At a distance from the Earth’s centre

PHYSICS · Subjective · CBSE · Physics · Gravitation · Gravitation

Problem

At a distance

r

from the Earth’s centre, the acceleration due to gravity is given by

g(r) = g_0 \left(\dfrac{R}{r}\right)^{2}

, where

g_0

is the surface value and

R

is the Earth’s radius. (i) Starting from this relation, derive a general formula for the altitude

h\,(>0)

at which the acceleration due to gravity falls to a fraction

k

of the surface value, i.e.

g(R+h) = k\,g_0, \quad 0<k<1.

(ii) Using your formula, calculate the altitude at which the gravitational acceleration equals

g_0/9

. (iii) A satellite is placed in a circular orbit at this altitude. Obtain an expression for its orbital speed in terms of

G

M

and

r

, and then compute the numerical value using

G = 6.67\times10^{-11}\,\text{N m}^2\text{/kg}^2

M = 6.0\times10^{24}\,\text{kg}

and

R = 6.4\times10^{6}\,\text{m}