An arrangedment illlustrated in figure consists of a horizontal uniform disc `D` of mass `m` and radius `R` and a thin rod `AO` whose torsional coefficient is equal to `k`. Find the amplitude and the energy of small torsional oscillationa if at the initial momentu the disc was deviated through an angle `varphi_(0)` from the equilibrium position and then imparted an angular velocity `varphi_(0)`.

Explanation

The K.E. of the disc is `(1)/(2)Idot(varphi)^(2)=(1)/(2)((mR^(2))/(2))dot(varphi^(2))=(1)/(4)m R^(2) dot(varphi^(2))`
The torsional potential energy is `(1)/(2)k varphi^(2)`. Thus the total energy is `:`
`(1)/(4)m R^(2)dot(varphi^(2))+(1)/(2)k varphi^(2)=(1)/(4)m R^(2) ddot(varphi_(0)^(2))+(1)/(2)k varphi_(0)^(2)`
By definition of the amplitude `varphi_(m), dot(varphi)=0` when `varphi=varphi_(m)` . Thus total energy is
`(1)/(2)k varphi_(m)^(2)=(1)/(4)m R^(2)dotvarphi_(0)^(2)+(1)/(2)k varphi_(0)^(2)`
or` varphi_(m)=varphi_(0)sqrt(1+(mR^(2))/(2k)(varphi_(0)^(2))/(varphi_(0)^(2)))`