Assuming the earth to be a sphere of uniform mass density, how much would a body weigh half way down to the center of the earth if it weighed 250 N on the surface?

The weight of a body is the force acting one it due to earth’s gravity and is the product of mass and acceleration due to gravity is given as

W = mg

Where W is the weight of a body of mass m on the surface of the earth, g is acceleration due to gravity on the surface of the earth

Now as the body goes inside surface of earth towards the centre of earth acceleration due to gravity decreases, so the weight of the body also decreases i.e.

W_d = mg_d

Where W_d is the weight of a body of mass m kept at a depth d inside the surface of the earth, Let g_d be the acceleration due to gravity at a at a depth d inside the surface of the earth

Now acceleration due to gravity at a depth d inside the surface of the earth is given by the relation

Where g_d is the acceleration due to gravity at a depth d inside the surface of the earth, g is acceleration due to gravity on earth’s surface and R is Radius of the Earth

The situation has been shown in figure

Now Let g_d be the acceleration due to gravity half way down to the centre of the earth and g be the acceleration due to gravity on the surface of the earth.

If R is the radius of the Earth and mass in half way down the earth then we have the depth of body from the surface of earth

d = R/2

Then using

We get

g_d = g/2

i.e. acceleration due to gravity half way down the surface of the earth is half of that on the surface of the earth

so the weight of the body at this depth will be

W_d = mg_d

W_d = m(g/2) = 1/2 mg

i.e. W_d = 1/2 W

now we are given the weight of the body on the surface of the earth as

W = 250 N

i.e. we have

W_d = 250/2 N

W_d = 125 N

So, the weight of the body at a depth equal to half the radius of the earth is 125 N.