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Review. The tank in Figure P15.13 is filled with water of depth d. At the bottom of one sidewall is a rectangular hatch of height h and width w that is hinged at the top of the hatch. (a) Determine the magnitude of the force the water exerts on the hatch. (b) Find the magnitude of the torque exerted by die water about die hinges.
(a)
The magnitude of the force the water exerts on the hatch.
Answer to Problem 14P
The magnitude of the force the water exerts on the hatch is
Explanation of Solution
The air outside and the water inside the tank exert atmospheric pressure so that only excess water pressure counts for the total force.
The diagram is shown below.
Consider a strip of hatch at a distance
Write the equation for the pressure due to the water at depth
Here,
Write the equation for the pressure in terms of force.
Here,
Rewrite the above equation for
Use the above equation to write the expression for the force exerted on the considered strip.
Here,
Write the equation for
Here,
Put equations (I) and (III) in equation (II).
Write the equation for the total force.
Here,
Put equation (IV) in the above equation and integrate it.
Conclusion:
Simplify equation (V) to find
Therefore, the magnitude of the force the water exerts on the hatch is
(b)
The magnitude of the torque exerted by the water about the hinges.
Answer to Problem 14P
The magnitude of the torque exerted by the water about the hinges is
Explanation of Solution
Write the equation for the total torque.
Here,
The lever arm of the force
Refer to the diagram and write the equation for
Put the above equation in equation (VI).
Put equation (IV) in the above equation and rearrange it.
Find the value of the above integral.
Conclusion:
Simplify equation (VII) to find
Therefore, the magnitude of the torque exerted by the water about the hinges is
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