Question

A body weighs 200 N on the surface of earth. How much does it weigh half way down to the centre of earth?

Answer 1

Hint: The weight of a body is equal to the product of its mass and the value of acceleration due to gravity at the place where it is measured. By obtaining the value of acceleration due to gravity at half the radius of earth, we can find out the weight of the given body.

Formula used:
The weight of an object is given as
$F = mg$
The variation of the acceleration due to gravity with depth towards the centre of earth is given as
$g' = g\left( {1 - \dfrac{h}{R}} \right)$

Complete step by step answer:
The weight of the given object at the surface of earth is given as
$F = 200N$
We know that on the surface of earth, the weight of a body is equal to the product of its mass and the value of acceleration due to gravity on the surface of earth.
$F = mg$
We know that the value of acceleration due to gravity on the surface of earth is $g = 10m/{s^2}$. Now we can calculate the mass of the given object in the following way.
$m = \dfrac{F}{g} = \dfrac{{200}}{{10}} = 20kg$
Now we know that the value of acceleration due gravity varies with depth from thr surface of earth through the following relation:
$g' = g\left( {1 - \dfrac{h}{R}} \right)$
Here h is the depth below the surface of earth while R represents the radius of earth. Now half way down the surface of earth means at a distance equal to half the radius of earth. The value of acceleration due to gravity at this distance is given as
$g' = g\left( {1 - \dfrac{1}{{2R}}} \right)$
Now the weight of the given object at this place will be given as
$F' = mg' = mg\left( {1 - \dfrac{1}{{2R}}} \right)$
The value of the radius of earth is $R = 6371km = 6371 \times 1000m$. Now inserting the known values, we get
$F' = 200\left( {1 - \dfrac{1}{{2 \times 6371000}}} \right) \simeq 200 \times 0.999 = 199.8N$
This is the required value of weight half way down to the centre of earth

Note:
It should be noted that the mass of an object is a constant quantity and does not change with the change in its position in the gravitational field of earth. The weight of an object signifies the gravitational force acting on an object and its value may change with change in its distance from the centre of earth.