Question

A round cone with half-angle α= 30° and the radius of the base R = 5.0 cm rolls uniformly and without slipping over a horizontal plane as shown in Fig. 1.8. The cone apex is hinged at the point O which is on the same level with the point C, the cone base centre. The velocity of point C is v = 10.0 cm/s. Find the moduli of 

(a) the vector of the angular velocity of the cone and the angle it forms with the vertical;

(b) the vector of the angular acceleration of the cone.

Answer 1

Let the axis of the cone (OC) rotates in anticlockwise sense with constant angular velocity vector ω and the cone itself about it's own axis (OC)  in clockwise sense with angular velocity vector ω_o(Fig.). Then the resultant angular velocity of the cone.

The vector ω_o which rotates about the OO' axis with the angular velocity vector ω, retains i magnitude. This increment in the time interval dt is equal to