Question

Derive the expression for variation in the value of g with the depth below the
surface of the earth also find what is the weight at the centre of the earth.​

Answer 1

Answer:

The acceleration due to gravity on the surface of the earth is

g=

R

2

GM

−−−−−−−−−−−−−−−−(1)

Consider p be the density of the material of the earth

mass=volume×density

M=

3

4

πR

3

×p

Now substituting in equation (1)

g=

R

2

G

×

3

4

πR

3

×p

=

3

4

πGRp

−−−−−−−−−−−(2)

Now, Consider the body be taken to the depth d below the surface of the earth.

Then, the acceleration due to gravity at the depth below the surface of the earth is

g

d

=

3

4

πG(R−d)p −−−−−−−−−(3)

Now, divide equation (3) by (2)

g

g

d

=

R

R−d

=(1−

R

d

)

g

d

=g(1−

R

d

)

This is an expression for the acceleration due to gravity at the depth below the surface of the earth and this expression shows the acceleration due to gravity decrease as we move down into the earth. and At the center of the earth d=R.

Hence, Acceleration due to gravity at the center of the earth is 0