Verified Solution • 2025–26 Session moderate

QJEEM24Phce88: JEE Main 2024 Physics - Class 11 JEEM24Phce88 Solution: Two planets and having masses and move around the sun in…

Two planets AA and BB having masses m1m_1 and m2m_2 move around the sun in circular orbits of r1r_1 and r2r_2 radii respectively. If angular momentum of AA is LL and that of BB is 3 L3 \mathrm{~L} , the ratio of time period (TATB)\left(\frac{T_A}{T_B}\right) is:

Multiple Choice Options

  • A (r2r1)32\left(\frac{r_2}{r_1}\right)^{\frac{3}{2}}
  • B 27(m1m2)327\left(\frac{m_1}{m_2}\right)^3
  • C (r1r2)3\left(\frac{r_1}{r_2}\right)^3
  • D 127(m2m1)3\frac{1}{27}\left(\frac{m_2}{m_1}\right)^3

Step-by-Step Verified Solution

amp;v1v2=r2r1.... (i)amp;m1v1r1=hamp;m2v2r2=32amp;⇒v1v2=13m2m1r2r1.... (ii)\begin{aligned}& \frac{v_1}{v_2}=\sqrt{\frac{r_2}{r_1}} \quad \text{.... (i)}\\& m_1 v_1 r_1=h \\& m_2 v_2 r_2=32 \\& \Rightarrow \frac{v_1}{v_2}=\frac{1}{3} \frac{m_2}{m_1} \frac{r_2}{r_1} \quad \text{.... (ii)}\end{aligned}

From (i) & (ii)

amp;r2r1=13m2m1r2r1amp;3m1m2=r2r1amp;T1T2=(r1r2)3/2=(m23m1)=127(m2m1)3\begin{aligned}& \sqrt{\frac{r_2}{r_1}}=\frac{1}{3} \frac{m_2}{m_1} \frac{r_2}{r_1} \\& \frac{3 m_1}{m_2}=\sqrt{\frac{r_2}{r_1}} \\& \frac{T_1}{T_2}=\left(\frac{r_1}{r_2}\right)^{3 / 2}=\left(\frac{m_2}{3 m_1}\right)=\frac{1}{27}\left(\frac{m_2}{m_1}\right)^3\end{aligned}

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