A rectangular tank 2m long, 1.5m wide and 1.5m deep is filled with oil of specific gravity 0.8. Find the force acting on the bottom of the tank when (a) the vertical acceleration 5m/s2 acts upwards (b) the vertical acceleration 5m/s2 acts downwards.

Problem Statement

A rectangular tank measuring 2 m in length, 1.5 m in width, and 1.5 m in depth is filled with oil of specific gravity 0.8. Determine the force acting on the bottom of the tank when:

• (a) A vertical acceleration of 5 m/s² acts upwards.
• (b) A vertical acceleration of 5 m/s² acts downwards.

Solution

1. The weight density of water is 9810 N/m³. For oil with specific gravity 0.8, the weight density is:

   γ_oil = 0.8 × 9810 = 7848 N/m³

2. The oil depth is h = 1.5 m. The hydrostatic pressure on the bottom under a vertical acceleration a_z is modified as:

   P = γ_oil h (1 + a_z/g)

3. (a) For a vertical acceleration of 5 m/s² acting upwards (a_z = 5 m/s²):

   P_A = 7848 × 1.5 × (1 + 5/9.81) ≈ 17772 N/m²

   The bottom area of the tank is:

   Area = Length × Width = 2 m × 1.5 m = 3 m²

   Therefore, the force acting on the bottom is:

   F_AB = P_A × Area = 17772 × 3 ≈ 53316 N

4. (b) For a vertical acceleration of 5 m/s² acting downwards (a_z = -5 m/s²):

   P_A = 7848 × 1.5 × (1 – 5/9.81) ≈ 5772 N/m²

   Thus, the force on the bottom is:

   F_AB = P_A × Area = 5772 × 3 ≈ 17316 N

Explanation

The pressure on the tank bottom is governed by the hydrostatic pressure of the oil, which is modified by the vertical acceleration. When the acceleration is upward, the effective gravitational force increases (g becomes g + 5 m/s²), resulting in higher pressure. Conversely, when the acceleration is downward, the effective gravity decreases (g becomes g – 5 m/s²), leading to lower pressure.

Physical Meaning

This problem illustrates the impact of vertical acceleration on fluid pressure within a container. The effective pressure is not only a function of the fluid’s depth but is also influenced by the acceleration acting on the system.

In practical terms, when the container accelerates upwards, the oil experiences an increased effective gravitational force, thereby increasing the pressure on the bottom. Conversely, when the container accelerates downwards, the effective gravitational force is reduced, decreasing the pressure.

Such analyses are vital in engineering applications where the dynamic behavior of fluids under acceleration needs to be understood, such as in transportation, process engineering, and safety assessments.