The surface of water in a water tank of cross section area \(750 \, \text{cm}^2\) on the top of a house is \(h \, \text{m}\) above the tap level. The speed of water coming out through the tap of cross section area \(500 \, \text{mm}^2\) is \(30 \, \text{cm/s}\). At that instant, \(\frac{dh}{dt}\) is \(x \times 10^{-3} \, \text{m/s}\). The value of \(x\) will be _________.
Show Hint
Remember the principle of continuity for fluid flow. Pay close attention to units and signs. The rate of decrease in height is equal in magnitude to the velocity at the top surface.
Correct Answer: 2
Approach Solution - 1
Step 1: Apply the Principle of Continuity
According to the principle of continuity, the volume flow rate is constant:
\[ A_1V_1 = A_2V_2 \]
where \(A_1\) and \(V_1\) are the area and velocity at the top surface of the water, and \(A_2\) and \(V_2\) are the area and velocity at the tap.
Step 2: Convert Units and Substitute Values
Given \(A_1 = 750 \, \text{cm}^2 = 750 \times 10^{-4} \, \text{m}^2\), \(A_2 = 500 \, \text{mm}^2 = 500 \times 10^{-6} \, \text{m}^2\), and \(V_2 = 30 \, \text{cm/s} = 0.3 \, \text{m/s}\):
\[ 750 \times 10^{-4} V_1 = 500 \times 10^{-6} \times 0.3 \]
\[ V_1 = \frac{500 \times 0.3 \times 10^{-6}}{750 \times 10^{-4}} = \frac{150 \times 10^{-6}}{750 \times 10^{-4}} = 2 \times 10^{-4} \, \text{m/s}. \]
Step 3: Relate \(V_1\) to \(\frac{dh}{dt}\)
The rate of change of height (\(\frac{dh}{dt}\)) of the water in the tank is equal to the velocity of the water at the top surface but with a negative sign, because the height is decreasing:
\[ \frac{dh}{dt} = -V_1 = -2 \times 10^{-3} \, \text{m/s}. \]
Step 4: Find the Value of \(x\)
Given \(\frac{dh}{dt} = x \times 10^{-3} \, \text{m/s}\), and we have found \(\frac{dh}{dt} = -2 \times 10^{-3} \, \text{m/s}\). Therefore, \(x = -2\). Since the magnitude is asked, take the absolute value:
\[ |x| = 2. \]
Conclusion: The value of \(x\) is 2.
Approach Solution -2
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Concepts Used:
Mechanical Properties of Fluid
The science of the mechanical properties of fluids is called Hydrostatics. A fluid is a substance that relents to the slightest pressure. Fluids are categorized into two classes famed by the names of liquids, and elastic fluids or gases, which later comprehend the air of the atmosphere and all the different kinds of air with which chemistry makes us acquainted.
Streamline Flow:
A streamline is a curve the tangent to which at any point provides the direction of the fluid velocity at that point. It is comparable to a line of force in an electric or magnetic field. In steady flow, the pattern of the streamline is motionless or static with time, and therefore, a streamline provides the actual path of a fluid particle.
Tube of Flow:
A tubular region of fluid enclosed by a boundary comprises streamlines is called a tube of flow. Fluid can never cross the boundaries of a tube of flow and therefore, a tube of flow acts as a pipe of the same shape.
Surface Tension and Viscosity:
The surface tension of a liquid is all the time a function of the solid or fluid with which the liquid is in contact. If a value for surface tension is provided in a table for oil, water, mercury, or whatever, and the contacting fluid is unspecified, it is safe to consider that the contacting fluid is air.