Problem Statement
In the figure, the areas of the plunger A and cylinder B are \( 38.7 \, \text{cm}^2 \) and \( 387 \, \text{cm}^2 \), respectively, and the weight of B is \( 4500 \, \text{N} \). The vessel and the connecting passages are filled with oil of specific gravity \( 0.75 \). What force \( F \) is required for equilibrium, neglecting the weight of A?
Solution
Given:
- Area of plunger A (\( A_A \)) = \( 38.7 \, \text{cm}^2 \)
- Area of cylinder B (\( A_B \)) = \( 387 \, \text{cm}^2 \)
- Weight of B (\( W_B \)) = \( 4500 \, \text{N} \)
- Specific weight of water (\( \gamma \)) = \( 9810 \, \text{N/m}^3 \)
- Specific weight of oil (\( \gamma_{\text{oil}} \)) = \( 0.75 \times 9810 = 7357.5 \, \text{N/m}^3 \)
Pressure Equation:
Using the pressure balance equation:
\( P_A + \gamma_{\text{oil}} h_{\text{oil from A to X_L}} = \frac{W_B}{A_B} \)
Substitute the values:
\( P_A + 7357.5 \times 4.8 = \frac{4500}{387 \times 10^{-4}} \)
Simplify:
\( P_A + 35316 = 116242.63 \)
\( P_A = 116242.63 – 35316 \)
Final Value:
\( P_A = 80963 \, \text{N/m}^2 \)
Force \( F \):
The force required for equilibrium is:
\( F = P_A \times A_A \)
Substitute the values:
\( F = 80963 \times 38.7 \times 10^{-4} \)
Final Value:
\( F = 313 \, \text{N} \)
Explanation
This problem uses hydrostatic principles to determine the force required for equilibrium:
- The pressure at A includes contributions from the oil column height and the pressure needed to balance the weight of B.
- The areas of the plunger and cylinder influence the pressure distribution, as seen in the equations.
- The resulting force is calculated using the pressure at A and the area of the plunger.
Physical Meaning
- Force Equilibrium: The calculated force \( F \) ensures equilibrium in the system by balancing the weight of B and the pressure contributions.
- Specific Weight: The specific weight of oil plays a critical role in determining the pressure changes due to the fluid column height.
- Hydrostatic Pressure: The pressure distribution in the system is governed by the principles of hydrostatics, accounting for fluid properties and geometry.
