Asume that as narrow tunnel is dug between two diametricaly opposite points of the earth. Treat the earth as a solid sphere of uniform density. Show that ilf a particle is relesed in this tunel, ilt wil execute a simple harmonic motion. Calculate the time period of this motion.

Explanation

Let at an instant, particle is at distance x from centre of earth.
The particle will be attracted only by sphere of radius x.
Let `M^'`=mass of sphere of radius x
`M_e:` mass of earth
`R_e:` radius of earth
`M^'=(M_e)/(4/3piR_e^3)*4/3pix^3=(M_ex^3)/(R_e^3)`
Force of attraction on m
`F=-(GM^'m)/(x^2)=-(GM_em)/(R_e^3)x`
`a=-(GM)/(R_e^3)x=-omega^2x`
`T=2pisqrt((R_e^3)/(GM_e))=2pisqrt((R_e^3)/(gR_e^2))=2pisqrt(R_e/g)`
`T=2pisqrt((R_e^3)/(GM_e))=2pisqrt((R_e^3)/(gR_e^3))=2pisqrt(R_e/g)`
`M_e=rho*4/3piR_e^3`
`rho:` density of earth
`T=2pisqrt((R_e^3)/(G*rho*4/3piR_e^3))=sqrt((4pi^2xx3)/(4pirhoG))=sqrt((3pi)/(rhoG))`
`T=2pisqrt((R_e^3)/(GM_e))=2pisqrt(R_e/g)=sqrt((3pi)/(rhoG))`