Question

A 20 m high dam is used to create a large lake. The lake is filled to a depth of 16 m as shown. The density of water is 1000 kg/m³ . 

(a) Calculate the absolute pressure at the bottom of the lake next to the dam. A release valve is opened 5.0 m above the base of the dam, and water exits horizontally from the valve. 

(b) Use Bernoulli’s equation to calculate the initial speed of the water as it exits the valve. 

(c) The stream below the surface of the dam is 2.0 m deep. Assuming that air resistance is negligible, calculate the horizontal distance x from the dam at which the water exiting the valve strikes the surface of the stream. 

(d) Suppose that the atmospheric pressure in the vicinity of the dam increased. How would this affect the initial speed of the water as it exits the valve? 

___It would increase. ____It would decrease. ____It would remain the same.

Answer 1

(a) P_abs = P_gauge + P_o 

P_abs = ρgh + 1.01 x 10⁵ 

P_abs = (1000)(9.8)(16) + 1.01 x 10⁵ = 260000Pa 

(b) Use Bernoulli,  … v₂ = √2gh … v₂ = √2(9.8)(11)… 

v₂ = 14.7m/s 

(c) Using projectile methods. 

d_y = v_iyt + ½at² 

 – 3 = 0 + ½ (–9.8) t² 

 t = 0.78 sec 

 d_x = v_xt = (14.7) (0.78) = 11.5 m 

(d) An increase in atmospheric pressure around the damn increases both P₁ and P₂ equally so there is no net effect on these terms in Bernoulli’s equation, which means the exit velocity would be the same.