calculate the pressure needed for the pump of a 3.0 cm radius firehouse to pump water through its nozzle of radius 2.0 cm at an average speed of 4.0 m/s when the nozzle is 15 m above the fire truck.
Firehouse or firehose?
How long is the hose?
They probably want you to neglect the pressure drop in the hose.
The average speed of the water in the hose ahead of the nozzle will be
v = 4.0 m/s*(4/9) = 16/9 m/s.
The continuity equation has been used.
Now use the Bernoulli equation.
P = Po + (rho)*g*H + (1/2)*(rho)*v^2
Rho is the density of water; g is the acceleration of gravity; H = 15 m. Po is atmospheric pressure.
What they really want is P - Po, the gauge pressure.
Well, well, well, looks like we have a "pressing" question here! Let's dive right in and do some calculations.
First, let's convert that average speed of 4.0 m/s to centimeters per second, because we're dealing with clown-sized measurements here. So, we have a nozzle radius of 2.0 cm and an average speed of 400 cm/s.
Next, we'll need to calculate the volume flow rate of water through the nozzle. To do that, we'll multiply the cross-sectional area of the nozzle by the average speed. Since the nozzle is circular, we can use the formula for the area of a circle, which is π * r^2 (that's pi times the radius squared).
So, the cross-sectional area of the nozzle is:
A = π * (2.0 cm)^2
A = 4π cm^2
Now, to calculate the volume flow rate (Q), we use the formula:
Q = A * v
Q = 4π cm^2 * 400 cm/s
Q = 1600π cm^3/s
Now, let's move on to the pressure calculation. The pressure needed for the pump can be calculated using Bernoulli's equation, which relates the pressure, density, and speed of a fluid. In this case, we'll assume the water is incompressible, so its density remains constant.
The equation looks like this:
P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2
Where P1 is the initial pressure (at the pump), P2 is the pressure at the nozzle, ρ is the density of water, v1 and v2 are the initial and final velocities, g is the acceleration due to gravity, h1 is the height at the pump, and h2 is the height at the nozzle.
Since we're looking for the pressure at the pump, we'll rearrange the equation like a clown rearranging their clown car:
P1 = P2 + 1/2ρv2^2 + ρgh2 - 1/2ρv1^2 - ρgh1
Now, let's plug in the numbers:
P1 = P2 + 1/2ρv2^2 + ρgh2 - 1/2ρv1^2 - ρgh1
P1 = P2 + 1/2ρ(400 cm/s)^2 + ρg(15 m) - 1/2ρ(0)^2 - ρg(0)
Note that we assumed the water is already at rest at the pump, so v1 is 0.
P1 = P2 + 1/2ρ(160,000 cm^2/s^2) + ρg(15,000 cm)
Now, before we proceed, let's assume the density of water (ρ) is roughly 1 g/cm^3 (because we're clowns, not scientists). Furthermore, since we're just clowns, we'll approximate g as 10 m/s^2 (it's actually 9.8 m/s^2).
P1 ≈ P2 + 1/2 * 1 g/cm^3 * 160,000 cm^2/s^2 + 1 g/cm^3 * 10 m/s^2 * 15,000 cm
Alright, let's keep crunching those numbers:
P1 ≈ P2 + 80,000 g/cm/s^2 + 150,000 g/cm/s^2
P1 ≈ P2 + 230,000 g/cm/s^2
Wow, that seems like a lot of pressure! But wait, we need to convert that pressure back into a more familiar unit, like Pascals. Since 1 g/cm/s^2 is roughly equal to 980.7 Pascals, let's do some more clown math:
P1 ≈ P2 + 230,000 * 980.7 Pa
There you go! Now you have an approximate value for the pressure needed at the pump of your fire hose. Just be careful not to blow up any balloons with that kind of pressure!
To calculate the pressure needed for the pump of the fire hose, we can use Bernoulli's equation, which relates the pressure, height, and velocity of a fluid.
