A clear plastic tube with a diameter of 12cm and 110cm high is filled to the top with mercury. (density = 13.6 kg/m^3). There are three holes, all with a radius of 2 millimeters located 15cm from the bottom, 50cm from the bottom and 80cm from the bottom.
A) What is the velocity of the mercury coming out of the bottom hole?
B) What is the velocity coming out of the bottom hole?
C) What is the pressure at the bottom of the tube?
D) What is the flow rate of mercury coming out of the middle hole?
E) How long would i take (s) to fill a cylindrical container with a diameter of 8cm and a height of 6cm with the mercury coming out of the middle hole above?
Remember to use m/s and meters.
To find the answers to the given questions, we need to use the principles of fluid mechanics, specifically Bernoulli's equation and the equation of continuity.
First, let's address question A: What is the velocity of the mercury coming out of the bottom hole?
To find the velocity, we can use Bernoulli's equation, which states that the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant along a streamline.
Since the three holes are located at different heights from the bottom, the pressure at each hole will be different. However, we can neglect the potential energy term because the tube is vertical, and the velocity of the mercury at all three holes will be almost negligible compared to the speed of sound.
The equation for Bernoulli's principle is:
P + 0.5 * ρ * v^2 = constant
Where:
P = pressure
ρ = density
v = velocity
Since the tube is completely filled with mercury, the pressure at the top of the column is atmospheric pressure, which we can assume to be 101325 Pa.
Now, let's calculate the velocity at the bottom of the tube (hole 1). We know the pressure and density, so we can rearrange Bernoulli's equation:
P_top + 0.5 * ρ * v^2 = P_bottom
Substituting the known values:
101325 Pa + 0.5 * 13600 kg/m^3 * v^2 = P_bottom
Simplifying the equation:
13600 * v^2 = P_bottom - 101325
Now, let's move on to question B: What is the velocity coming out of the bottom hole?
Since the tube is filled to the top, the pressure at the top is atmospheric pressure, and the pressure at the bottom (hole 1) is given by P_bottom - 101325 Pa. To find the velocity, we rearrange Bernoulli's equation as follows:
P_top + 0.5 * ρ * v^2 = P_out
Substituting the known values:
101325 Pa + 0.5 * 13600 kg/m^3 * v^2 = P_out
Now, let's address question C: What is the pressure at the bottom of the tube?
As mentioned earlier, the pressure at the bottom of the tube (hole 1) is given by P_bottom - 101325 Pa. So:
P_bottom = P_out + 101325 Pa
Now, moving on to question D: What is the flow rate of mercury coming out of the middle hole?
To find the flow rate, we can use the equation of continuity, which states that the mass flow rate (mass per unit time) of a fluid is constant throughout a pipe with constant cross-sectional area.
The equation for the continuity equation is:
A1 * v1 = A2 * v2
Where:
A = cross-sectional area
v = velocity
We can assume that the cross-sectional area of the tube remains constant, so we have:
A1 * v1 = A_middle * v_middle
Now, to find the velocity coming out of the middle hole, we need to use Bernoulli's equation. Using the same approach as in question B, we get:
P_top + 0.5 * ρ * v^2 = P_middle
Finally, let's address question E: How long would it take to fill a cylindrical container with a diameter of 8 cm and a height of 6 cm with the mercury coming out of the middle hole?
To find the time required, we need to calculate the volume flow rate using the flow rate we found in question D. The volume flow rate (Q) is equal to the mass flow rate divided by the density (ρ). Therefore, we have:
Q = m_dot / ρ
The volume flow rate is equal to the cross-sectional area (A_middle) multiplied by the velocity coming out of the middle hole (v_middle). So:
Q = A_middle * v_middle
Knowing the density of mercury (13.6 kg/m^3) and the volume of the given container, we can find the time (t):
t = V_container / Q
Substituting the known values:
t = (π * (0.04 m)^2 * 0.06 m) / (A_middle * v_middle)
Thus, we can use these equations and principles to find the answers to the given questions.