In the figure shown, the spring are connected to the rod at one end and at the midpoint. The rod is hinged at its lower end. Rotational `SHM` of the rod (Mass `m`, length `L`) will occur only if-

Answer

B, C

Explanation

`theta = (x_(1))/(l//2), x_(1) = (thetal)/(2), f_(1) = (Kthetal)/(2)`
`theta = (x_(2))/(l), x_(2) = thetal, l_(2) = kthetal`
`tau = (L)/(2) xx F_(1) + L xx F_(2) ge mg sin theta (L)/(2)`
`rArr L/2 (KthetaL)/(2) + LKthetaL ge mgtheta L/2`
`rArr (5)/(4)KthetaL^(2) ge mg"(thetal)/(2) rArr K ge (2mg)/(5L)`
(ii) If the rod is displacement through an angle `theta`, then
`-(kLtheta)L - ((kL)/(2)theta) L/2+Mg(L/2)theta = (ML^(2))/(3)alpha(k = (Mg)/(L))`
`rArr (9ktheta)/(4M) = alpha rArr omega 3/2sqrt((k)/(m))`