Correct Answer - Option 4 : 19600 km
The correct answer is option 4) i.e. 19600 km
CONCEPT:
-
Acceleration due to gravity on the surface of Earth of mass M and radius Re is denoted by g.
- It has an approximated uniform value of 9.8 m/s2 on the surface of Earth.
- The acceleration due to gravity at a depth 'd' below the surface of Earth is given by
\(⇒ g' = g(1- \frac{d}{R_e})\)
- The acceleration due to gravity at a height 'h' above the surface of Earth is given by
\(⇒ g'' = g(1+ \frac{h}{R_e})^{-2}\)
-
Weight: Weight is defined as the force with which an object is pulled towards the Earth due to gravity.
It is given by:
Weight, W = mg
Where m is the mass of the object and g is the acceleration due to gravity.
CALCULATION:
Let the weight of the body of mass m be W' and W'' at height h' and h'' above the earth respectively.
W' = mg' and W'' = mg''
Given that:
h'' = 100 km
\(W' = \frac{1}{16}W'' ⇒ mg' =\frac{1}{16}mg''\)
\(⇒ g' =\frac{1}{16}g''\)
\(⇒ g(1+ \frac{h'}{R_e})^{-2} =\frac{1}{16}\times g(1+ \frac{h''}{R_e})^{-2}\)
\(⇒ \frac{(1+ \frac{h'}{R_e})^{-2}}{(1+ \frac{h''}{R_e})^{-2}} =\frac{1}{16}\)
\(⇒ (\frac{R_e\ +\ h''}{R_e\ +\ h'} )^2=\frac{1}{16}\)
\(⇒ (\frac{R_e\ +\ h''}{R_e\ +\ h'} )=\frac{1}{4}\)
\(⇒ 4R_e +4h''=R_e +h'\)
\(⇒ 3(6400) +4(100)= h'\)
⇒ h' = 19600 km
