Q) A point particle is acted upon by a restoring force $-kx^3$. The time period of oscillation is $T$, when the amplitude is $A$. The time period for an amplitude $2A$ will be

(A) $T$ (B) $T/2$ (C) $2T$ (D) $4T$

Now I know that if a force function is given in SHM questions, the time period can be simply written as $2\pi \sqrt{m\over F'(0)}$ where $F'(0)$ is the first derivative of the force function given at zero. However, this technique doesn't quite work here because $F'(0)$ becomes zero making $T$ infinite. Also, how do I draw a relation between the amplitude and time period?

  • $\begingroup$ The H in SHM is misleading; your force is anharmonic. $\endgroup$
    – J.G.
    Jul 25 at 20:30
  • $\begingroup$ Oh I see. Actually, this question was under the category of SHM in the test paper I took it from. Thank you for clarifying though. Can you still provide me with the solution to the question ? $\endgroup$ Jul 25 at 20:47
  • $\begingroup$ @HarshDarji "Can you still provide me with the solution to the question?" Hi, kindly note that this site is not for homework help. $\endgroup$ Jul 26 at 7:10
  • $\begingroup$ I know that, but don't you think I had a major concept issue with this question and that I NEEDED the solution of the Q to clear it ? $\endgroup$ Jul 26 at 10:23

Let's generalize the exponent from $3$ to $p$, for an instructive comparison to the harmonic case, $p=1$. For a mass-$m$ particle, the equation of motion is $\ddot{x}=-\frac{k}{m}x^p$.

We're given that some solution is of amplitude $A$ and period $T$, i.e. can be written as $x=Af(y)$ with $y:=t/T$,where $f$ is a function of amplitude $1$ and period $1$. The EOM becomes$$-\frac{1}{f^p}\frac{d^2f}{dy^2}=\frac{k}{m}A^{p-1}T^2$$(proof is a by-substitution exercise). Note the left-hand side has period $1$ in $y$.

If $p=1$, scaling $A$ doesn't change $T$; this is the familiar fact that an SHM period depends on only on $k/m$, not $A$. But if $p\ne1$, when we replace $A$ with $cA$ for $c>0$ we need to multiply $T$ by $c^{2/(1-p)}$. I leave you to substitute this problem's values of $c,\,p$ to work out which answer is correct.


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