A very light rod of length `l` pivoted at `O` is connected with two springs of stiffness `k_1` & `k_2` at a distance of a & l from the pivot respectively. A block of mass ma attached with the spring `k_(2)` is kept on a smooth horizontal surface. Find the angular frequency of small oscillation of the block `m`.

Explanation

Let the block be pulled towards right (fig a) through a distance x given as

`x = x_(B) + x_(cB)` ….(i)
where `x_(cB) =` displacement of c (the block) relative to B.
Thus `x_(CS) = (F)/(k_(2))` ….(ii)
& `x_(B) = ((F.)/(K_(1))) (l)/(a)`....(iii)
F &F. can be related by taking the moment of these forces about O, that yields
`tau_(0) = F.a - Fl`
`rArr l_(0) (d^(2) theta)/(dt^(2)) = F.a - Fl`, Since the rod is very light its `M l_(0)` about O is approx equal to zero

`rArr F. = F (l//a)` ...(iv)
Using (iii) & (iv)
`rArr x_(B) = (F)/(k_(1)) ((l)/(a))^(2)` (v)
Using (i), (ii) & (v)
`rArr x = (F)/(k_(1)) ((l)/(a))^(2) + (F)/(k_(2))`
As force F is opposite to displacement x, then
`rArr F = (k_(1) k_(2))/(k_(2) (l//a)^(2) + k_(1)) x`
`rArr m omega^(2)x = (k_(1) k_(2))/(k_(2)(l//a)^(2) + k_(1)) x`
`rArr omega = sqrt((k_(1) k_(2) a^(2))/(m(k_(1) a^(2) + k_(2) l^(2))))`