# How to solve SHM questions using Taylor's Series? [closed]

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?

• The H in SHM is misleading; your force is anharmonic.
– J.G.
Jul 25, 2021 at 20:30
• 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 ? Jul 25, 2021 at 20:47
• @HarshDarji "Can you still provide me with the solution to the question?" Hi, kindly note that this site is not for homework help. Jul 26, 2021 at 7:10
• 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 ? Jul 26, 2021 at 10:23

## 1 Answer

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.