A rod of mass M and length K is hinged at its one end n carris a block f mass m at its other end. A spring of force constant `k_(1)` is installed at distance a form the hinge and another of force constant `k_(2)` at a distance b as shown in the figure. If the whole arrangement rests on a smoth horizontal table top. Find the frequency of vibrations.

Answer

A, B, C

Explanation

For small angular displacement `theta`, net torque towards mean position is
`tau = (k_(1)atheta) a + (k_(2)btheta) b rArr Ialpha = (k_(1)a^(2) + K_(2)b^(2))theta`
`rArr (mL^(2) + 1/3 ML^(2)) alpha = (k_(1)a^(2) + k_(2)b^(2))theta`
`rArr alpha = (k_(1)a^(2) + k_(2)b^(2))/(L^(2)(m+(M)/(3))) theta :. omega^(2) = (k_(1)a^(2) + k_(2)b^(2))/(L^(2) (m+(M)/(3)))`
Hence frequency
`f = (omega)/(2pi) = (1)/(2pi) sqrt((k_(1)a^(2) - k_(2)b^(2))/(L^(2)(m+(M)/(3))))`