Respuesta :
Answer:[tex]P_2=2.246\times 10^5 Pa[/tex]
Explanation:
Given
Flow rate [tex]Q=7200 cm^3/s[/tex]
At one Point [tex]r_1=4 cm[/tex]
[tex]A_1=\frac{\pi d_1^2}{4}[/tex]
[tex]A_1=\frac{\pi 64}{4}=16\pi cm^2[/tex]
[tex]P_1=2.4\times 10^5 Pa[/tex]
At second Point
[tex]r_2=2 cm[/tex]
[tex]A_2=\frac{\pi d_2^2}{4}[/tex]
[tex]A_2=4\pi cm^2[/tex]
Density of water [tex]\rho =10^3 kg/m^3[/tex]
As Flow is constant therefore
[tex]Q=A_1v_1=A_2v_2[/tex]
[tex]v_1=\frac{Q}{A_1}=\frac{7200}{16\pi }=143.22 cm/s\approx =1.43 m/s[/tex]
[tex]v_2=\frac{Q}{A_2}=\frac{7200}{4\pi }=572.88 cm/s\approx 5.72 m/s[/tex]
Applying Bernoulli's theorem
[tex]P_1+\frac{\rho v_1^2}{2}+\rho gh_1=P_2+\frac{\rho v_2^2}{2}+\rho gh_2[/tex]
as pipe is horizontal therefore [tex]h_1-h_2=0[/tex]
thus [tex]P_2=P_1+\frac{\rho }{2}\left [ v_1^2-v_2^2\right ][/tex]
[tex]P_2=2.40\times 10^5+\frac{10^3}{2}\left [ 1.43^2-5.72^2\right ][/tex]
[tex]P_2=2.40\times 10^5-0.1533\times 10^5[/tex]
[tex]P_2=2.246\times 10^5 Pa[/tex]
Water’s absolute pressure as it flows through second point of a golf course sprinkler system is [tex]2.246\times10^5\rm Pa[/tex].
What is continuity equation for pipe flow?
The continuity equation for pipe flow is says that the product of cross sectional area of the pipe and velocity of the water flow is constant throughout the pipe.
Let two points in pipe ([tex]P_1[/tex] ), with cross section [tex]A_1[/tex] and velocity [tex]v_1[/tex] and the point [tex]P_2[/tex] with cross section [tex]A_2[/tex] and velocity [tex]v_2[/tex]. Thus by continuity equation,
[tex]v_1A_1=v_2A_2=Q[/tex]
Q is the discharge rate.
As, flow rate is 7200 cm3/s and the radius of the pipe at point one is 4.00 cm. Thus the velocity at point one from continuity equation,
[tex]v_1=\dfrac{7200}{{\pi \times4^2}}\\v_1=1.43\rm m/s[/tex]
As, flow rate is 7200 cm3/s and the radius of the pipe at point two is 2.00 cm. Thus the velocity at point two from continuity equation,
[tex]v_2=\dfrac{7200}{{\pi \times2^2}}\\v_2=5.72\rm m/s[/tex]
As the height of both the point is same, thus from the Bernoulli theorem the pressure at point two for the same height can be given as,
[tex]p_2=p_1+\dfrac{\rho}{2}(v_1^2-v_2^2)[/tex]
As density of the water is 1000 kg/m³ and the pressure at point one is [tex]2.40\times10^5[/tex] Pa. Thus, put the values in the above equation to find the value of pressure at point 2 as,
[tex]p_2=2.4\times10^5+\dfrac{1000}{2}(1.43^2-5.72^2)\\p_2=2.246\times10^5\rm Pa[/tex]
Thus, the water’s absolute pressure as it flows through second point of a golf course sprinkler system is [tex]2.246\times10^5\rm Pa[/tex].
Learn more about the continuity equation here;
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