# Square

Time of emptying a square, rectangular or circular tank through an orifice at its bottom

### Key Facts

**Gyroscopic Couple**: The rate of change of angular momentum () = (In the limit).

- = Moment of Inertia.
- = Angular velocity
- = Angular velocity of precession.

**Blaise Pascal**(1623-1662) was a French mathematician, physicist, inventor, writer and Catholic philosopher.

**Leonhard Euler**(1707-1783) was a pioneering Swiss mathematician and physicist.

## Overview

Consider a square, rectangular or circular tank of uniform cross-sectional area, containing some liquid and having an orifice at its bottom. Let,- A = Surface area of the tank
- = Initial height of the liquid
- = Final height of the liquid
- a = Area of the orifice

*h*above the orifice. We know that the theoretical velocity of the liquid at this instant, After a small interval of time

*dt*, let the liquid level fall down by the amount

*dh*. Therefore volume of the liquid that has passed in time

*dt*,

The value of

We know that the volume of liquid that has passed through the orifice in time *dh*is taken as negative, as its value will decrease with the increase in discharge.*dt*, = Coefficient of discharge Area Theoretical velocity Time Equating equations (2) and (3) Now the total time

*T*required to bring the liquid level from to may be found out by integrating the equation (4) between the limits to i.e., Taking minus out from the bracket (as is greater than )

If the tank is to be completely emptied, then putting = 0 in this equation, we get

Example:

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##### Example - Time of emptying circular tank through an orifice at its bottom

Problem

A circular water tank of 4m diameter contains 5m deep water. An orifice of 400mm diameter is provided at its bottom. Find the time taken for water level fall from 5m to 2m. Take = 0.6

Workings

Given,

- Diameter of circular tank, = 4m
- Diameter of orifice, = 400mm = 0.4m
- = 5m
- = 2m
- = 0.6

Solution

Time taken to fall the water level = 61.9 s