Question

The radii of orbits of two satellites revolving
around the earth are in the ratio 3: 8. Compare
their (i) critical speeds (ii) period of revolution​

Answer 1

The radii of orbits of two satellites revolving  around the earth are in the ratio 3: 8 have a critical speeds is $=(\frac{3}{8})^{\frac{5}{2} }$ and period of revolution is $=(\frac{3}{8 } )^{\frac{3}{2} }$

1. By using the Kepler's third law
2. $T^{2} =r^{3}$ here T is the period of revolution and r is the semi major axis or radius
3. Given is  $\frac{r_{1} }{r_{2} } = \frac{3}{8}$
4. $(\frac{T1}{T2} )^2=(\frac{r_{1} }{r_{2} } )^{3}$ here T1 and T2 is period of revolution by satellite 1 and 2 respectively.
5. $(\frac{T1}{T2} )=(\frac{3}{8 } )^{\frac{3}{2} }$ or can be written as
6. $(\frac{w2}{w1} )=(\frac{3}{8 } )^{\frac{3}{2} }$ where $w = Vr$
7. $(\frac{V2r_{2} }{V1r_{1} } )=(\frac{3}{8 } )^{\frac{3}{2} }$
8. the comparison of critical  speed of two satellites is
9. $\\ (\frac{V2 }{V1 } )=\frac{r_{1} }{r_{2} } (\frac{3}{8 } )^{\frac{3}{2} }=(\frac{3}{8 } )^{\frac{3}{2}+1 }=(\frac{3}{8})^{\frac{5}{2} }$