A satellite is launched into a circular orbit of radius $R$ around earth while a second satellite is…

A satellite is launched into a circular orbit of radius $R$ around earth while a second satellite is launched into a orbit of radius $1.02 R$. The percentage difference in the time periods of the two satellites is :
  1. 0.7
  2. 1.0
  3. 1.5
  4. 3

Solution

From Kepler's law, $T^2 \propto R^3 \Rightarrow \frac{T_1^2}{T_2^2}=\frac{R_1^3}{R_2^3}$ $\frac{T_1}{T_2}=\left(\frac{R_1}{R_2}\right)^{3 / 2}=\left(\frac{R}{1.02 R}\right)^{3 / 2}$ $\Rightarrow \quad \frac{T_1}{T_2}=\left(\frac{50}{51}\right)^{3 / 2}$ $\Rightarrow \quad \frac{T_1}{T_2}=\frac{50}{51}\left(\frac{50}{51}\right)^{\frac{1}{2}}$ $\Rightarrow \quad \frac{T_1}{T_2}=\frac{50}{51} \times 0.99$ $\frac{T_1}{T_2}=\frac{0.97}{1} \Rightarrow T_1=0.97 x, T_2=x$ Difference $=T_2-T_1$ $=x-0.97 x$ $=0.03 x$ $\%$ difference $=\frac{0.03 x}{x} \times 100$ $=3 \%$

Asked in: AP EAMCET 2003

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