An object of mass 10g executes SHM along the $x$-axis with frequency of $\left(\dfrac{10}{\pi}\right)$Hz. At the point $x = 2\ cm$ the object has KE 1.2 J and PE 0.4 J. The amplitude of oscillation is

Given, Kinetic Energy ($KE$) = $1.2$ J and Potential Energy ($PE$) = $0.4$ J at displacement $x = 2$ cm.

The total energy ($E$) of the object executing SHM is the sum of its kinetic and potential energies:

$E = KE + PE = 1.2 + 0.4 = 1.6$ J

In simple harmonic motion, the potential energy at a displacement $x$ and the total energy are given by:

$PE = \dfrac{1}{2} m \omega^2 x^2$

$E = \dfrac{1}{2} m \omega^2 A^2$

Taking the ratio of total energy to potential energy:

$\dfrac{E}{PE} = \dfrac{A^2}{x^2}$

Substituting the given values:

$\dfrac{1.6}{0.4} = \dfrac{A^2}{2^2}$

$4 = \dfrac{A^2}{4}$

$A^2 = 16$

$A = 4$ cm

Answer: $4\ \text{cm}$