Answer to Question #315590 in Calculus for Aune

Question

A cone of radius 𝑟 centimeters and height ℎ centimeters is lowered point first at

a rate of 1 cm/s into a tall cylinder of radius 𝑅 centimeters that is partially filled with

water. How fast is the water level rising at the instant the cone is completely

submerged

Answer 1

Potential energy is converted into kinetic energy of translational and rotational motion.

$m*g*h=m*v²/2 + J*ω²/2$

moment of inertia of the cylinder

$J=m*r²/2$

Angular velocity at the end of the inclined plane

$ω=v/r\, \\ m*g*h=m*v²/2 + (m*r²/2)*(v²/(2*r²))\\ g*h=v²/2 + v²/4=(3/4)*v²\\ v=2*\sqrt{g*h/3}$

Answer: $v=2*\sqrt{g*h/3}$