A tank of liquid accelerates horizontally to the right with acceleration ax. How does the free surface of the liquid behave?

Introduction:
When a container of liquid is accelerated horizontally, the free surface tilts due to the superposition of gravitational and inertial (accelerative) fields. Recognizing the direction of tilt is a classic statics-of-fluids concept.

Given Data / Assumptions:

• Acceleration to the right with magnitude ax.
• Incompressible liquid, open to atmosphere.
• Quasi-static condition (no sloshing transients considered).

Concept / Approach:
In the accelerating frame, a pseudo-acceleration ax acts opposite to motion. The free surface becomes perpendicular to the resultant of gravity g (downward) and the pseudo-acceleration ax (to the left). This makes the surface a plane, not a curve, tilting so that it is lower at the front (direction of motion) and higher at the back.

Step-by-Step Solution:
1) Resultant field has components: horizontal = ax (leftward in the accelerating frame), vertical = g (downward).2) The free surface is orthogonal to the resultant field vector.3) Geometry shows the surface drops toward the direction of acceleration and rises opposite to it.4) Therefore, with acceleration to the right, the front (right) end falls and the back (left) end rises.

Verification / Alternative check:
The slope of the free surface satisfies tan(theta) = ax / g, where theta is the inclination relative to the horizontal. Positive ax yields a downward slope in the forward direction, confirming the behavior described.

Why Other Options Are Wrong:

• Remains horizontal: Only true for zero horizontal acceleration.
• Becomes curved: The surface is planar under uniform acceleration, not curved.
• Falls on the back end: Reverses the correct tilt direction.

Common Pitfalls:

• Mixing inertial and accelerating frames and reversing the tilt.
• Assuming curvature due to viscosity; viscosity does not control static free-surface shape in this case.

Final Answer:
Falls on the front (right) end and rises at the back