The main water line enters a house on the first floor. The line has a gauge pressure of 1.70 105 Pa.
(a) A faucet on the second floor, 6.5 m above the first floor, is turned off. What is the gauge pressure at this faucet?
(b) How high could a faucet be before no water would flow from it, even if the faucet were open?
(a) Well, if the faucet on the second floor is turned off, it's not feeling any pressure at all. It's just enjoying its vacation. So, the gauge pressure at this faucet would be a big fat zero! Just remember to turn it on if you want some water.
(b) Ah, the height limit for faucets! Well, water might be a great companion, but it's not much of a climber. As you go higher, the pressure decreases due to gravity giving it a hard time. So, the height limit would depend on the strength of the water pressure and the determination of that poor little water droplet. But if you ask me, I don't think I've ever seen a faucet on top of Mount Everest. So, let's just say it's pretty high up there.
To answer these questions, we can use the concept of hydrostatic pressure. The hydrostatic pressure in a fluid at a given height is given by the equation:
P = P₀ + ρgh
Where:
P is the gauge pressure at a certain height h above the reference height
P₀ is the gauge pressure at the reference height
ρ is the density of the fluid
g is the acceleration due to gravity
In this case, the reference height is the first floor, where the gauge pressure is given as 1.70 x 10^5 Pa.
(a) To find the gauge pressure at the faucet on the second floor (6.5 m above the first floor), we need to calculate the hydrostatic pressure at that height.
The density of water is approximately 1000 kg/m³, and the acceleration due to gravity is approximately 9.8 m/s².
Using the equation, we have:
P = P₀ + ρgh
P = 1.70 x 10^5 Pa + (1000 kg/m³)(9.8 m/s²)(6.5 m)
P ≈ 1.70 x 10^5 Pa + 63,700 Pa
P ≈ 2.34 x 10^5 Pa
Therefore, the gauge pressure at the faucet on the second floor is approximately 2.34 x 10^5 Pa.
(b) To determine the maximum height at which water will stop flowing from a faucet, we need to consider when the hydrostatic pressure will drop to zero, which means the gauge pressure will also drop to zero.
Setting P = 0 in the equation, we have:
0 = 1.70 x 10^5 Pa + (1000 kg/m³)(9.8 m/s²)h
h = -1.70 x 10^5 Pa / [(1000 kg/m³)(9.8 m/s²)]
h ≈ -1.70 x 10^5 m³ / [9.8]
h ≈ -1.73 x 10^4 m
The negative sign indicates that the height must be negative for the gauge pressure to be zero. Therefore, the maximum height at which water will stop flowing from a faucet is approximately 17,300 m above the first floor.
Note: In reality, there may be other factors such as pipe friction and flow rate that affect the actual height to which water can be supplied.
To answer these questions, we need to understand the concept of gauge pressure and how it relates to the height of the water column.
(a) To find the gauge pressure at the faucet on the second floor, we need to consider the difference in height between the first and second floors. The gauge pressure at a certain point is given by the formula:
P = ρgh
where P is the gauge pressure, ρ is the density of the fluid (water), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height difference between the reference point (in this case, the first floor) and the point where the gauge pressure is to be calculated (in this case, the second floor).
Given:
P1 = 1.70 * 10^5 Pa (gauge pressure at the first floor)
h = 6.5 m (height difference between the first and second floors)
Since the gauge pressure at the first floor is 1.70 * 10^5 Pa, we can use this as the reference point and calculate the gauge pressure at the second floor by considering the height difference:
P2 = P1 + ρgh
To find ρgh, we need the density of water. At room temperature, the density of water is approximately 1000 kg/m^3.
ρgh = (1000 kg/m^3) * (9.8 m/s^2) * (6.5 m)
Substituting this value into the previous equation, we can calculate the gauge pressure at the second floor:
P2 = 1.70 * 10^5 Pa + (1000 kg/m^3) * (9.8 m/s^2) * (6.5 m)
(b) To determine the maximum height at which water can flow from a faucet even if it is open, we need to consider the maximum height difference that can be supported by the pressure of the water. If the gauge pressure drops to zero, no water will flow.
Using a similar approach as before, we need to determine the height at which the gauge pressure is reduced to zero.
P = ρgh
At this maximum height, the gauge pressure is zero, so the equation becomes:
0 = (1000 kg/m^3) * (9.8 m/s^2) * h_max
Solving for h_max:
h_max = 0
This means that the maximum height from which water can flow is zero. In other words, the faucet needs to be at the same level as the surface of the water source or lower in order for water to flow.
I hope this helps! Let me know if you have any further questions.
(a) Subtract (density)*g*h
from the pressure in the floor below.
h = 6.5 m
density = 1000 kg/m^2
g = 9.8 m/s^2
(b) The pressure at a height H above the first floor is
1.70*10^5 N/m^2 - (density)*g*H
Set (density)*g*H = 1.70*10^5 N/m^2
and solve for H