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Question

# A light rod of length l pivoted at O is connected with two springs of stiffness k1 & k2 at a distance of a & l from the pivot respectively. A block of mass m attached with the spring k2 is kept on a smooth horizontal surface. Find the angular frequency of small oscillations of the block m.

A
ω=k21a2m(k1a2+k2l2)
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B
ω=k22a2m(k1a2+k2l2)
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C
ω=k1k2a2m(k1a2+k2l2)
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D
None of these
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Solution

## The correct option is C ω=√k1k2a2m(k1a2+k2l2) Let the block be pulled towards right through a distance x given as, x=XB+XCB....(i) where, XCB = displacement of block C relative to B FBD of the system is shown in figure below XCB=Fk2....(ii) Extension in spring 1, XA=F′k1 We know, XAXB=al [from the diagram, using similarity of triangles] ⇒XB=(F′k1)la....(iii) F & F′ can be related by taking the moments of these forces about O, which yields τ0=F′a−Fl ⇒I0d2θdt2=F′a−Fl Since the rod is light, its MOI I0 about O is equal to zero. ⇒F′=F(la)....(iv) Using (iii) & (iv) ⇒XB=Fk1(la)2.....(v) Using (i), (ii) & (v), x=Fk1(la)2+Fk2 ⇒F=k1k2k2(la)2+k1x ⇒mω2x=k1k2k2(la)2+k1x ⇒ω=√k1k2a2m(k1a2+k2l2)

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