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The question is:

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Now my question is that , why will we take the tension in the string , or in fact , the string to be making the same angle with respect to the surface as the fluid is making with the horizontal? I know the surface must make a right angle to the net acceleration experienced by it in an accelerating frame however why is it necessary for the string to be also make the same angle with the horizontal?

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    $\begingroup$ Intuitive answer: someone inside the fluid cannot tell if the surface is oblique due to gravity or to acceleration or to a combination of both, the pressure will be constant across parallel lines to the fluid surface, in the same way that in a body of water that is not accelerated. $\endgroup$
    – Pato Galmarini
    Commented Jan 25 at 19:09

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Compose (aka sum) the opposite of the acceleration $-\mathbf{a}$ with gravity $\mathbf{g}$,

\begin{equation} \tilde{\mathbf{g}} = \mathbf{g} - \mathbf{a} \end{equation}

If you need pressure distribution in the water, use Stevino's law with pressure gradient $\nabla p = \rho \tilde{\mathbf{g}}$, and remember to prescribe the atmospheric pressure $P_a$ on the free surface of water.

That's it!

You'll have a linear distribution of pressure water, whose gradient points in direction orthogonal w.r.t. the free surface, so Archimedes's force acts in that direction as well. And the "weight" felt by the ball is $m \tilde{\mathbf{g}}$

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  • $\begingroup$ So, everytime a force is acting internally in an accelerating fluid , will it always have the same angle as that made by the fluid with respect to the horizontal? $\endgroup$
    – Adhway
    Commented Jan 25 at 19:14
  • $\begingroup$ "everytime" is a dangerous word. Now I'm busy. I'll answer to this comment later, if nobody has answered before $\endgroup$
    – basics
    Commented Jan 25 at 19:19
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Imagine a simpler case where the is no sideways acceleration.

The container holds water. Without the container exerting forces, the water would accelerate downward. There is another way of thinking about it. You could put a container of water on a rocket in space. The rocket accelerates the container upward. You would not be able to tell the difference in pressure and forces on wooden balls and such.

The ball feels an "upward" force parallel to the rocket force. This originates in pressure differences in layers of water. Deeper layers have more water on them, which press harder than on shallow layers. So water presses harder on the bottom of the ball than on the top.

The surface of the water is perpendicular to the force. If one side of the surface was higher, water could flow downhill.

So the string and surface are perpendicular.

The situation is the same if you add a second engine on the side of the rocket. The total force is at an angle. The string and surface align with the total force.

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