Question

Consider the earth as a uniform sphere of mass $M$ and radius $R$ . Imagine a straight smooth tunnel made through the earth which connects any two points on its surface. Determine the time that a particle would take to go from one end to the other through the tunnel.

• A. $2\pi \sqrt{\frac{\textit{R}^{3}}{\textit{GM}}}$
• B. $\pi \sqrt{\frac{\textit{R}^{3}}{\textit{GM}}}$
• C. $\frac{\pi }{2}\sqrt{\frac{\textit{R}^{3}}{\textit{GM}}}$
• D. None of these

Answer 1

Answer: (B)

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)
Therefore, $\quad \mathrm{F}_x=-\mathrm{F} \sin \theta=-\frac{\mathrm{GMmr}}{\mathrm{R}^3}\left(\frac{x}{\mathrm{r}}\right)$
$ = - \frac{\text{GMmr}}{\text{R}^{3}} \cdot x$
Since, F_x ∝ - x, motion is simple harmonic in nature. Further,
$ \text{m} a_{x} = - \frac{\text{GMm}}{\text{R}^{3}} \cdot x \text{or} a_{x} = - \frac{\text{GM}}{\text{R}^{3}} \cdot x$
∴ Time period of oscillation is,
$\mathrm{T}=2 \pi \sqrt{\left|\frac{x}{a_x}\right|}=2 \pi \sqrt{\frac{\mathrm{R}^3}{\mathrm{GM}}}$
The time taken by particle to go from one end to the other is $\frac{T}{2} .$
$\mathrm{t}=\frac{\mathrm{T}}{2}=\pi \sqrt{\frac{\mathrm{R}^3}{\mathrm{GM}}}$