Question

derive the expression for Acceleration​

Answer 1

Answer:

Acceleration due to gravity at a depth d 

Consider earth to be a homogeneous sphere of radius R and mass M with centre at O . 

Let g be the value of acceleration due to gravity at a point A on the surface of earth then

g=r2GM      ........(1)

If ρ is uniform density of material of the earth, then 

M=34πR3ρ

Putting value of M in equation (1) and equating it, we get:

∴g=R2G×34πR3ρ=34πGRρ        .......(2)

Let g′ be the acceleration due to gravity at the point B at a depth d below the surface of earth. The distance of the point B from the centre of the earth is (R–d). The earth can be supposed to be made of a smaller sphere of radius (R–d) and a spherical shell of thickness d. 

The body at B is inside the spherical shell of thickness d. The force on body of mass m at B due to spherical shell is zero. 

The body at B is outside the surface of smaller sphere of radius (R–d). The force on the body of mass m at B is due to smaller sphere of earth of radius (R–d) is just as if the entire mass M′ of the smaller sphere of earth is concentrated at the centre O.

∴g′=(R−d)2GM′

and M′=34π(R−d)3ρ

∴g′=g(1−Rd)

Hence, as depth increases, value of acceleration due to gravity decreases

Explanation:

hope its helpful