consider a cylinder of radius R. At its bottom, there is a hole of radius r. the cylinder is filled upto the height h and the hole is opened. If t is the time in which the cylinder is emptied, then find the relation between t and h.

Explanation

let at an instant, x be the height of the liquied column in the cylinder. The velocity of efflux is,
`v=sqrt(2 g x)`
let dx be the fall in height of the liquid column in the cylinder in time dt. Therefore, volume of the liquid flowing out per unit time is
`V= pi R^(2) xx (dx)/(dt) = upsilon xx pi r^(2)`
or `pi R^(2) (dx)/(dt) = sqrt(2gx) xx pi r^(2)`
or `(R^2)/(r^2) (dx)/(sqrt(2gx)) =dt`
Integrating it within the limits h to 0, we have
`(R^2)/(r^(2)sqrt(2gh)) int_(h)^(0) (dx)/(sqrt(x)) = int_(0)^(t) dt`
or `(R^2)/(r^(2)sqrt(2g)) xx 2sqrt(h) = t or t prop sqrt(h)`.