00:01
Hello and welcome to this video solution of numerate.
00:04
Here it is given that the density rho of a fluid varies with depth h although its bulk modulus e can be assumed to be constant.
00:13
Now you have to determine how the pressure varies with height h.
00:18
The density of the fluid at the surface is rho naught.
00:21
So what you can start by is writing the expression of bulk modulus e v let's say that is equal to negative differential pressure over dv by v right.
00:40
So this is the expression of fluid sorry this is a compressibility and you know that the volume is equal to mass over the density right.
00:54
So what you have now is dv by v that is equal to you can take the derivative of it from here.
01:10
So it's negative m over rho square d rho over m by rho right.
01:23
If you take the derivative of this equation then you will be having this expression right and this is nothing but negative d rho by rho itself fine and from here if you just replace dv by v then e v will be equal to minus d p by d rho by rho itself right.
01:50
This will be plus now because both are having negative sign.
01:54
Now at the surface where rho or what you can do is at surface we consider p to be equal to zero and rho to be equal to rho naught right.
02:10
Here you will be having if you just take the integral e v integral of d rho over rho that will be equal to d p right...