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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=Fk1
We know, XAXB=al
[from the diagram, using similarity of triangles]
XB=(Fk1)la....(iii)
F & F can be related by taking the moments of these forces about O, which yields
τ0=FaFl
I0d2θdt2=FaFl
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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