This is a very unrealistic but interesting 'thought experiment' involving SHM. Imagine boring a straight tunnel from one place $(\mathrm{A})$ on the surface of the earth to another place (B) diametrically opposite, and then releasing a ball of mass $m$ at point $A$ (Figure 1.28 ). To answer the following questions, you need to know that gravitation implies that, when the ball is at the position shown in Figure 1.28 , it experiences a force of gravitation from the mass inside the dotted circle only. Further, this mass inside the dotted circle may be considered to be concentrated at the center of the earth. Assume that the density of the earth is uniform.
(a) Denoting the mass of the earth by $M$ and its radius by $R$, derive an expression for the mass inside the dotted circle (of radius $x$ ).
(b) Derive an expression for the gravitational force on the ball when at the position shown in Figure $1.28,$ a distance $x$ from the centre.
(c) Hence deduce that the motion of the ball is simple harmonic.
(d) Determine the period of the motion.
(e) Evaluate this period using $M=6.0 \times 10^{24} \mathrm{kg}$
$R=6.4 \times 10^{6} \mathrm{m}$ and $G=6.67 \times 10^{-11} \mathrm{Nkg}^{-2} \mathrm{m}^{2}$.
(f) Compare the period of this motion with the period of rotation of a satellite around the earth in a circular orbit of radius $R$.