At what height from the surface of earth the gravitational potential and the value of $$g$$ are $$ - 5.4 \times {10^7}\,J\,k{g^{ - 1}}$$ and $$6.0\,m{s^{ - 2}}$$ respectively?
Take the radius of earth as $$6400\,km$$ :
A.
$$2600\,km$$
B.
$$1600\,km$$
C.
$$1400\,km$$
D.
$$2000\,km$$
Answer :
$$2600\,km$$
Solution :
As we know, gravitational potential $$\left( v \right)$$ and acceleration due to gravity $$\left( g \right)$$ with
height
$$\eqalign{
& V = \frac{{ - GM}}{{R + h}} = - 5.4 \times {10^7}\,......\left( {\text{i}} \right) \cr
& {\text{and}}\,\,g = \frac{{{\text{ }}GM{\text{ }}}}{{{{\left( {R + h} \right)}^2}}} = 6\,......\left( {{\text{ii}}} \right) \cr} $$
Dividing (i) by (ii)
$$\eqalign{
& \frac{{\frac{{ - GM}}{{R + h}}}}{{\frac{{GM}}{{{{\left( {R + h} \right)}^2}}}}} = \frac{{ - 5.4 \times {{10}^7}}}{6} \Rightarrow \frac{{5.4 \times {{10}^7}}}{{\left( {R + h} \right)}} = 6 \cr
& \Rightarrow R + h = 9000\,km\,{\text{so,}}\,h = 2600\,km \cr} $$
Releted MCQ Question on Basic Physics >> Gravitation
Releted Question 1
If the radius of the earth were to shrink by one percent, its mass remaining the same, the acceleration due to gravity on the earth’s surface would-
If $$g$$ is the acceleration due to gravity on the earth’s surface, the gain in the potential energy of an object of mass $$m$$ raised from the surface of the earth to a height equal to the radius $$R$$ of the earth, is-
A geo-stationary satellite orbits around the earth in a circular orbit of radius $$36,000 \,km.$$ Then, the time period of a spy satellite orbiting a few hundred km above the earth's surface $$\left( {{R_{earth}} = 6400\,km} \right)$$ will approximately be-