An open tank has a vertical partition and on one side contains gasoline with a density ρ = 700 kg/m3 at a depth of 4 m, as shown in Fig. P2.99. A rectangular 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?
The Correct Answer and Explanation is :
To solve the problem, we use the principles of hydrostatic pressure and moments to determine the depth hh at which the gate begins to open.
Hydrostatic Pressure
- Gasoline Side: The pressure on the gasoline side at any depth yy is: Pg=ρggyP_g = \rho_g g y where ρg=700 kg/m3\rho_g = 700 \, \text{kg/m}^3, g=9.81 m/s2g = 9.81 \, \text{m/s}^2, and yy is the depth. The total hydrostatic force on the gate due to gasoline is: Fg=ρgg∫04y dy=ρgg[y22]04=ρgg422=56,000 N.F_g = \rho_g g \int_0^4 y \, dy = \rho_g g \left[\frac{y^2}{2}\right]_0^4 = \rho_g g \frac{4^2}{2} = 56,000 \, \text{N}. The force acts at a depth of 23⋅4=2.67 m\frac{2}{3} \cdot 4 = 2.67 \, \text{m} from the bottom.
- Water Side: As water is added, the pressure due to water at depth hh is: Pw=ρwgyP_w = \rho_w g y where ρw=1000 kg/m3\rho_w = 1000 \, \text{kg/m}^3. The total hydrostatic force on the gate due to water is: Fw=ρwg∫0hy dy=ρwgh22.F_w = \rho_w g \int_0^h y \, dy = \rho_w g \frac{h^2}{2}. The force acts at h3\frac{h}{3} from the bottom.
Equilibrium Condition
The gate begins to open when the moments about the hinge balance. The moment due to gasoline is: Mg=Fg⋅2.67.M_g = F_g \cdot 2.67.
The moment due to water is: Mw=Fw⋅h3.M_w = F_w \cdot \frac{h}{3}.
Setting moments equal: Fg⋅2.67=Fw⋅h3.F_g \cdot 2.67 = F_w \cdot \frac{h}{3}.
Substitute FgF_g and FwF_w: (56,000)⋅2.67=(1000⋅9.81⋅h22)⋅h3.(56,000) \cdot 2.67 = (1000 \cdot 9.81 \cdot \frac{h^2}{2}) \cdot \frac{h}{3}.
Simplify: 149,520=4905⋅h36.149,520 = 4905 \cdot \frac{h^3}{6}.
Solve for h3h^3: h3=149,520⋅64905=183.h^3 = \frac{149,520 \cdot 6}{4905} = 183. h=1833≈5.73 m.h = \sqrt[3]{183} \approx 5.73 \, \text{m}.
Final Answer
The gate begins to open when the water depth reaches approximately 5.73 m.
Explanation
The problem relies on balancing hydrostatic forces and their moments about the hinge. The force from gasoline is constant, while the water force increases as h2h^2. The gate opens when the moments due to these forces are equal. This calculation shows the critical depth hh at which the equilibrium is disrupted.