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Theory of Compound Pendulum, , Any rigid body capable of oscillating freely about, a horizontal axis passing through it is called as, compound pendulum., , Consider a rigid body of mass M capable of, oscillating about a horizontal axis XY passing through, the point O and perpendicular to the plane of paper.
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If body is at rest,, its centre of gravity G is, below the point O. Let, distance between G & O, be l., , If the body is, displaced through an, angle 0, G occupies new, position G’. Now. there, will be restoring couple, which tends to bring G’, to G ie. its original, position and it is given, by, , = Mg X AG’, , , , = Mgl sin@, , This couple produces an angular acceleration,, , a If J is the moment of inertia of the body about XY,, t, , then couple = / “, 2 153 = Mg | sin@, =Mgl6 6 is small, ie oe = au! (@)oncancte (4), = ae y wat is constant, , Since angular acceleration is proportional to the, angular displacement, the pendulum performs simple, harmonic motion and its period is given by, , displacement, Wel leeceleration
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Let /, be the moment of inertia of the body about, the parallel axis passing through centre of gravity G,, then using parallel axis theorem, we have, moment of, inertia about XY axis, | = /, + MU?, , l= MK? +M/? (Where K is radius of-gyration *), , “Radius of gyration is distance between centre of, gravity and axis of rotation and also it Is defined as, root mean square distance of all particles from axis of, rotation, } ., , , , , , , , , , Tm M(K? +1?) (3), Substituting / in eq” (2), we get, M(K? + |?), , Mgl, , t4 jar, , T = 2n, , K?4+2, gl, , , , =2r, , , , , , l, = 27 lt pl rioreay anne CA), , Comparing above equation with time Period ‘of, simple pendulum (2n EE ) length L of simple pendulum, , is equal to length of compound pendulum: we = ?), called as length of equivalent simple pendulum, is, , abt eee, , eh, ef, Fo, ies ot
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Centre of suspension (0), centre of oscillation (0’), and their interchangeability, , The point 0’ at a distance, equal to the length of an, equivalent simple pendulum from, , the point of suspension 0 is called, as centre of oscillation., , let OG = 1, & 0'G = 1p., Hence 00’ = |, + 1, ='L, , Let T be time period about, the point of suspension 0., , , , K2, Tt + ly, T=2n Pye (5), , , , When the pendulum is inverted and suspended, about the point of oscillation 0’. Let T’ be its time period., , K2, , ee +1,, T' = 20 Fee PN (6))*. : hg, , We know that, length of equivalent.simple, pendulum is {, K2, bart alt laces (7), Bh, , , , K2, —=l,, , 1, , Kk? = hl,, , Adding /? on both side, , Ket leh +l”, , Dividing above eq" by lz , we get
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L, , From eg? (7) & eq" (8), K? K*, , ——-+ L, = — + L, , L L, , Thus, the time period compound pendulum, about the centre of suspension is same as that about, centre of oscillation. That is centre of suspension and, centre of oscillation is interchangeable.
