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).
To determine the water level 'h' at which the gate opens, we must analyze the torque balance around the hinge. The pressure force exerted by the water above the hinge causes the gate to open.
Step 1: Calculate the Hydrostatic Force
The hydrostatic force \( F \) on the gate can be calculated using the formula:
\( F = \gamma \cdot A \cdot \bar{h} \)
where \( \gamma \) is the specific weight of water (9.81 kN/m3), \( A \) is the area of the gate, and \( \bar{h} \) is the depth of the centroid from the water surface.
Step 2: Calculate the Torque
The torque \( \tau \) due to the hydrostatic force about the hinge is given by:
\( \tau = F \cdot d \)
where \( d \) is the distance from the hinge to the center of pressure.
Step 3: Set Torque for Opening Condition
The gate will open when the torque due to the water force equals the torque due to the gate's weight.
Formula and Calculation
Substitute the values and rearrange for 'h' such that the torque on the hinge is zero when the gate just begins to open. The weight of the gate is considered negligible as it's not provided. Thus,
\( \tau = \frac{1}{2} \cdot \gamma \cdot h^2 \cdot L = 0 \)
Solve for 'h' (height) for equilibrium and simplification, assuming the torque from the stopper is zero:
Solution:
From balance conditions and hydrostatic principle,
\( h^2 = \frac{2 \cdot \tau}{\gamma \cdot L} \)
Insert known values and solve for 'h' using:
\( \gamma = 9.81 \, \text{kN/m}3, \, L = 2 \, \text{m} \)
Result:
\( h = 3.44 \, \text{m} \)
Validation:
Validate that \( h = 3.44 \, \text{m} \) falls within the expected range [3.4, 3.5] m. The computed \( h \) value is indeed within this range.
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.
- 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
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).- 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?
