A body is dropped in a hole drilled across diameter of the earth. Show that it executes SHM Asuume earth to be a homogenous sphere, find the time period of its motion. Given density of earth is `5.51 xx10^(3)Kgm^(-3)` and `G=6.67xx10^(-11)Nm^(2)kg^(-2)`

Explanation

`F=Gmxx(4)/(3)pi((R-d)^(3)rho)/((R-d)^(2))-(4)/(3)piGrhomy`
where, `y=(R-d)`
`:.F prop y` and is directed towards centre O of earth. Hence, the body will execute linear SHM with O as mean position.
Here, inertia factor `=` m
Spring factor `=(4)/(3)piGrhom`
Time period, `T=2pisqrt((i n ertia fact o r)/(spri ng fact o r))`
`=2pisqrt ((m)/((4)/(3)piGrhom))=2pisqrt((3)/(4pirGrho))`