Answer:
The frequency
= 521.59 Hz
The rate at which the frequency is changing = 186.9 Hz/s
Explanation:
Given that :
Diameter of the tank = 44 cm
Radius of the tank =
=
= 22 cm
Diameter of the spigot = 3.0 mm
Radius of the spigot =
=
= 1.5 mm
Diameter of the cylinder = 2.0 cm
Radius of the cylinder =
=
= 1.0 cm
Height of the cylinder = 40 cm = 0.40 m
The height of the water in the tank from the spigot = 35 cm = 0.35 m
Velocity at the top of the tank = 0 m/s
From the question given, we need to consider that the question talks about movement of fluid through an open-closed pipe; as such it obeys Bernoulli's Equation and the constant discharge condition.
The expression for Bernoulli's Equation is as follows:
![P_1+\frac{1}{2}pv_1^2+pgy_1=P_2+\frac{1}{2}pv^2_2+pgy_2](https://tex.z-dn.net/?f=P_1%2B%5Cfrac%7B1%7D%7B2%7Dpv_1%5E2%2Bpgy_1%3DP_2%2B%5Cfrac%7B1%7D%7B2%7Dpv%5E2_2%2Bpgy_2)
![pgy_1=\frac{1}{2}pv^2_2 +pgy_2](https://tex.z-dn.net/?f=pgy_1%3D%5Cfrac%7B1%7D%7B2%7Dpv%5E2_2%20%2Bpgy_2)
![v_2=\sqrt{2g(y_1-y_2)}](https://tex.z-dn.net/?f=v_2%3D%5Csqrt%7B2g%28y_1-y_2%29%7D)
where;
P₁ and P₂ = initial and final pressure.
v₁ and v₂ = initial and final fluid velocity
y₁ and y₂ = initial and final height
p = density
g = acceleration due to gravity
So, from our given parameters; let's replace
v₁ = 0 m/s ; y₁ = 0.35 m ; y₂ = 0 m ; g = 9.8 m/s²
∴ we have:
v₂ = ![\sqrt{2*9.8*(0.35-0)}](https://tex.z-dn.net/?f=%5Csqrt%7B2%2A9.8%2A%280.35-0%29%7D)
v₂ = ![\sqrt {6.86}](https://tex.z-dn.net/?f=%5Csqrt%20%7B6.86%7D)
v₂ = 2.61916
v₂ ≅ 2.62 m/s
Similarly, using the expression of the continuity for water flowing through the spigot into the cylinder; we have:
v₂A₂ = v₃A₃
v₂r₂² = v₃r₃²
where;
v₂r₂ = velocity of the fluid and radius at the spigot
v₃r₃ = velocity of the fluid and radius at the cylinder
![v_3 = \frac{v_2r_2^2}{v_3^2}](https://tex.z-dn.net/?f=v_3%20%3D%20%5Cfrac%7Bv_2r_2%5E2%7D%7Bv_3%5E2%7D)
where;
v₂ = 2.62 m/s
r₂ = 1.5 mm
r₃ = 1.0 cm
we have;
v₃ =
![(\frac{1.5mm^2}{1.0mm^2} )](https://tex.z-dn.net/?f=%28%5Cfrac%7B1.5mm%5E2%7D%7B1.0mm%5E2%7D%20%29)
v₃ = 0.0589 m/s
∴ velocity of the fluid in the cylinder = 0.0589 m/s
So, in an open-closed system we are dealing with; the frequency can be calculated by using the expression;
![f=\frac{v_s}{4(h-v_3t)}](https://tex.z-dn.net/?f=f%3D%5Cfrac%7Bv_s%7D%7B4%28h-v_3t%29%7D)
where;
= velocity of sound
h = height of the fluid
v₃ = velocity of the fluid in the cylinder
![f=\frac{343}{4(0.40-(0.0589)(0.4)}](https://tex.z-dn.net/?f=f%3D%5Cfrac%7B343%7D%7B4%280.40-%280.0589%29%280.4%29%7D)
![f= \frac{343}{0.6576}](https://tex.z-dn.net/?f=f%3D%20%5Cfrac%7B343%7D%7B0.6576%7D)
= 521.59 Hz
∴ The frequency
= 521.59 Hz
b)
What are the rate at which the frequency is changing (Hz/s) when the cylinder has been filling for 4.0 s?
The rate at which the frequency is changing is related to the function of time (t) and as such:
![\frac{df}{dt}= \frac{d}{dt}(\frac{v_s}{4}(h-v_3t)^{-1})](https://tex.z-dn.net/?f=%5Cfrac%7Bdf%7D%7Bdt%7D%3D%20%5Cfrac%7Bd%7D%7Bdt%7D%28%5Cfrac%7Bv_s%7D%7B4%7D%28h-v_3t%29%5E%7B-1%7D%29)
![\frac{df}{dt}= -\frac{v_s}{4}(h-v_3t)^2(-v_3)](https://tex.z-dn.net/?f=%5Cfrac%7Bdf%7D%7Bdt%7D%3D%20-%5Cfrac%7Bv_s%7D%7B4%7D%28h-v_3t%29%5E2%28-v_3%29)
![\frac{df}{dt}= \frac{v_sv_3}{4(h-v_3t)^2}](https://tex.z-dn.net/?f=%5Cfrac%7Bdf%7D%7Bdt%7D%3D%20%5Cfrac%7Bv_sv_3%7D%7B4%28h-v_3t%29%5E2%7D)
where;
(velocity of sound) = 343 m/s
v₃ (velocity of the fluid in the cylinder) = 0.0589 m/s
h (height of the cylinder) = 0.40 m
t (time) = 4.0 s
Substituting our values; we have ;
![\frac{df}{dt}= \frac{343*0.0589}{4(0.4-(0.0589*4.0))^2}](https://tex.z-dn.net/?f=%5Cfrac%7Bdf%7D%7Bdt%7D%3D%20%5Cfrac%7B343%2A0.0589%7D%7B4%280.4-%280.0589%2A4.0%29%29%5E2%7D)
= 186.873
≅ 186.9 Hz/s
∴ The rate at which the frequency is changing = 186.9 Hz/s when the cylinder has been filling for 4.0 s.