decide which hole (1 or 2) is located nearest the surface of the water. (b) Calculate the ratio of the radius of the larger hole to the radius of the smaller hole, rA/rB.

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Answer from: Poohpooh8548

A) The hole located nearest the surface is; hole 2

B) rA/rB = 1.565

We are told that the center of one of the holes is located 6 times as far beneath the surface of the water as the other. Thus;

h2 = 6h1

A) Since the volume flow rate of water coming out of both holes is the same, then it means that the small hole must be located below the large hole. B) Using Bernoulli's theorem at hole 1 and 2, we have;

P1 + ρgh1 = P_atm + ½ρ(v1)² - - - (eq 1)

P2 + ρg6h1 = P_atm + ½ρ(v2)² - - - (eq 2)

We are told that the top is open to the atmosphere and as such;

P1 = P2 = P_atm

Thus,equations 1 and 2 give us;

ρgh1 = ½ρ(v1)²

gh1 = (v1)²

v1 = √(2gh1)

Similarly;

v2 = √(12gh1)

Now, we know from continuity equation and conservation of energy that;

Q = A1v1 = A2v2

Now, A1 = π(r1)²

And A2 = π(r2)²

Thus;

π(r1)²v1 = π(r2)²v2

This reduces to;

(r1)²v1 = (r2)²v2

Plugging in the relevant values;

(r1/r2)² = v2/v1

(r1/r2)² = [√(12gh1)]/[√(2gh1)]

(r1/r2)² = √6

(r1/r2)² = 2.44948974

r1/r2 = √2.44948974

r1/r2 = 1.565

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