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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=a2

            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, lIC.G.+Mh2

=mr22+m(r2)2

=mr2 (12+14)=34mr2

            T=2π Imgl=2π 3mr24mgl

=2π 3r24gr/2=2π 3r2g

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