PE of a particle is  U(x)=ax2−bx. Then the time period of small oscillation.

by question PE=U(x)=ax2−bx---(1) to find force F=dUdx hence differentiate eqn(1) wrt x F=-2ax3+bx2---(2) at mean postion F=0, apply to eqn(2) 2ax3=bx2 ⇒x=2ab---(3) to find force constant, K=d2Udx2x=2ab K=6ax4-2bx3x=2ab∵dFdx=ddx-2ax3+bx2=ddx-2ax-3+bx-2=6ax4-2bx3 K=6a2ab4-2b2ab3 K=b48a3  =ω2m;  ω=angular velocity;m=mass;T=time period ω=b48a3m=2πT T=2π8a3mb4