Q.The free surface of oil in a tanker, at rest, is horizontal. If the tanker starts accelerating the free surface will be titled by an angle $\theta$. If the acceleration is $a$ m s$^{-2}$, what will be the slope of the free surface?.

The key idea here is the relative equilibrium of a fluid in an accelerating frame. The free surface of a liquid in an accelerating container will orient itself perpendicular to the direction of the effective gravitational field.

1. When the tanker accelerates horizontally with acceleration $a$, every fluid particle experiences a pseudo force $ma$ in the direction opposite to the acceleration.
2. Each fluid particle is also subject to its weight $mg$ acting vertically downwards.
3. The net effective force on a fluid particle is the vector sum of its weight $mg$ and the pseudo force $ma$. The free surface of the liquid will be perpendicular to this resultant effective force. …

When a liquid is at rest in a container, its free surface is always horizontal. This is because the liquid surface adjusts itself to be perpendicular to the local gravitational field. In other words, the net force acting on any particle on the surface, due to gravity and pressure, is perpendicular to the surface.

When the tanker accelerates, the liquid inside also accelerates. To analyze the forces on the liquid from the perspective of an observer inside the accelerating tanker (a non-inertial frame of reference), we must introduce a pseudo force. This pseudo force acts on every particle of the liquid, in a direction opposite to the tanker's acceleration. The free surface of the liquid will then orient itself perpendicular to the resultant of the actual gravitational force and this pseudo force. This resultant force effectively acts as a new, tilted gravitational field.

Here's how we determine the slope:

1. Choose the Frame of Reference:

   We analyze the situation from the non-inertial frame of reference of the accelerating tanker. This means we consider the tanker to be at rest, but we must account for the acceleration by introducing a pseudo force.

2. Identify Forces on a Fluid Particle:

   Consider a small mass $m$ of oil on the free surface.

   • Gravitational Force ($F_g$): This acts vertically downwards, with magnitude $mg$.
   • Pseudo Force ($F_p$): Since the tanker is accelerating horizontally with acceleration $a$, a pseudo force acts on the mass $m$ in the opposite direction to the acceleration. Its magnitude is $ma$, and it acts horizontally.

3. Determine the Resultant Force (Effective Gravity):

   The gravitational force $F_g$ and the pseudo force $F_p$ are perpendicular to each other. Their vector sum gives the net force acting on the particle in this non-inertial frame. Let's call this resultant force $F_{net}$.

This resultant force acts as the "effective gravity" in the accelerating frame.

4. Orientation of the Free Surface:

The free surface of the liquid will always be perpendicular to the direction of the effective gravity. If the effective gravity makes an angle $\theta$ with the vertical, then the free surface will make the same angle $\theta$ with the horizontal.

Imagine a right-angled triangle formed by the force vectors:
*   The vertical side represents $mg$.
*   The horizontal side represents $ma$.
*   The hypotenuse represents $F_{net}$. …

In tanker's accelerating frame, effective force = real weight mg (down) + pseudo-force ma (horizontal, opposite acceleration). Liquid surface settles perpendicular to t …