Question

A tank has a gate that automatically opens if the water levelhis high enough. The gate has a squarecross section of side1m and can rotate about an axis going through the point O. Calculate the minimumwater level that would trigger the opening of the floodgate.

Answer 1

The gate will open if the height of water is equal to or more than 0.337m.

Step-by-step explanation:

From the diagram attached, (as seen from the reference question found on google)

The forces are given as

Force on OA

`F_1=P A_1\\F_1=\rho g \bar{h} A_(OA)`

Here

• ρ is the density of water.
• g is the gravitational acceleration constant
• 
  `\bar{h}` is the equivalent height given as

`\bar{h}=h+(0.6)/(2)\\\bar{h}=h+0.3`

• 
  `A_(OA)` is the area of the OA part of the door which is calculated as follows:

`A_(OA)=L* W\\A_(OA)=1* 0.6\\A_(OA)=0.6 m^2`

The Force is given as

`F_1=0.6\rho g[h+0.3]`

Force on OB

`F_2=P A_2\\F_2=\rho g \bar{h} A_(OB)`

Here

• ρ is the density of water.
• g is the gravitational acceleration constant
• 
  `\bar{h}` is the equivalent height given as

`\bar{h}=h+0.6+(0.4)/(2)\\\bar{h}=h+0.8`

• 
  `A_(OA)` is the area of the OB part of the door which is calculated as follows:

`A_(OB)=L* W\\A_(OB)=1* 0.4\\A_(OB)=0.4 m^2`

The Force is given as

`F_2=0.4\rho g[h+0.8]`

Now the moment arms are given as

`\bar{y}_a=\bar{h}+\frac{I}{A\bar{h}}\\\bar{y}_a=h+0.3+((1)/(12)* 0.6^3 * 1)/(0.6 *[h+0.3])\\\bar{y}_a=h+0.3+(0.03)/(h+0.3)`

`\bar{y}_b=\bar{h}+\frac{I}{A\bar{h}}\\\bar{y}_b=h+0.8+((1)/(12)* 0.4^3 * 1)/(0.4 *[h+0.8])\\\bar{y}_b=h+0.8+(0.0133)/(h+0.8)`

Taking moment about the point O as zero

`F_1(h+0.6-\bar{y}_a)=F_2(\bar{y}_b-h+0.6)\\F_1(h+0.6-h-0.3-(0.03)/(h+0.3))=F_2(h+0.8+(0.0133)/(h+0.8)-h-0.6)\\F_1(0.3-(0.03)/(h+0.3))=F_2(0.2+(0.0133)/(h+0.8))\\0.6\rho g[h+0.3](0.3-(0.03)/(h+0.3))=0.4\rho g[h+0.8](0.2+(0.0133)/(h+0.8))\\0.6[h+0.3](0.3-(0.03)/(h+0.3))=0.4[h+0.8](0.2+(0.0133)/(h+0.8))\\0.18h -0.054-0.018=0.08h+0.064+0.00533\\h=0.337 m`

So the gate will open if the height of water is equal to or more than 0.337m.