Find the time period of small oscillations of the following system. `a`. A meter stick suspended through the `20 cm` mark. `b` A ring of mas m and radius `r` suspended through a point on it periphery. `c` A uniform square plate of edge a suspended through a corner. `d`. A uniform disc of mass m and radius r suspended through a point `r/2` away from the center.

Answer

A, B, C, D

Explanation

a. M.I. about the pt `a=1=I.C.G+mh^2`
`=(Mi^2)/12+mh^2`
`=(Mi^2)/12+M(0.3)^2`
`=M(1/12+1.09)`
`=M((1+1.08)/12)`
`=M(2.08/12)`
`T=2pi(l/mgl)`
`=2pisqrt((2ma^2)/(3mgl))`
`=2pisqrt(2a^2sqrt2)/(3ga))`
`=2pisqrt((8a)/(3g))`
`d h=r/2`
`l=r/2= Dist. between C.G. and suspension point
M.I., about `A, l-I_(C.G.)+Mh^2`
`(mr^2)/2+m(r/2)^2`
`=mr^2(1/2+1/4)=3/4mr^2`
`:. T=2pisqrt(I/(mgl))=2pisqrt((3mr^2)/(4mgl))`
`=2pisqrt((3r^2)/(4gr/2))=2pisqrt((3r)/(2g))`