Question

Be able to derive and interpret an expression for the acceleration of an oscillating mechanical system. In Figure 3, the smaller pulley is attached to the ground by a spring. a) M1, R Figure 3 Show that if the mass is pulled down and then released, the system will oscillate with SHM with a frequency given by 2k 2n (M, + M2 + 2m) The moment of inertia of a disc of radius R and mass m is I =mR?. Assume that the rope is always taut and the system is conservative. Use any method you like to prove this. b) Determine if the setup shown in Figure 4 behaves similarly. Assume that the rope is always taut. Mi, M2, R Figure 4 c) Explain what affect friction would have on the acceleration on the system in Figure 3. In Figure 3, if the mass m is attached to the ground with a spring having spring constant k, d) what would be the new frequency of oscillation?

Answer 1

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2 DOF system - there will be a f natural frecercy for system, EO -7777.. Sola: Cang Newstans serand low [ fa inocs ra] I m* + KCY-RG,)= 0 for Rotahing bodies IÔ, +I,Ö+ K RO, -40). Te calculate Naturel free I'm Ö 7587 TK - RKITZ to 5 € +72]10] 1 RK LO, when F= R=1 . Im o Tri] [k o (+12) ||0Ik To find wni x= Asin wt = B sinul 53 - Aw? sin wt Ö,:- Bu? sinu). mwirk -K TAR I-K. -CIHT.)W+K JIB! Te! → (- mauftk) [-C173'+ k] –K? = 0 Cm)(1+2)w e-ma? ke-C1+T2)? 14 +kº – ?, → mC It Iz) wt - [mw²t ItI z] u²k=o c? [m C. I +12) "?- Cmač+ IntI2)]]> Scanned with CamScanner TATAS

Wie . g= 0 mu?+ 4+ Ia - McIt Is I 3 Sustem be haves diffrently Effeg of fiction on acclerahen Sirro for fag 3. . Il Ö, -- wao " Since: (02= SHM ) leti 10, Xcoswt 0,- Kwe coswt JO -W? 02] > aceleration cp System :: 0 = - 1 2m+ M₂ +M, L if acceration is to be determined using a fiction, the fichen in Suetem ods as a resistive clampor which is diredly Proportional to velocity. & system tends to come to on equilibrium after some time - known as Critically demped susten New Eou will be mä R+ I, 7, + I, •;+ KCV? 6.) + 4.0, +f0]. 1. P, & h are coeff of fiction which acst as demping Constant & energ will be absorbed by them then Nature Requency is to be colectated indirectly as Enery here is not constant now; Seanned with CamScanner

X Semi EOM: Cucing Previous @ah) mi0,+ M2R? O, + M RYO, + KYRO, + KCROR zo .:[2 mp+My RP+ MiRr70 +2KXR+KR)0, = 0 Wn ckyR+ KR2)2 V 2mp + My R2+ M, Rx when =R=1 298) – 4] กะ 2m+M+M, Scanned with CamScanner