A horizontal pipe, 10 cm in diameter, has a smooth reduction to a pipe 5 cm in diameter. If the pressure of
the water in the larger pipe is 8.00 104 Pa and the pressure in the smaller pipe is 6.00 104 Pa, what is the
flow speed of the water (in m/s) in the smaller pipe?
a. 0.0625 m/s
b. 6.53 m/s
c. 0.94 m/s
d. 42.67 m/s
e. 2 m/s
b. 6.53 m/s
To find the flow speed of the water in the smaller pipe, we can use Bernoulli's equation, which relates the pressure and velocity of a fluid.
Bernoulli's equation is given as:
P₁ + 1/2 ρv₁² = P₂ + 1/2 ρv₂²
where:
P₁ = pressure in the larger pipe
P₂ = pressure in the smaller pipe
v₁ = velocity in the larger pipe
v₂ = velocity in the smaller pipe
ρ = density of water (assumed constant)
Given:
P₁ = 8.00 * 10^4 Pa
P₂ = 6.00 * 10^4 Pa
The diameter of the larger pipe is 10 cm, which means its radius is 10/2 = 5 cm = 0.05 m.
The diameter of the smaller pipe is 5 cm, which means its radius is 5/2 = 2.5 cm = 0.025 m.
Since both pipes are horizontal, the height difference is not considered in this problem, so we can ignore the potential energy term in Bernoulli's equation.
Therefore, the equation becomes:
P₁ + 1/2 ρv₁² = P₂ + 1/2 ρv₂²
Since the flow is incompressible, the density ρ is the same at both points and cancels out from the equation.
Simplifying the equation:
P₁ + 1/2 v₁² = P₂ + 1/2 v₂²
Rearranging the equation:
1/2 v₂² = (P₁ - P₂) + 1/2 v₁²
Solving for v₂:
v₂² = 2(P₁ - P₂) + v₁²
Substituting the given values:
v₂² = 2(8.00 * 10^4 - 6.00 * 10^4) + v₁²
v₂² = 2(2.00 * 10^4) + v₁²
Next, we need to find v₁². Since v₁ is the flow speed in the larger pipe, it can be determined using the equation:
A₁v₁ = A₂v₂
where:
A₁ = area of the larger pipe
A₂ = area of the smaller pipe
Since the pipes have circular cross-sections, the area of a pipe can be calculated using the formula:
A = πr²
Substituting the values:
A₁v₁ = A₂v₂
πr₁²v₁ = πr₂²v₂
(π(0.05)²)v₁ = (π(0.025)²)v₂
(0.0025)v₁ = (0.000625)v₂
Simplifying:
v₁ = (0.000625/0.0025)v₂
v₁ = 0.25v₂
Substituting this expression for v₁² into the earlier equation:
v₂² = 2(2.00 * 10^4) + (0.25v₂)²
v₂² = 4.00 * 10^4 + (0.0625)v₂²
Multiplying through by 1,000 to get rid of decimal points:
1,000v₂² = 40,000 + 62.5v₂²
Rearranging and simplifying:
937.5v₂² = 40,000
v₂² = 40,000 / 937.5
v₂² ≈ 42.67
Taking the square root of both sides:
v₂ ≈ √(42.67)
v₂ ≈ 6.53 m/s
Therefore, the flow speed of the water in the smaller pipe is approximately 6.53 m/s. So, the correct answer is option b) 6.53 m/s.
To find the flow speed of water in the smaller pipe, we can apply Bernoulli's equation, which relates the pressure and speed of a fluid.
Bernoulli's equation states that the total energy per unit mass of a fluid particle remains constant along a streamline. It can be expressed as:
P1 + (1/2)ρv1^2 + ρgh1 = P2 + (1/2)ρv2^2 + ρgh2
Where:
P1 and P2 are the pressures at points 1 and 2 respectively.
v1 and v2 are the flow speeds at points 1 and 2 respectively.
ρ is the density of the fluid.
g is the acceleration due to gravity.
h1 and h2 are the heights at points 1 and 2 respectively.
In this problem, the two points are located at the larger and smaller pipes, so we can assume that both points are at the same height, and h1 = h2 = 0.
The equation can be simplified to:
P1 + (1/2)ρv1^2 = P2 + (1/2)ρv2^2
Now, let's substitute the given values into the equation.
P1 = 8.00 × 10^4 Pa
P2 = 6.00 × 10^4 Pa
v1 is unknown
v2 is unknown
Since the fluid is incompressible, the density (ρ) cancels out.
Therefore, the equation becomes:
P1 + (1/2)v1^2 = P2 + (1/2)v2^2
Now, let's solve this equation to find the flow speed (v2) in the smaller pipe.
P1 + (1/2)v1^2 - P2 = (1/2)v2^2
(1/2)v2^2 = P1 + (1/2)v1^2 - P2
v2^2 = 2(P1 - P2) + v1^2
v2^2 = 2(8.00 × 10^4 Pa - 6.00 × 10^4 Pa) + v1^2
v2^2 = 2(2.00 × 10^4 Pa) + v1^2
v2^2 = 4.00 × 10^4 Pa + v1^2
Now, we need to solve for v2, the flow speed in the smaller pipe. To do this, we need to know the value of v1, the flow speed in the larger pipe.
Unfortunately, the problem doesn't provide us with the value of v1. Therefore, we cannot determine the flow speed in the smaller pipe (v2).
Given the options, it is not possible to determine the flow speed in the smaller pipe based on the information provided.