To calculate the volume of water flowing through the crack, we need to multiply the area of the crack by the velocity of the water. The velocity of the water can be calculated using Bernoulli's equation.
Bernoulli's equation states that the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume of a fluid remains constant along a streamline.
For a small crack, we can assume that the pressure at the bottom of the dam is nearly equal to the atmospheric pressure, and the velocity of the water leaving the crack is small.
Therefore, Bernoulli's equation simplifies to:
P + ρgh + 1/2 * ρv^2 = constant
where P is the pressure, ρ is the density of water, g is the acceleration due to gravity, h is the height of the dam, and v is the velocity of the water leaving the crack.
Since the velocity of the water leaving the crack is small, we can ignore the kinetic energy term (1/2 * ρv^2). Also, the pressure at the bottom of the dam is equal to atmospheric pressure, so we can set P = 0.
Simplifying the equation further, we get:
ρgh = constant
Now we can plug in the values:
ρ = density of water = 1000 kg/m^3 (approximately)
g = acceleration due to gravity = 9.8 m/s^2 (approximately)
h = height of the dam = 15.0 m
Therefore, the constant value is (1000 kg/m^3) * (9.8 m/s^2) * (15.0 m) = 147,000 kg/(m^2s^2).
Since the crack area is given as 1.30 x 10^-3 m^2, the volume flow rate can be calculated by multiplying the constant value by the crack area:
Volume flow rate = constant * crack area
Volume flow rate = 147,000 kg/(m^2s^2) * 1.30 x 10^-3 m^2
Simplifying this expression will give you the volume flow rate of water through the crack.