A plunger weighing 314 kN is balanced in a cylindrical vessel of diameter 1.5 m and filled with an oil (specific gravity 0.9) as shown in the following figure (not to the scale).
If a pressure gauge is connected with the vessel using 1.5 cm diameter tube, the reading of the gauge will be ________ (in kPa, rounded off to two decimal places).
Given Data
- Weight of the plunger, \( W = 314 \text{ kN} = 314,000 \text{ N} \)
- Diameter of the vessel, \( D = 1.5 \text{ m} \)
- Specific gravity of oil, \( S = 0.9 \)
- Density of water, \( \rho_{w} = 1000 \text{ kg/m}^3 \)
- Acceleration due to gravity, \( g \approx 9.81 \text{ m/s}^2 \)
- Height of the pressure gauge above the plunger base (from diagram), \( h = 1.5 \text{ m} \)
Step-by-Step Solution
1. Calculate the Area of the Plunger Base
The pressure exerted by the weight of the plunger is distributed over its circular base area \( A \):
$$ A = \frac{\pi D^2}{4} $$ $$ A = \frac{\pi \times (1.5 \text{ m})^2}{4} \approx 1.7671 \text{ m}^2 $$
2. Calculate the Pressure at the Plunger Base
The pressure directly beneath the plunger \( P_{\text{plunger}} \) is due to its weight:
$$ P_{\text{plunger}} = \frac{W}{A} $$ $$ P_{\text{plunger}} = \frac{314,000 \text{ N}}{1.7671 \text{ m}^2} \approx 177,687.65 \text{ Pa} $$ Converting to kPa: $$ P_{\text{plunger}} \approx 177.69 \text{ kPa} $$
3. Determine the Pressure at the Gauge
According to the principles of hydrostatics, the pressure decreases as we move upwards in a fluid. From the diagram, the pressure gauge is located at a height of \( h = 1.5 \text{ m} \) above the level of the plunger base. The pressure at the gauge \( P_{\text{gauge}} \) is:
$$ P_{\text{gauge}} = P_{\text{plunger}} - (\rho_{\text{oil}} \cdot g \cdot h) $$
First, find the density of the oil:
$$ \rho_{\text{oil}} = S \times \rho_{w} = 0.9 \times 1000 \text{ kg/m}^3 = 900 \text{ kg/m}^3 $$
Now, calculate the pressure drop due to the height difference:
$$ \Delta P = 900 \text{ kg/m}^3 \times 9.81 \text{ m/s}^2 \times 1.5 \text{ m} = 13,243.5 \text{ Pa} $$ $$ \Delta P = 13.24 \text{ kPa} $$
Subtracting this from the base pressure:
$$ P_{\text{gauge}} = 177.69 \text{ kPa} - 13.24 \text{ kPa} = 164.45 \text{ kPa} $$
Final Answer
The reading of the pressure gauge is approximately 164.44 kPa (accepted range: 162 to 166 kPa).
Important Questions from Hydrostatic Pressure
A cylindrical roller gate of diameter 6.0 m and length 10.0 m is placed on a dam to store water behind it, as shown in figure.
The magnitude of the resultant force due to water acting on that gate when the water is about to spill is _______ $\times 10^6$ N (rounded off to two decimal places).
Consider acceleration due to gravity ($g$) = 9.81 m/s$^2$;
density of water ($\rho$) = 1000 kg/m$^3$; and $\pi$ = 3.14.
With reference to the gate shown in the figure, the gate will start opening automatically when the water level ‘h' above the hinge is _________ m (rounded off to 2 decimal places).
- Two ends of a differential mercury manometer are connected at two points on a pipe carrying oil. The manometer shows difference in mercury level of $20$ cm. The specific gravity of oil and mercury are $0.8$ and $13.6$, respectively. The density of water is $1000$ kg m$^{-3}$ at $4$ °C and acceleration due to gravity (g) is $9.81$ m s$^{-2}$. At the same two points in pipe, the difference of pressure in N m$^{-2}$ is
- Consider the following statements regarding the pressure measured at a point in a fluid.
S1: Absolute pressure is gauge pressure plus atmospheric pressure.
S2: Gauge pressure is measured by considering absolute zero pressure as datum.
S3: Absolute pressure is measured by considering atmospheric pressure as datum.
S4: Vacuum pressure is negative pressure obtained by subtracting absolute pressure from atmospheric pressure.
Which ONE of the following options lists all the TRUE statements?
