Question

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

• A. $0.5\%$
• B. $1.0\%$
• C. $1.5\%$
• D. $3.0\%$

Answer 1

The correct option is C $1.5\%$

From the Kepler's laws of period, Time period of satellite is $T^2=\left(\frac{4{\pi }^2}{GM}\right)R^3$

$\Rightarrow T^2\propto R^3$

$\Rightarrow \frac{T_2^2}{T_1^2}=\frac{R_2^3}{R_1^3}$ $\Rightarrow T_2=T_1{\left(\frac{R_2}{R_1}\right)}^{\frac{3}{2}}$

$\Rightarrow T_2=T_1{\left(\frac{1.01R}{R}\right)}^{\frac{3}{2}}$ $\Rightarrow T_2=1.015T_1$

$\Rightarrow \Delta T$% $=\left(\frac{T_2-T_1}{T_1}\right)\times 100=1.5$%