A pipe of constant diameter d = 10.5 cm carries water up a hill h = 38.1 m high. How much pressure is required at the bottom of the hill if the water is to reach the top?
Any help would be great
To determine the pressure required at the bottom of the hill for the water to reach the top, you can use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid flowing in a pipe.
Bernoulli's equation: P1 + 1/2 ρv1^2 + ρgh1 = P2 + 1/2 ρv2^2 + ρgh2
Where:
P1 and P2 are the pressures at points 1 and 2 in the pipe.
ρ is the density of water.
v1 and v2 are the velocities of the water at points 1 and 2.
g is the acceleration due to gravity.
h1 and h2 are the heights of points 1 and 2.
In this case, point 1 is at the bottom of the hill and point 2 is at the top. At the bottom of the hill, the velocity is zero, so v1 = 0. At the top of the hill, the height is h = 38.1 m, so h2 = 38.1 m. We need to find the pressure at the bottom, P1.
Since the pipe has constant diameter, the velocity of the water is the same at all points in the pipe. So we can write v1 = v2. Now let's simplify Bernoulli's equation:
P1 + ρgh1 = P2 + ρgh2
Since v1 = v2 and v1 = 0, we can see that the velocity terms cancel out. Now the equation becomes:
P1 + ρgh1 = P2 + ρgh
Rearranging the equation, we get:
P2 - P1 = ρg(h2 - h1)
Since P1 is the pressure at the bottom, P1 = 0 (atmospheric pressure) and P2 is the pressure at the top. Therefore:
P2 = ρg(h2 - h1)
Now we can substitute the known values:
d = 10.5 cm = 0.105 m (convert cm to m)
ρ = density of water = 1000 kg/m^3
g = acceleration due to gravity = 9.8 m/s^2
h1 = 0 m (bottom of the hill)
h2 = 38.1 m (top of the hill)
Plugging these values into the equation, we get:
P2 = (1000 kg/m^3) * (9.8 m/s^2) * (38.1 m - 0 m)
Calculating this, we find:
P2 = 376380 Pa
Therefore, the pressure required at the bottom of the hill for the water to reach the top is approximately 376380 Pa.