Question

A liquid is poured into a vessel at rest with the hole in a wall closed by a valve. It is filled to height \( H \). The distance of the hole from the top surface is \( h \). What is the horizontal acceleration required to move the vessel so that the liquid does not come out when the valve is opened (given \( l = \) length of the base)?

• A. \( 2gh \)
• B. \( g \)
• C. \( \frac{1}{gH} \)
• D. \( \frac{2gh}{l} \)

Hint:

When dealing with a liquid in a moving container, the horizontal acceleration is related to the height of the liquid and the dimensions of the container. Ensure that the liquid’s surface is not disturbed by the movement.

Answer 1

The Correct Option is D

The liquid will not come out of the hole when the horizontal acceleration of the vessel causes the liquid to remain stationary relative to the vessel.
This condition can be met if the effective acceleration due to gravity in the horizontal direction is balanced by the force due to the liquid column's height. The horizontal acceleration required can be found using the following relation derived from the balance of forces: \[ a_{\text{horizontal}} = \frac{2gh}{l} \]
Thus, the correct answer is (d).