A planet of mass M, has two natural satellites with masses $$m1$$ and $$m2$$. The radii of their circular orbits are $$R1$$ and $$R2$$ respectively. Ignore the gravitational force between the satellites. Define $$v1$$,$$L1$$,$$K1$$ and $$T1$$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of $$satellite1$$; and $$v2$$,$$L2$$,$$K2$$ and $$T2$$ to be the corresponding quantities of satellite $$2$$.Given $$m1m2=2$$ and $$R1R2=14$$, match the ratios in $$List-I$$ to the numbers in $$List-II$$.

Question

A planet of mass M, has two natural satellites with masses $$m1$$  and  $$m2$$. The radii of their circular  orbits  are  $$R1$$ and $$R2$$ respectively.  Ignore  the  gravitational  force  between  the satellites. Define $$v1$$,$$L1$$,$$K1$$ and $$T1$$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of $$satellite1$$; and $$v2$$,$$L2$$,$$K2$$ and $$T2$$ to be the corresponding quantities of satellite $$2$$.Given  $$m1m2=2$$ and  $$R1R2=14$$, match the ratios in $$List-I$$ to the numbers in $$List-II$$.

Infinity Learn · Accepted Answer

$$GMm1R12=m1V12R1$$      ⇒$$V12=GMR1GMm2R22=m2V22R2$$⇒$$V22=GMR2V12V22=R2R1=4$$                 (P) $$V1V2=2(Q)$$ $$L=mvR$$                   $$L1L2=m1v1R1m2v2R2=2$$×$$2$$×$$14=1$$ (R) $$K=12mv2$$               $$K1K2=m1v12m2v22=2$$×(2)2=8$$ $$(S) $$T=2$$$$π$$$$R/V$$            $$T1T2=R1v1$$×$$v2R2=R1R2$$×$$v2v1=14$$×$$12=18$$

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