Question
A small block with mass m is placed inside an inverted cone this is rotating about a vertical axis such that the time for one revolution of the cone is T. The walls of the cone make an angle beta with the vertical. The coefficient of stat friction between the block and the cone is mu s. If the block is to remain at a constant height h above the apex of the cone, what are the maximum and minimum values of T? Answer must be in terms of h, beta, mu s, and g.
Solution is below, I need to know how to get there.
A small block with mass m is placed inside an inverted cone this is rotating about a vertical axis such that the time for one revolution of the cone is T. The walls of the cone make an angle beta with the vertical. The coefficient of stat friction between the block and the cone is mu s. If the block is to remain at a constant height h above the apex of the cone, what are the maximum and minimum values of T? Answer must be in terms of h, beta, mu s, and g. Solution is below, I need to know how to get there.
Explanation / Answer
the block is at the height of h. so that radius of the circular motion. r=h*tan0. centripetal force. F=mw^2*r=mw^2*h*tan0. gravity force P=mg. normal force. N=mg*sin0+F*cos0. "displace-tend" force. mg*cos0-F*sin0. we have that [mgcos0-Fsin0]=(mgsin0+Fcos0)*k. if mgcos0>Fsin0. so mg(cos0-sin0*k)=F*(sin0+cos0*k). so that mg*(cos0-sin0*k)/(sin0+cos0*k)=mw^2*h*tan0. so w^2=(g/htan0)*(cos0-sin0*k)/(sin0+cos0*k) from this you have the first T then assume mgcos0