Answer:
the gauge pressure of the water on the second floor is Pg2 = 403765 Pa = 4.03 * 10⁵ Pa
Explanation:
According to Bernoulli's equation ( conservation of mechanical energy, therefore we neglect the friction inside the pipes)
P1 + ρ*g*h1 + 1/2*ρ*v1² = P2 + ρ*g*h2 + 1/2*ρ*v2²
where ρ= density of water , h=height , P= absolute pressure, g= gravity ( 9.8 m/s²)
since the gauge pressure Pg is
Pg = P - Pa (atmospheric pressure)
Pg1 +Pa + ρ*g*h1 + 1/2*ρ*v1² = Pg2 + Pa + ρ*g*h2 + 1/2*ρ*v2²
Pg1 + ρ*g*h1 + 1/2*ρ*v1² = Pg2 + ρ*g*h2 + 1/2*ρ*v2²
since our reference point to measure heights can be chosen, we choose h1=0 and h2= 4m. Thus
Pg1 + 1/2*ρ*v1² = Pg2 + ρ*g*h2 + 1/2*ρ*v2²
therefore
Pg2 = Pg1 + 1/2*ρ*v1² -ρ*g*h2 - 1/2*ρ*v2² = Pg1 -ρ*g*h2 -1/2*ρ*(v2²-v1²)
replacing values and assuming ρ=1000 kg/m³
Pg2 = Pg1 -ρ*g*h2 -1/2*ρ*(v2²-v1²) = 4.5 * 10⁵ Pa - 1000kg/m³*9.8 m/s²*4m -1/2* 1000 kg/m³ * [(4.4 m/s)² - (2.3 m/s)² ] = 403765 Pa = 4.03 * 10⁵ Pa
Pg2 =4.03 * 10⁵ Pa
