Question

A satellite is revolving around planet of mass $2M$ in orbit of radius $R$ with time period is $T_1$. Another satellite is revolving around planet of mass $4M$ in orbit of radius $2R$, with time period $T_2$. Find $\frac{T_1}{T_2}$.

• A. $\frac{1}{\sqrt{2}}$
• B. $\sqrt{2}$
• C. $\frac{1}{2}$
• D. $\frac{1}{2\sqrt{2}}$

Answer 1

The Correct Option is C

Step 1: State the formula for the time period of a satellite.
$T = 2\pi \sqrt{\frac{r^3}{GM_p}}$, which implies $T \propto \sqrt{\frac{r^3}{M_p}}$.

Step 2: Set up the ratio for $T_1$ and $T_2$.
$\frac{T_1}{T_2} = \sqrt{\frac{r_1^3 / M_{p1}}{r_2^3 / M_{p2}}} = \sqrt{\left( \frac{r_1}{r_2} \right)^3 \cdot \frac{M_{p2}}{M_{p1}}}$.

Step 3: Substitute the given values.
$r_1 = R$, $r_2 = 2R$.
$M_{p1} = 2M$, $M_{p2} = 4M$.
$\frac{T_1}{T_2} = \sqrt{\left( \frac{R}{2R} \right)^3 \cdot \frac{4M}{2M}} = \sqrt{\frac{1}{8} \cdot 2} = \sqrt{\frac{1}{4}} = \frac{1}{2}$.

Final Answer: Option (3).