Accelerating a water-filled rectangular tank horizontally What constant horizontal acceleration is required so that the free surface of water in a rectangular tank makes a 45° slope to the horizontal?

Introduction / Context:
When a tank of liquid is accelerated horizontally, the free surface tilts due to the superposition of gravitational and inertial accelerations. The angle of the surface is a basic result from hydrostatics in a non-inertial frame and is widely applied in tanker design and instrument calibration.

Given Data / Assumptions:

• Rectangular tank, partially filled with water.
• Uniform horizontal acceleration a.
• No sloshing transients (quasi-static condition); steady tilt achieved.

Concept / Approach:
In a uniformly accelerating frame, the effective body force on the fluid is the vector sum of gravity (g vertically downward) and inertial acceleration (a directed opposite to motion). The free surface becomes perpendicular to the resultant. The tangent of the free-surface angle θ to the horizontal satisfies tan θ = a / g.

Step-by-Step Solution:

Verification / Alternative check:
For θ smaller than 45°, a < g; for θ larger than 45°, a > g. The 45° case is the midline where horizontal and vertical accelerations are equal in magnitude.

Why Other Options Are Wrong:
a = g/2 would give θ ≈ 26.6°, and a = 2 g, 2.5 g, 3 g give θ much steeper than 45°. Only a = g satisfies tan θ = 1.

Common Pitfalls:
Confusing degrees and radians; forgetting that the free surface is always perpendicular to the effective gravity vector in steady acceleration.

Final Answer:
g