Two artificial satellites of masses  $m_1$ and $m_2$ are moving with speeds $v_1$ and $v_2$ in orbits of radii $r_1$ and $r_2$ , respectively. If r₁>r₂ then which of the following statements is true

 

According to kepler’s third law of time period –

  ${\left(\frac{T_A}{T_B}\right)}^2={\left(\frac{r_A}{r_B}\right)}^3\Rightarrow \frac{r_A}{r_B}=4^{1/3}$  

$\Rightarrow r_4^3=4r_B^3$  

T² ∝ R³
⇒ (T? /T? )² = (R? /R? )³ ⇒ T? = 9³/² T = 27T

Time period of Satellite = T = $\frac{2\pi R}{v}={\left(\frac{4{\pi }^2{R}^3}{GM}\right)}^{\frac{1}{2}}$  

$\Rightarrow T\alpha R^{\frac{3}{2}}\Rightarrow \frac{T_2}{T_1}=\frac{{\left(1.02R\right)}^{\frac{3}{2}}}{R^{\frac{3}{2}}}={\left(1.02\right)}^{\frac{3}{2}}={\left(1+0.02\right)}^{\frac{3}{2}}$     

The percentage difference in the time periods of the two satellites = $\frac{\Delta T}{T_1}$ × 100

= (0.03) × 100 = 3%