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
IIT 1998,AIIMS 2008, Diffcult
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(a) $U = k|x{|^3} $
$\Rightarrow F= - \frac{{dU}}{{dx}} = - 3k|x{|^2}$ ...(i)
Also, for $SHM $ $x = a\sin \omega \,t$ and $\frac{{{d^2}x}}{{d{t^2}}} + {\omega ^2}x = 0$
$ \Rightarrow $ acceleration $ = \frac{{{d^2}x}}{{d{t^2}}} = - {\omega ^2}x$
$\Rightarrow F = ma$
$ = m\frac{{{d^2}x}}{{d{t^2}}} = - m{\omega ^2}x$ ...(ii)
From equation (i) & (ii) we get $\omega = \sqrt {\frac{{3kx}}{m}} $
$ \Rightarrow T = \frac{{2\pi }}{\omega } = 2\pi \sqrt {\frac{m}{{3kx}}} = 2\pi \sqrt {\frac{m}{{3k(a\sin \omega \,t)}}} $
$ \Rightarrow T \propto \frac{1}{{\sqrt a }}$.
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