Gravitation Question 5
Question 5 - 2024 (29 Jan Shift 1)
At what distance above and below the surface of the earth a body will have same weight, (take radius of earth as R.)
(1) $\sqrt{5} \mathrm{R}-\mathrm{R}$
(2) $\frac{\sqrt{3} R-R}{2}$
(3) $\frac{R}{2}$
(4) $\frac{\sqrt{5} R-R}{2}$
Show Answer
Answer: (4)
Solution:
$g_{p}=\frac{g R^{2}}{(R+h)^{2}}$
$g_{q}=g\left(1-\frac{h}{R}\right)$
$g_{p}=g_{q}$
$\frac{g}{\left(1+\frac{h}{R}\right)^{2}}=g\left(1-\frac{h}{R}\right)$
$\left(1-\frac{h^{2}}{R^{2}}\right)\left(1+\frac{h}{R}\right)=1$
Take $\frac{\mathrm{h}}{\mathrm{R}}=\mathrm{x}$
So
$\mathrm{x}^{3}-\mathrm{x}+\mathrm{x}^{2}=0$
$\mathrm{x}=\frac{\sqrt{5}-1}{2}$
$\mathrm{h}=\frac{\mathrm{R}}{2}(\sqrt{5}-1)$
