  Question

# Find the time period of small oscillations of the following systems. (a) A metre stick suspended through the 20 cm mark. (b) A ring of mass m and radius r suspended through a point on its 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 r2 away from the centre.

Solution

## (a) M.I. about the pt. a=1=I.C.G.+mh2 =Mi212+mh2 =Mi212+M (0.3)2 =M(12+0.09) =M(1+1.0812) =M(2.0812) T=2x √lmgl =2π √2.08 mm×9.8×0.3 (1 = dist. between C.G. and pt. of suspension) (b) Moment of inertia about A, I=I.C.G.+mh2+mr2 =2mr2 ∴     Time period =2π √Imgl =2π √2mr2mgr=2π √2rg (c) lxx   (corner)  =(a2+a23)=2m3a2 In the  ΔABC, l2+l2=a2 ∴            l=a√2 ∴            T=2π √Imgl =2π √2ma23mgl =2π √2a2 √23ga =2π √√8a3g (d)           h=r2, l=r2 = Dist. between C.G. and suspension point M.I., about A, l−IC.G.+Mh2 =mr22+m(r2)2 =mr2 (12+14)=34mr2 ∴            T=2π √Imgl=2π √3mr24mgl =2π √3r24gr/2=2π √3r2g  Suggest corrections   