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A planet of mass m moves around the Sum along an elliptical path with a period of revolution T. During the motion, the planet's maximum and minimum distance from Sum is R and (R)/(3) respectively. If T^(2)=alphaR^(3), then the magnitude of constant alpha will be |
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Answer» `(10)/(9).(PI)/(Gm)` Maximum and minimum distance of the planet from Sum is R and `(R)/(3)`, respectively. `:.` Semi-major AXIS of elliptcal path of planet around the Sum, `a=(R+(R)/(3))/(2)=(2R)/(3)` `:.` Time period of planet is given by `T=2pisqrt((a^(3))/(Gm))=2pisqrt((((2R)/(3)))/(Gm))^(3)` `T=2pisqrt((8R^(3))/(27G m))` Square on the both sides we get `T^(2)=4pi^(2).(8R^(3))/(27Gm)` `T^(2)=(32pi^(2))/(27Gm).R^(3)=alphaR^(2)` `:.alpha=(32pi^(2))/(27Gm)` HENCE, the MAGNITUDE of constant `alpha` will be `(32)/(27).(pi^(2))/(Gm)` |
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