An open rectangular tank 1.5mx1mx1.2m high is completely filled with water when at rest. Determine the volume spilled after the tank acquired a linear uniform acceleration of 0.6 m/s2 in the horizontal direction.

Problem Statement

An open rectangular tank with dimensions 1.5 m × 1 m × 1.2 m (length × width × height) is completely filled with water when at rest. Determine the volume of water spilled after the tank acquires a uniform horizontal acceleration of 0.6 m/s².

Solution

1. Given the horizontal acceleration:

   a_x = 0.6 m/s²

2. The free surface of the water will tilt so that its slope is determined by the ratio of the horizontal acceleration to gravitational acceleration. Thus,

   tanθ = a_x / g = 0.6 / 9.81 ≈ 0.0611
   θ ≈ arctan(0.0611) ≈ 3.50°

3. Over the length of the tank (1.5 m), this tilt produces a vertical difference in the water level. Let DE represent this difference:

   DE = (Length) × tanθ = 1.5 × 0.0611 ≈ 0.091 m

4. The spilled water forms a triangular prism along the edge of the tank. The area of the triangular cross‑section is given by:

   Area = 0.5 × (Base) × (Height) = 0.5 × 1.5 × 0.091 ≈ 0.06825 m²

   Since the width of the tank is 1 m, the volume spilled is:

   Volume Spilled = Area × Width ≈ 0.06825 × 1 ≈ 0.06825 m³

Explanation

When the tank accelerates horizontally, the water inside tends to remain at rest relative to an inertial frame. This causes the free surface of the water to tilt, aligning perpendicular to the net acceleration (the vector sum of gravity and the horizontal acceleration). The tilt is quantified by the angle θ, which is determined by the ratio a_x/g.

The vertical difference in the water level over the tank’s length can be calculated using the tangent of the tilt angle. The spilled water forms a triangular prism whose cross‑sectional area is half the product of the tank’s length and the vertical difference.

Physical Meaning

This problem illustrates how inertial effects due to horizontal acceleration affect the free surface of a fluid. In an accelerating tank, the free surface is no longer horizontal; it tilts so that the surface is perpendicular to the effective gravitational field (the sum of actual gravity and the inertial acceleration).

The tilt causes a redistribution of the fluid, leading to spillage from one side of the tank. Understanding these effects is essential in the design of liquid containers in vehicles and other dynamic systems, ensuring safety and preventing loss of fluid.