Probability — practice question

Particle $P$ travels inside a smooth hollow cone with its vertex pointing downward, its axis vertical, and a semi-vertical angle of $45^\circ$. A light inextensible string, parallel to the cone’s surface, links $P$ to the vertex. $P$ moves at constant angular speed in a horizontal circle of radius $0.67\,\text{m}$ (see diagram). The tension in the string is equal to the weight of $P$.