A body is droppped in a hole drilled across a diameter of the earth. Show that it executes SHM. Assume the earth to be homogenous sphere.

Explanation

Consider earth to a sphperical body of radius R with centre at O. Let a hole be made alon the diameter of earth. Let g be the acceleration due to gravity on the surface of earh at A
Consider a body of mass m be dropped in a hole and at an instant, it reaches at P, i.e., a distance d below the surface of earth. Figure, If `g^(')` is the acceleration due to gravity at P,
Then, `g^(')=g(1-(d)/(R))=g((R-d)/(R))`..(i)
If y is the distance of the body from the centre of the earth (i.e., displacement from mean position ), then, `y=R-d`
Putting this value in (i) we get
`g^(')=gxxy//R`
`:.` Force on the body at P will be
`F=mg^(')=(mg)/(R)y ` ..(ii)
Form (ii), `Fpropy` and this force F is directed towards the mean positino O, therefore, the body dropped in hole will execute linear SHM
Here, inertia factor `=` mass of body `=` m,
spring factor, `k=(mg)/(R)`
[By comparing (ii) with `F=-ky]`
`:.` Time period, `T=2pisqrt((i n ert ia fact o r)/(spri ng fact o r))`
`=2pisqrt((m)/(mg//R))=2pisqrt((R)/(g))`