If the earth were a homogeneous sphere of radius R and a straight hole bored in it through its centre, show that a particle dropped into the hole will execute shm and find out its period.

If the earth were a homogeneous sphere of radius R and a straight hole

1.	If the earth were a homogeneous sphere of radius R and a straight hole bored in it through its centre, show that a particle dropped into the hole will execute shm and find out its period.
Answer» Solution :Acceleration due to gravity at the surface of the earth `g= (GM )/R^2` M-Mass of the earth and R-radius G-Gravitational constant If `rho`is the DENSITY of earth then `M = 4/3piR^2rho` `:. g = 4/3 piRrhoG ""...(1)` At a depth h below the surface of the earth `g.=4/3 pi(R-h)rhoG""...(2)` Dividing eq. (2) by eq. (1) `(g.)/g=(4/3pi(R-h)rhoG)/(4/3piRrhoG)` `g.=g""((R-h))/R` Let X be the distance of the particle from the centre of the earth. Then R-h = x Then acceleration `=(d^2x)/(dt^2) =g.` i.e.`(d^2x)/(dt^2) = -(gx)/R` Negative sign is put to show that acceleration acts towards the centre of the earth which is opposite to the direction of increasing x. `(d^2x)/(dt^2)+g/Rx=0` `omega^2=g/R,omega=sqrt(g/R)` Hence the particle DROPPED into the hole execute SHM Priod `T=(2pi)/omega=2pisqrt(R/g)`