Solve the previous problem if the pulley has a moment of inertia I about its axis and the string does not slip over it.

Solve the previous problem if the pulley has a moment of inertia I abo

1.	Solve the previous problem if the pulley has a moment of inertia I about its axis and the string does not slip over it.
Answer» Solution :LET us slove the problem by ENERGY method, iniltial EXTENSION of the spring in the mean position `delta=(mg)/k` During oscilating at ny positin x below the equlibrium positon let tehh velocilty of m be v and angular velocity of the puley be OMEGA. If r is the radius of the pulley, then `v=romega` At any instant total energy =constant (For SHM) `:. 1/2 mv^2+1/2Iomega^2+1/2k[(x+delta)^2-delta^2]-mgx` =constant `rarr(1/2mv^2+(1/2)Iomega^2+(1/2)kx^2+kxd-mgx` ltbrgeconstant `rarr (1/2)mv^2+(1/2)1(v^2+r^2)+(1/2)kx^2` =constant `(delta=mg/k)` Taking derivative ofboth sides with respect of t `mv.(dv)/(dt)1/r^2v.(dv)/(dt)+kx(dx)/(dt)=0` `rarr a((m+1)/r^2)=-kx` `(:.=(dx)/(dt)and a =(dv)/(dt))` `rarr a/r=k/((m+1)/r^2)=omega^2` `rarr T=2pisqrt((m+1/r^2)/k)`