Question

A satellite is launched into a circular orbit of radius ' $R$ ' around earth while a second satellite is launched into an orbit of radius $1.02R$. The percentage difference in the time periods of two satellites is_________

Answer 1

Answer: 3

$T^2=K{R}^3$
$\Rightarrow 2\frac{dT}{T}=3\frac{dR}{R}$
$\Rightarrow (\%$ change in Time period $=\frac{3}{2}(\%$ change in $R)$
$\Rightarrow \frac{3}{2}(2\%)=3\%$
Alternate method
$T^2\propto R^3$
$T\propto R^{\frac{3}{2}}$
$\therefore T=k{R}^{\frac{3}{2}}$
For radius $R,T=k{R}^{\frac{3}{2}}$
$\therefore \frac{dT}{dR}=\frac{3}{2}k{R}^{\frac{1}{2}}$
When radius $=1.02R$
$\therefore dR=1.02R-R=0.02R$
$\therefore \frac{dT}{dR}=\frac{3}{2}k{R}^{\frac{1}{2}}$
$\therefore dT=\frac{3}{2}k{R}^{\frac{1}{2}}\times dR$
$\therefore dT=1.5k{R}^{\frac{1}{2}}\times 0.02R$
$\therefore dT=0.03k{R}^{\frac{3}{2}}$
$\therefore dT=0.03T$
Percentage change $=\frac{dT}{T}\times 100$
$=\frac{0.03T}{T}\times 100$
$=3$