Question

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

Answer 1

Weight of a body of mass m at the Earth’s surface, \(W\) = \(mg\) = \(250 \,N\) 
Body of mass m is located at depth, \(d\) = \(\frac{1}{2} R_e\)
Where,
\(R_e\) = Radius of the Earth
Acceleration due to gravity at depth g (d) is given by the relation: 
\(g' \bigg(1-\frac{d}{R_e}\bigg)g\)

= \(\bigg(1-\frac{R_e}{ 2 \times R_e}\bigg)g = \frac{1}{2}g\)

Weight of the body at depth d, 
\(W'\) = \(mg'\)
= \(m \times \frac{1}{2} g\)= \(\frac{1}{2} mg \)= \(\frac{1}{2} W\)

= \(\frac{1}{2} \times 250\) = \(125 \;\text N\)