A body of mass m, is attached to a vertical rod of mass M nad length L, hung from a pivoted supprot. A springs of constant K fixed to a support on the left as shown and is attached to the rod at a distance form the pivot. The frequency of the oscillation is:

A body of mass m, is attached to a vertical rod of mass M nad length L

1.	A body of mass m, is attached to a vertical rod of mass M nad length L, hung from a pivoted supprot. A springs of constant K fixed to a support on the left as shown and is attached to the rod at a distance form the pivot. The frequency of the oscillation is:
Answer» `(1)/(2pi) sqrt((K)/((M+2m)))` `(1)/(2pi)sqrt((K)/(((M)/(3)+2m)))` `2pi sqrt((K)/(((M)/(3)+2m)))` `2pi sqrt((M+2m)/(K))` Solution : As the rod is displaced by `y` towards the spring, the spring will GET compressed by `y` and the angular shift, `tan theta = (y)/(x), (1)/(2)I omega^(2) +(1)/(2)Ky^(2) +(1)/(2)mv^(2) =Constant` `(1)/(2) [(ML^(2))/(3) +mL^(2)]omega^(2) +(1)/(2)Ky^(2) +(1)/(2)mv^(2)=Constant` `(1)/(2) [(M)/(3)+m] v^(2) +(1)/(2)Ky^(2) +(1)/(2)mv^(2) =Constant` Differentitating w.r.t. TIME, we get, `(1)/(2)[(M)/(3)+m]2va +(1)/(2)K2yv +(1)/(2)m2va = 0` `a = (Ky)/([(M)/(3)+m+m]) :. omega =sqrt((K)/(((M)/(3)+2m)))` So, choice (c ) is correct and CHOICES (a) and (b) are incorrect. Choice (d) is not possible since, it is incorrect dimensionally.