Water is retained by a curved gate `AB`. Compute the horizontal and vertical component of the resultant of the hydrostatic pressure dsitribution on the gate `AB`, which is a quarter of a cylinder. Assume that the gate is `4m` wide.
Water is retained by a curved gate `AB`. Compute the horizontal and vertical component of the resultant of the hydrostatic pressure dsitribution on the gate `AB`, which is a quarter of a cylinder. Assume that the gate is `4m` wide.
Text Solution
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a. Horizontal component. The horizontal component of the resultant force on the gate is equal to the force on the vertical projection `BA'` of the gate. The pressure at a depth `y` from the free surface of the water is `P=rhogy`. The force on the strip of area `4dy` of the gate is ,
`dF_(H)=P(4dy)=rhogy(4dy)=4rhoydy`
Total horizontal force is
`F_(H)=4rhogint_(10)^(15) ydy=4rhog|y^2/2|_10^15`
`=4xx1000xx10((15^(2)-10^(2))/2)=250xx10^(4)N`
b. Vertical component : It is equal to the weight of the water above the curved surface. Dividing the volume above the curved surface into a rectangular parallelepiped and a quarder cylinder, we thus have
`F_(v)=` weight of water in `DEFB+` Weight of water in `ACBF`
`=rho[V_(1)+V_(2)]g=1000[10xx4xx4+(pi(5)^(2)xx4)/4]xx10`
`[8xxpi]xx10^(4)N`
After calculating the values of `F_(v)` and `F_(H)` we can calculate the resultant force on the gate `F=sqrt(F_(H)^(2)+F_(V)^(2))`
`dF_(H)=P(4dy)=rhogy(4dy)=4rhoydy`
Total horizontal force is
`F_(H)=4rhogint_(10)^(15) ydy=4rhog|y^2/2|_10^15`
`=4xx1000xx10((15^(2)-10^(2))/2)=250xx10^(4)N`
b. Vertical component : It is equal to the weight of the water above the curved surface. Dividing the volume above the curved surface into a rectangular parallelepiped and a quarder cylinder, we thus have
`F_(v)=` weight of water in `DEFB+` Weight of water in `ACBF`
`=rho[V_(1)+V_(2)]g=1000[10xx4xx4+(pi(5)^(2)xx4)/4]xx10`
`[8xxpi]xx10^(4)N`
After calculating the values of `F_(v)` and `F_(H)` we can calculate the resultant force on the gate `F=sqrt(F_(H)^(2)+F_(V)^(2))`
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