Question

Two artificial satellites $P$ and $Q$ are revolving round the earth in circular orbits. If the ratio of their radii is $1:4$ and ratio of their masses is $3:1$ then the ratio of their time periods will be

• A. $\frac{1}{8}$
• B. $8$
• C. $4$
• D. $3$

Answer 1

Answer: (A)

$\frac{1}{8}$

The time period of a satellite is given by $T=2\pi {\left(\frac{r^3}{GM}\right)}^{0.5}$ i.e. $T=K(r{)}^{1.5}$ because remaining are constant in this case here.
So, for the first satellite $T_1=K{r_1}^{1.5}$
similarly foe the second satellite $T_2=K{r_2}^{1.5}$.
Given that $\frac{r_1}{r_2}=\frac{1}{4}$.
Taking the ratio of timeperiods, we get $\frac{T_1}{T_2}=\frac{K{r_1}^{1.5}}{K{r_1}^{1.5}}={\left(\frac{1}{4}\right)}^{1.5}=\frac{1}{8}$