Physics - SHM Question with Solution | TestHub

$$U=\{\begin{array}{ll}k{x}^3& x\ge 0\\ -k{x}^3& x<0\end{array}$$

$$F=\{\begin{array}{ll}-k{x}^2& x\ge 0\\ k{x}^2& x<0\end{array}$$

As $F(0)=0\Rightarrow$ m.p. $=0$. For one-fourth oscillation, from $x=-a$ to $x=0$:

$$a_x=k{x}^2$$

$$V\frac{dv}{dx}=k{x}^2$$

$${\int }_0^{v}vdv={\int }_a^{x}k{x}^{2}dx$$

$$\frac{v^2}{2}=\frac{k{x}^3}{3}+\frac{k{a}^3}{3}$$

$$v=\sqrt{\frac{2k{x}^3+2k{a}^3}{3}}$$

$$\frac{dx}{dt}=x^{3/2}\sqrt{\frac{2k}{3}}$$

$${\int }_0^{T/4}\sqrt{\frac{2k}{3}}dt={\int }_{-a}^0{x}^{-3/2}dx$$

$$\sqrt{\frac{2k}{3}}\frac{T}{4}={\left[\frac{x^{-1/2}}{-1/2}\right]}_{-a}^0$$

$$T\propto \sqrt{\frac{3}{2k}}\cdot \frac{4}{\sqrt{a}}$$