A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy is U(x)=k|x|3, where k is a positive constant. If the amplitude of oscillation is a, then its time period T is

U=k|x|3⇒F=−dUdx=−3k|x|2    ........(i) Also, for SHM, x=asin⁡ωt and d2xdt2+ω2x=0⇒ Acceleration a=d2xdt2=−ω2x⇒F=ma=md2xdt2=−mω2x     .......(ii) From eqs. (i) and (ii), we get ω=3kxm⇒T=2πω=2πm3kx=2πm3k(asin⁡ωt)⇒T∝1a