A beaver dam forms a pond with a surface level 2.0 m above the bottom of the dam

Question

A beaver dam forms a pond with a surface level 2.0 m above the bottom of the dam. The dam has developed a leak at the very bottom through a circular hole of radius 3.0 cm. You can safely assume that the cross-sectional area of the surface of the pond is much, much larger than the cross-sectional area of the hole. (a) At what speed does water exit the hole? (b) If the pond holds 2.0 times 10^4 L of water, and assuming that the speed you found in (a) remains constant as the pond drains, estimate how long it takes to drain the pond completely? (c) Is the time that you found in (b) an underestimate or an overestimate of the time actually required to drain the pond? Explain.

Explanation / Answer

(A) Applying bernoulli's equation,

at top of surface = at location of hole

P1 + rho g h1 + rho v1^2 /2 = P2 + rho g h2 + rho v2^2 /2

P_atm + (1000 x 9.8 x 2) + 0 = P_atm + 0 + (1000 v^2 / 2)

v = 6.26 m/s .........Ans

(b) volume flow rate = A v

= pi (0.03^2) (6.26)

= 0.0177 m^3 / s

2 x 10^4 L = 20 m^3

t = 20 m^3/ 0.0177 = 1130 sec or 18.8 min

(c) as level decreases, waterspeed will also decrease.

that results in less volume flow rate with time.

so actually it will take more time .

so (b) is an undersetimate of the time.

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