Suppose at some instant the particle is at radial distance r from centre of earth O. Since, the particle is constrained to move along  the tunnel, we define its position as distance x from C. Hence, equation of motion of the particle is,

 ma_x = F_x

The gravitational force on mass m at distance r is,



$\text{F}=\frac{\text{GMmr}}{{\text{R}}^3}$     (towards O)

$\text{Therefore,}{\text{F}}_x=-\text{F sin}\text{θ}=-\frac{\text{GMmr}}{{\text{R}}^3}\left(\frac{x}{\text{r}}\right)$

$=-\frac{\text{GMmr}}{{\text{R}}^3}·x$

Since, F_x ∝ - x, motion is simple harmonic in nature. Further,

$\text{m}{a}_x=-\frac{\text{GMm}}{{\text{R}}^3}·x\text{or}{a}_x=-\frac{\text{GM}}{{\text{R}}^3}·x$

∴   Time period of oscillation is,

$\text{T}=2\text{π}\sqrt{\left|\frac{x}{a_x}\right|}=2\text{π}\sqrt{\frac{{\text{R}}^3}{\text{GM}}}$

The time taken by particle to go from one end to the other is $\frac{\mathrm{T}}{2}\mathrm{.}$

$\text{t}=\frac{\text{T}}{2}=\text{π}\sqrt{\frac{{\text{R}}^3}{\text{GM}}}$