Question

Hydrostatic in bottle

1. pierce three holes of the same size in bottle in different heights with a soldering iron.

2. pierce three other holes which have the same size but are bigger than the step 1) holes next to them at the same heights

3. Cover the hole with tape and fill it with water.

4. Observe each stream that comes out when a blocked hole is opened.

In that case, there are difference of the theoretical value if hydrostatic is assumed and observed.

Why error did happen? Do the sizes of the hole relate with error? Show the relationship between the hole size and error.

Answer 1

This is the 1st case. As we all know if we have water upto 'H' height in a bottle and hole at height 'h'. Then velocity of efflux

V= (2g(H-h))^1/2

Also time of flight,

T= (2h/g)^1/2

And for getting range we can multiply velocity with time

R= 2(h(H-h))^1/2

Here we are considering bernoullis law. According to that we can say pressure will be same at both surface and at hole(atmospheric pressure ). But we can also find a error with this conclusion also. Since it is accelerating. It can be considered as the 1st source of error.

Now in second condition we are making big holes compared to 1st set up and at same height( horizontally )

Now we can see the size of hole effecting the range of water flow. At first glance we might wordered at this error. By looking at the derived equation. But the truth is even a simple change in the structure of hole effects the Velocity of efflux, time of flight and finally the range.  

Flow rate= volume lost/ time lost

If the hole is big the volume of water through the hole will be more. So it effect the flow rate and make changes in range. That is the error.

For more clarification,

Error in flow rate= ( Time error+ volume error) x( flow rate)

Also, Error = 2 l * area/ circumference

We are widely using this feature for large scale industrial applications.