Q: A water clock in ancient Greek is designed as a closed vessel with a small orifice " O ". The time is determined according to the level of water in the vessel. What should be the shape of the vessel for the time scale to be uniform.
NOTE: 1. The orifice " O " is enlarged only for clarity.
2. The curve may " NOT " look like as it is shown in figure.
Find mathematical equation governing the curve " AOB ".
71My attempt:
I got the idea that for the time to uniform, the amount of water falling through the orifice should be constant with time. But since pressure is varying with the time and also the slope is changing (and also Cross Section) we need to derive the equation of a curve suitable to this. But how do i approach this mathematically is unknown to me.
1is it like y=kx4 ???
you can take water filled upto height y,where radius is x.
apply A.v= constant
and bernoulli's principle.
21yes,i also think y= kx^4
@Vivek '' I got the idea that for the time to uniform, the amount of water falling through the orifice should be constant with time.''
This is not correct. Take O as origin(figure).The velocity of water along y direction should be constant.ie dy/dt = k. The key equation is
dy* (pi)x2 = [√(2gy)]dt*A (where A is the small area of orifice O)
so dy/dt = [√(2gy)]*A(pi)x2 = k
which gives y = c x4
71Subhodip.. Now that it's correct. Please latexify your solution and elaborate more so that anyone who sees this post could understand.