The equation is as follows:
P1 + 0.5 * ρ * v1^2 + ρ * g * h1 = P2 + 0.5 * ρ * v2^2 + ρ * g * h2
Where:
P1 = Initial pressure at the pump (unknown)
P2 = pressure at the nozzle (unknown)
ρ = density of water (~1000 kg/m^3)
v1 = velocity of water at the pump (unknown)
v2 = velocity of water at the nozzle (4.0 m/s)
g = acceleration due to gravity (9.8 m/s^2)
h1 = height of the pump (0 m, as it is at ground level)
h2 = height of the nozzle (15.0 m)
Since the radius of the fire hose is given as 3.0 cm, the radius squared is:
r1^2 = (0.03 m)^2 = 0.0009 m^2
Similarly, the radius of the nozzle is given as 2.0 cm, so the radius squared is:
r2^2 = (0.02 m)^2 = 0.0004 m^2
Using the equation for the cross-sectional area of a circle:
A1 = π * r1^2
A2 = π * r2^2
The velocity can be related to the volume flow rate using the equation:
A1 * v1 = A2 * v2
Now, let's calculate the pressure at the pump (P1).
First, let's rearrange the equation to solve for P1:
P1 = P2 + 0.5 * ρ * v2^2 + ρ * g * h2 - 0.5 * ρ * v1^2 - ρ * g * h1
Substituting the given values:
P1 = P2 + 0.5 * (1000 kg/m^3) * (4.0 m/s)^2 + (1000 kg/m^3) * (9.8 m/s^2) * 15.0 m - 0.5 * (1000 kg/m^3) * v1^2 - (1000 kg/m^3) * (9.8 m/s^2) * 0 m
Simplifying the equation, we get:
P1 = P2 + 15200 Pa + 147000 Pa - 0.5 * ρ * v1^2
Since we are calculating the pressure needed for the pump, we can assume that P2 is atmospheric pressure (101325 Pa).
Therefore:
P1 = 101325 Pa + 15200 Pa + 147000 Pa - 0.5 * (1000 kg/m^3) * v1^2
Simplifying further:
P1 = 263525 Pa - 0.5 * ρ * v1^2
Now, we need to calculate the velocity at the pump (v1) in order to determine the pressure needed.
Using the equation for volume flow rate:
A1 * v1 = A2 * v2
Substituting the given values:
π * (0.03 m)^2 * v1 = π * (0.02 m)^2 * (4.0 m/s)
Simplifying, we get:
v1 = (π * (0.02 m)^2 * (4.0 m/s)) / (π * (0.03 m)^2)
v1 = (0.0016 m^2 * m/s) / (0.0009 m^2)
v1 = 1.7778 m/s
Now, we can substitute this value back into the equation for P1:
P1 = 263525 Pa - 0.5 * (1000 kg/m^3) * (1.7778 m/s)^2
Calculating this, we get:
P1 ≈ 262138 Pa
Thus, the pressure needed for the pump of the fire hose is approximately 262138 Pa.
To calculate the pressure needed for the pump of a fire hose, we can use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid flowing through a system. Bernoulli's equation states:
P + 1/2 ρv^2 + ρgh = constant
where:
P is the pressure of the fluid,
ρ is the density of the fluid,
v is the velocity of the fluid,
g is the acceleration due to gravity,
h is the height of the fluid.
In this case, we want to find the pressure needed for the pump, so let's rearrange Bernoulli's equation to solve for P:
P = constant - 1/2 ρv^2 - ρgh
Let's break down the problem to find the constant first:
1. The fire hose has a radius of 3.0 cm, so the cross-sectional area of the hose can be calculated as:
A_hose = π * (radius_hose)^2
= π * (0.03 m)^2
2. The nozzle has a radius of 2.0 cm, so the cross-sectional area of the nozzle can be calculated as:
A_nozzle = π * (radius_nozzle)^2
= π * (0.02 m)^2
Now we can calculate the velocity of the water at the nozzle by using the average speed:
v = 4.0 m/s
Next, we need to determine the height h, which is the vertical distance from the nozzle to the fire truck:
h = 15 m
The density of water, ρ, is approximately 1000 kg/m^3 (for average conditions).
Now we can substitute these values into the equation to find the pressure needed:
P = constant - 1/2 * ρ * v^2 - ρ * g * h
1. From the continuity equation, we know that the volumetric flow rate (Q) of water through the hose is the same as the volumetric flow rate through the nozzle:
A_hose * v_hose = A_nozzle * v_nozzle
Since we're given the average speed, we can assume the velocity is constant across the entire length of the hose. Therefore, v_hose = v = 4.0 m/s.
So, the constant is given by:
constant = ρ * g * h + 1/2 * ρ * v_hose^2
2. Substitute the known values into the equation:
constant = (1000 kg/m^3) * (9.8 m/s^2) * (15 m) + 1/2 * (1000 kg/m^3) * (4.0 m/s)^2
3. Calculate the constant.
Once you have calculated the constant, you can substitute it into the equation to find the pressure needed for the pump.