(a) A uniform disc of radius `5*0cm` and mass `200g` is fixed at its centre to a metal wire, the other end of which is fixed with a clamp. The hanging disc is rotated about the wire through an angle and is released. If the disc makes torsional oscillations with time period `0.20s`, find the torsional constant of the wire.
(b) In the previous part, another disc is placed over the first one and the time period of the system become `0.65`. Find the M.I. of the second disc about the wire.

Explanation

(a)
M.I. of disc, `I=1/2mr^2=1/2xx0.2xx(5xx10^-2)^2`
`=25xx10^-5kgm^2`
`T=0.20s`
Time period
`T=2pisqrt(I/k)`, k: torsional constant
`k=(4pi^2I)/(T^2)=(4xx(3.14)^2xx25xx10^-5)/((0.2)^2)`
`=0.25kg-m^2//s^2`
(b) `T=2pisqrt(I/k)`
`T^'=2pisqrt((I+I^')/(k))`
(ii), (i)
`T^'/T=sqrt((I+I^')/(I))`
`(0.6)/(0.2)=(1+I^'/I)^(1/2)`
`1+I^'/I=9`
`I^'=8I=8xx25xx10^-5`
`=2xx10^-3kgm^2`