Question

Problem 1.11 The general form for the energy of a simple harmonic oscillator is E = mass (velocity)? + stiffness (displacement)? Set up the energy equations for the oscillators in Figure 1.1(a), (b). (c), (d), (e), (f) and (g), and use the expression DE = 0 dr to derive the equation of motion in each case.

please also include the explanations and write in paper. thank you

Answer 1

d) Simple pendulum If e is the Length of the storing then Kinctic energy > T = mi 2 em (eo) ? mêö² 0 mg (OA- oc) ecoso lo е. B le- ecoso) mg 2 2 nge ele - COSO 050) E Ttv Ź méo + inge (i-coso) Hence, the in eg of motion JE gives at 3 Ź mě (20) + mge (o tasino) me o - nge sino 5) con gesino o $s -) + g sino e This is the ear of motion (ö +10 - 0) (0-0) comparing with da we et t wa get 9 е.

If we ce b) twisted the disc by an angle o from o from the equil position, the wire will ol extert torque, given by GU Za -Co, And KE = 10 -jedo - feodo v= Z co + k : E = K K.E tu 2 1 0 + Ź co + n motion (CE de elt f da elt" an - f o i U14

c) Here ） Kinetic energy T = 2 må S 000000000 And V = -S.s.de -1-5+ să te E = Ttv = 1/2mă + să te The egne motion mi S2 O ww S m 옮 1) Here 2 T T= 2 m ² I e x² lx a V ze E = mă + 1 Equ MR + 122 ma + 1 ش 2 co 2T em

e) If d is the total length of the eiquid there inside the tube then 27. WINTTI T= mi = Ž Peña a density of the requid V = magar ale ale mg doc = / P (22) gdx 2 pq 2 ak =pg&tk E = Ttv Ź pe am + Pgn tk Eqr of motion at we pen + apga Suppose have lequid inside a tube and ke push the léquid so inside the tube and so form S.H.M, + 28 they will 2 stast oscillating E = zg OU e

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