quiquenobi2v6

2021-12-15

An open tank has a vertical partition and on one side contains gasoline with a density $p=700\frac{kg}{{m}^{3}}$ at a depth of 4m. Arectangular gate that is 4 m high and 2 m wide and hinged at one end is located in the partition. Water is slowly added to the empty side of the tank. At what depth, h, will the gate start to open?

esfloravaou

Beginner2021-12-16Added 43 answers

The depth of water possible that will cause for the gate to open can be solved by summing up the forces acting on
the wall to the hinge.

The forces acting on the wall are the weight and the hydrostatic forces cause by the water and the gasoline.

Use$\gamma$ for water which is equal to $9.80\frac{kN}{{m}^{3}}$ trom Table 1.5.

The specific weight of the gasoline can be solved by multiplying its density by the acceleration due to gravity which is equal to$9.81\frac{m}{{s}^{2}}$

${\gamma}_{gas}=p\left\{gas\right\}\times g$

$=700\frac{kg}{{m}^{3}}\times 9.81\frac{m}{{s}^{2}}$

$=6867\frac{N}{{m}^{3}}$

$=6.87\frac{kN}{{m}^{3}}$

The hydrostatic force on the gate can be solved using the formula${F}_{R}=\gamma {h}_{c}A$ where $\gamma$ is the specific weight of liquid, $h}_{c$ is the centroid of the gate with liquid and A is the area of the gate with liquid.

Solve for the hydrostatic force exerted by the gasoline.

${F}_{R1}={\gamma}_{gas}{h}_{c}A$

$=6.87\frac{kN}{{m}^{3}}\left(\frac{4m}{2}\right)(4m\times 2m)$

$=109.92kN$

Solve for the hydrostatic force exerted by the water.

${F}_{R2}={\gamma}_{water}{h}_{c}A$

$=9.80\frac{kN}{{m}^{3}}\left(\frac{h}{2}\right)(h\times 2m)$

Sum up the moment of the two hydrostatic forces.

Note that the location of the hydrostatic force for rectangular plane surfaces can be found$\frac{1}{3}$ of the height of the liquid from the base.

${F}_{R1}\times \frac{1}{3}\cdot 4m={F}_{R2}\times \frac{1}{3}\cdot h$

$=109.92kN\times \frac{1}{3}\cdot 4m=9.80{h}^{2}\frac{kN}{{m}^{2}}\times \frac{1}{3}\cdot h$

$146.56kN-m=3.27{h}^{3}\frac{kN}{{m}^{2}}$

$\frac{146.56kN-m}{3.27\frac{kN}{{m}^{2}}}$

$44.87{m}^{3}={h}^{3}$

$\sqrt{3}\left\{44.87{m}^{3}\right\}=\sqrt{3}\left\{{h}^{3}\right\}$

$3.55m=h$

The forces acting on the wall are the weight and the hydrostatic forces cause by the water and the gasoline.

Use

The specific weight of the gasoline can be solved by multiplying its density by the acceleration due to gravity which is equal to

The hydrostatic force on the gate can be solved using the formula

Solve for the hydrostatic force exerted by the gasoline.

Solve for the hydrostatic force exerted by the water.

Sum up the moment of the two hydrostatic forces.

Note that the location of the hydrostatic force for rectangular plane surfaces can be found

intacte87

Beginner2021-12-17Added 42 answers

where g refers to gasoline..

where w refers to water...

where h is depth of the water...

For Equilibrium,

so that,

with

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