A particle of mass m moves in one dimension, whose potential energy varies as  Ux=−ax2+bx4 , where a and  b are positive constants. The angular frequency of small oscillations about the minima of the potential energy is equal to

Potential energyUx=−ax2+bx4 Force  F=−dUdx=2ax−4bx3At mean position  F=0⇒2ax=4bx3  ∴x=a2b d2Udx2=−2a+12bx2=-2a+6a=4a   i.e +ve⇒U is minima at  x=a2b∴  Force constant Keff=Fxor dFdx⇒Keff=d2udx2=2a−12bx2 ⇒K=2a−12b×a2b=4a Now,F=ma'=-kx⇒a'=-4am.xThis represents equation of SHM.⇒ω=4am=2am