A particle is allowed to fall through a smooth tunnel drille

A particle is allowed to fall through a smooth tunnel drilled through the Earth on an oblique

path. Ignoring any eects due to the Earths rotation, show that the particle undergoes simple

harmonic motion and nd its period. You should dene the angle between the tunnel and

Gravitational force on particle m ,

F = G Me m / r^2

as particle goes inside, we have to consider the mass of earth below the particle.

so with distance (radius) mass of earth will change.

Mass of earth w.r.t r (considering uniform density rho )

M(r) = rho ( 4 pi r^3 / 3) = (4 pi rho r^3 / 3)

F = - G M(r) m / r^2

F = - G (4 pi rho r^3 / 3) m / r^2 =- (4 G pi rho m / 3) r

this condition for SHM , F = - kx

where k = (4 pi G rho m / 3)

hence its proved that particle will SHM.

and mw^2 = k

m w^2 = (4 pi G rho m / 3)

w = sqrt(4 pi G rho / 3)

Time period = 2pi / w = 2pi sqrt(3 / (4 pi G rho )

T = pi sqrt(3 / (G pi rho ))