What pressure difference is required between the ends of a 2.0--long, 1.0--diameter horizontal tube for 40 water to flow through it at an average speed of 4.0 ?

I know we have to find delta P, but i'm stuck on how to do that.
Help would be great

To find delta P, use

(deltaP)=8(pi)(eta)(L*v)/A

Where eta is the coefficient for viscosity. For 40C water, it is .07e-3 Pa*s.

To find the pressure difference (ΔP) required for water to flow through the tube, you can use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid.

Bernoulli's equation states:
P1 + 0.5ρv1^2 + ρgh1 = P2 + 0.5ρv2^2 + ρgh2

Here's how you can use this equation to solve the problem step by step:

Step 1: Convert the length of the tube to meters.
Given: Length of the tube = 2.0 m

Step 2: Calculate the cross-sectional area of the tube.
Given: Diameter of the tube = 1.0 m
The area (A) of a circular tube can be calculated using the formula:
A = πr^2
where r is the radius of the tube. In this case, the radius (r) is half the diameter, so r = 1.0/2 = 0.5 m.
Substituting these values into the formula:
A = π(0.5)^2 = π(0.25) ≈ 0.785 m^2

Step 3: Calculate the volume flow rate (Q) of the water.
The volume flow rate is given by the formula:
Q = Av
where A is the cross-sectional area of the tube and v is the speed of the water.
Substituting the known values:
Q = 0.785 m^2 × 4.0 m/s = 3.14 m^3/s

Step 4: Calculate the density (ρ) of water.
The density of water is approximately 1000 kg/m^3.

Step 5: Rearrange Bernoulli's equation to solve for ΔP.
ΔP = P2 - P1 = 0.5ρv1^2 - 0.5ρv2^2

Step 6: Plug in the known values into the equation.
ΔP = 0.5 × 1000 kg/m^3 × (0 - (4.0 m/s)^2)

Step 7: Simplify and calculate ΔP.
ΔP = 0.5 × 1000 kg/m^3 × (-16.0 m^2/s^2)
= -8,000 Pa

Therefore, a pressure difference of -8,000 Pa (or 8,000 Pa lower pressure at the end of the tube compared to the beginning) is required for water to flow through the tube at an average speed of 4.0 m/s. Note that the negative sign indicates a decrease in pressure.

To find the pressure difference (ΔP) required for water to flow through a tube, you can use Bernoulli's equation. Bernoulli's equation describes the relationship between pressure, velocity, and height for a fluid flowing through a confined space.

1. Identify the known values:
- Tube length (L) = 2.0 m
- Tube diameter (D) = 1.0 m
- Water flow speed (v) = 4.0 m/s
- Water density (ρ) = 1000 kg/m^3 (typical value for water)

2. Calculate the cross-sectional area of the tube:
- Tube radius (r) = D/2 = 0.5 m
- Tube cross-sectional area (A) = πr^2

3. Determine the mass flow rate (ṁ) of water:
- Mass flow rate (ṁ) = ρAv

4. Apply Bernoulli's equation:
Bernoulli's equation states: P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2
- P1 and P2 are the pressures at the two different points (the ends of the tube in this case)
- ρ is the density of the fluid (water in this case)
- v1 and v2 are the velocities of the fluid at the two different points
- h1 and h2 are the heights of the fluid at the two different points (which can be considered equal here as the tube is horizontal)

Bernoulli's equation can be simplified in this case due to the equal heights:
P1 + 1/2ρv1^2 = P2 + 1/2ρv2^2

5. Rearrange the equation to solve for the pressure difference (ΔP):
ΔP = P2 - P1 = 1/2ρ(v2^2 - v1^2)

6. Substitute the known values into the equation and calculate the pressure difference:
ΔP = 1/2 * 1000 kg/m^3 * (4.0^2 - 0) m^2/s^2
ΔP = 1/2 * 1000 kg/m^3 * 16.0 m^2/s^2
ΔP = 8000 N/m^2 or 8000 Pa (Pascal)

Therefore, a pressure difference of 8000 Pascal (Pa) is required between the ends of the tube for water to flow through it at an average speed of 4.0 m/s.