Question 43: Gravitation

Question

When one moves from a point 16 km below the earth's surface to a point 16 km above the earth's surface. The change in g is approximately $\alpha \%$. The value of $\alpha$ is $\_\_\_\_$ .
(Take radius of the earth $=6400 \mathrm{~km}$.)

Accepted Answer

For small heights and depths compared to the radius of Earth, NCERT uses these approximations:
At a depth $d$ below the surface:
$$ g_d = g\left(1-\frac{d}{R}ight) $$
At a height $h$ above the surface:
$$ g_h = g\left(1-\frac{2h}{R}ight) $$
Here,
$$ d = 16\ 	ext{km}, \quad h = 16\ 	ext{km}, \quad R = 6400\ 	ext{km} $$
Now,
$$ g_{	ext{below}} = g\left(1-\frac{16}{6400}ight) $$
and
$$ g_{	ext{above}} = g\left(1-\frac{2	imes 16}{6400}ight) $$
So the change in $g$ while moving from 16 km below the surface to 16 km above the surface is
$$ \Delta g = g_{	ext{below}} - g_{	ext{above}} $$
$$ = g\left(1-\frac{16}{6400}ight) - g\left(1-\frac{32}{6400}ight) $$
$$ = g\left(\frac{16}{6400}ight) $$
Hence percentage change is
$$ \frac{\Delta g}{g}	imes 100 = \frac{16}{6400}	imes 100 $$
$$ = 0.25\% $$
So,
$$ \alpha = 0.25 $$
Therefore, the correct option is:
$$ \boxed{0.25} $$

Answer

0.12

Answer

0.25

Answer

0.50

Answer

0.75

Gravitation

When one moves from a point 16 km below the earth's surface to a point 16 km above the earth's surface. The change in g is approximately \alpha \%. The value of \alpha is \_\_\_\_ . (Take radius of the earth =6400 \mathrm{~km}.)

Choose the correct answer: