1. A solid cone of angle 20, base ro, and density p. is rotating with initial angular velocity mo inside a conical seat, as shown below. The clearance h is filled with oil of viscosity u. Neglecting air drag, derive an analytical expression for the cone's angular velocity o(t) if there is no applied torque. @(t) Base radius ro Oil 20 h

3.14. Find the vorticity in polar coordinates for the following velocity com- ponents (a) v=rsin 0, ve = 2r cos 0 (b) v = cos 0, 11=0

A cube of side (a) and mass (M) is initially sitting fully submerged at the bottom of a container filled with a liquid of kinematic viscosity v and density p. The container has a square cross-section of side (a+a/5) and the cube is sitting right at the middle of the container base. (a) A force (F) starts pulling the cube up at a constant velocity (U). Develop an expression for the force in terms of (U, M. a. g, p and v). You may assume that the velocity in the gap between the cube's sides and the container walls is linear. The expression for (F) is to be valid as long as the cube remains submerged. (b) After the cube reaches the water surface, it continues to be pulled up by the same force. Develop a differential equation for the variation with time of the fraction of the cube that is submerged in water.