## Monday, December 29, 2014

### Problems on Bernoulli's theorem and Its Applications

Problem and solution

A rectangular vessel when full of water and it takes 10 minutes to be emptied through a small hole. If the same vessel is only half filled , calculate the time taken to empty the vessel?

Basing on the concept of Bernoulli’s Theorem it is proved that the time taken to empty the tank is the difference between the Square route of heights of the fluid filled.

We can solve one more problem basing on the same concept.

Water in a tank flowing through a hole of diameter 2 cm under a constant pressure difference of 10 cm water column. What is the rate of the flow of the water through the hole ?

Problem and solution

A hole is made at the bottom of the tank filled with water. If the total pressure at the bottom of the tank is three  tmes of atmospheric pressure what is the velocity of the efflux?

The velocity of the water with which it comes out through the hole is similar to the velocity of a freely falling body. The pressure due to 10 meter of water  is mathematically equal to one atmospheric pressure. It is proved in the following diagram.

Problem and solution

in compressible liquid flows in a horizontal tube as shown. Find the velocity of the fluid?

To explain this concept we shall use the equation of continuity. As per this concept the mass of the fluid that enters through the system is equal to the mass of the fluid that exits through the system in one second.

We have one more problem to solve in this attached paper.

An aeroplane of certain mass and certain area of cross-section can experience a certain pressure in the up thrust.

As there is no information is given in terms of velocities we have to deal it only in terms of pressure as pressure is defined as the force per unit area.

Problem and solution

square hole having a certain length is made at the depth y and a circular hole is made at a depth of 4y from the surface of the water tank. If equal amount of the water comes out of the vessel through both the holes, find the radius of the circular hole in terms of the length of the Squire hole?

This problem also can be solved basing on the law of equation of continuity. The concept is simple. The mass of the fluid that enters through one hole per second shall be equal to the mass of the fluid that enters through the other hole also.

Problem and solution

Water is moving with the speed of 5 m/s through a pipe with a cross-sectional area of 4 cm Squire. The water gradually decreased to 10 meters high it as the pipe increases the area to 8 cm Squire. If the pressure at the upper surface is given what is the pressure at the Lower surface?

We can use both equation of continuity and the Bernoulli’s Theorem to solve the problem as shown below.

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