# Derivation of acceleration due to gravity1)at an altitude2)at a depth inside the earth

2
do u need the value only

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do u need the value only

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F2=GM1 M2/R²

F= gravitational force

g= gravitational accelaration

R= distance means height

both shows force so equal them

F1= F2

M1g=GM1M2/R²

G=gR²/M2

g=GM2/R²

now part 2nd

we have mg=Gm1m2/r²

Density = mass / volume

mass=volume×density

volume of sphere = 4/3r³

density =D

Mass=4/3r³×D

we have already proved that g=GM2/R²

put the value of M2 we drived

g=G4/3R³/R²

g=G4/3R

When the object is at a depth d below the surface of earth, it is attracted by the sphere of radius (R-d) only as the attraction due to the rest of earth cancel out. (At the centre of earth, the object would be attracted equally in all directions and the net force experienced by the object and hence the value of g would be zero)

Fdepth=Mg1=gm1m2/(R-d)²

M1=4/3(R-d)³

g1=4/3(R-d)³/(R-d)²

g1=4/3(R-d)

hence -------------g1=4/3π(R-d)

----------------------g=4/3πR

g1/g=4/3π(R-d)/4/3πR

g1=g{(R-d)R}.............................hence prove