Question

A satellite is launched into a circular orbit of radius $R$ around the earth. A second satellite is launched into an orbit of radius $1.01R$. The time period of the second satellite is larger than that of the first one by approximately

• A. $0.5\%$
• B. $1.5\%$
• C. $1\%$
• D. $3\%$

Answer 1

Answer: (B)

$1.5\%$

Time period $T^2=\frac{4{\pi }^2}{GM}{R}^3$
i.e $T^2\propto R^3$
$\Rightarrow T^2=k{R}^3$ ; $k=$constant

Differentiating with rest to $R$,
$2T\Delta T=k3R^2\Delta R$
$\Rightarrow 2\frac{\Delta T}{T}=3\frac{\Delta R}{R}$

From the given data, $\Delta R=1.01R-R=0.01R$
$\therefore 2\frac{\Delta T}{T}=3\times 0.01$
$\Rightarrow \frac{\Delta T}{T}=\frac{0.03}{2}=0.015$
Percentage change is $1.5\%$