Fill in the blanks:
$1.$ If a tunnel is dug along the diameter of the Earth and a body is dropped freely in it,the motion of this body is .........,assuming there is no frictional force of the medium acting on it.
$2.$ When a body executes simple harmonic motion and makes $\frac{1}{2\pi}$ oscillations,its phase increases by ......... $rad$.
$3.$ For a body executing simple harmonic motion,the phase increases by ......... every second.
$4.$ The ratio of force constants of two springs is $1:2$ and the ratio of their mechanical energies is $2:9$. The ratio of the amplitudes of the two bodies suspended at the ends of the springs is .........

(A) $1.$ The motion is Simple Harmonic Motion $(SHM)$.
$2.$ Since $1$ oscillation corresponds to a phase change of $2\pi \ rad$,then $\frac{1}{2\pi}$ oscillations correspond to $\frac{1}{2\pi} \times 2\pi = 1 \ rad$.
$3.$ The phase $\theta$ is given by $\theta = \omega t + \phi$. The rate of change of phase is $\frac{d\theta}{dt} = \omega$. Thus,the phase increases by $\omega$ every second.
$4.$ The mechanical energy $E$ is given by $E = \frac{1}{2} k A^2$. Given $\frac{k_1}{k_2} = \frac{1}{2}$ and $\frac{E_1}{E_2} = \frac{2}{9}$.
$\frac{E_1}{E_2} = \frac{k_1}{k_2} \times \left(\frac{A_1}{A_2}\right)^2 \Rightarrow \frac{2}{9} = \frac{1}{2} \times \left(\frac{A_1}{A_2}\right)^2$.
$\left(\frac{A_1}{A_2}\right)^2 = \frac{2}{9} \times 2 = \frac{4}{9}$.
Therefore,$\frac{A_1}{A_2} = \frac{2}{3}$.