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het the solution of the equalion is,, Xn A et, putting these values in differuntial equation °f SH.M., we jet -, dert (a*+ w*) = 0, Since A ext *o, that means '+ w* -o antiw, Thus the two possilula solutions of differeutial equation, are-, X : A, etat + 1.e-(wt, * = A, ( Cos wt + i Sin wt ) +^ ( cos wt - i sinot), X: (A, + A,) cos wt + i(^,- Ae) Sinwt, R sin, R Cos p, Now, let, A,+ A2, %3D, i (A, - A,), Thus,, R Sinp cos wt + Rcos p Sinwt, R Sin (wt +), Solution of second order, differemtial equation., TIME PERIOD, one complete yede -, For, a, T = 2T, T: 21 m, VELOCITY AND ACCELERATION, The displacement equation of s.H.M. is, given by, R Sin ( wt + ), Now differntiating the equation with respect bo t -, de, v = Rw Cos (wt +), dt, V : RW, Ji- Sin+(wt+ 4), Sa, again diffuenbiabing, the equation, we jet -, = « = -R o* Sin (wt ++), dt, Required condition for S.H.M., P The velocity of partiele is maximum at z=0, thus, max, = Rw
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A.I The differeutial equation of simple harmonic mobon is. given by -, dy, 10og, dt, the fre queency, time period .of motion. [ Ha, 2 Sec], calculate, and, - U= 6x*49 3/kg, simple harmonic. Calculate time period of, [ see ], and its velocely, R.2 Poteutial energy of A, vibrating body is given by, Show that, this, motion, is, motion., 83 A parbide exccutis S.H.M., along, straight line, when passing, through point 3 im and 4cm from the centre of its, path is 16 cm/s aud 12 cm/s respeetively . Find, the amplitude and time-, 1.57 sec], [ 5 cm,, period of motion., line. when the, Q.4 A particle is, mouing, straight, with, S.H.M. in a, distance of the particle from equilibrium position has the values ,, values of velocity, and *, the corresponding, are v, and v. show that, ci) The period of s.H.M. is, 2T | -지, cii) The maximum velocity during, Vmaz =, this motion is, x - x,-, POSITION AVERAGE, ENERGY, energy of oseilatör is -, Imco ca^-x*) de, The, kinetie, average, KE, a, a, a, KE, mw, 20, KE, I mw a, %3D, 3, The, Ihe average potembial energy of system is -, Imi, %3D, muta", Thus, the position average kinetic ehergy and potential enery, wist not, L.e.,, Ke + V, be, Some., The botal energy of.oscillahór i -, Ky + V maa +Lmo at, mota, E :, 5
