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Physics

Mechanics

Easy

Question

Kepler's third law states that square of period of revolution(T) of a planet around the sun, is proportional to third power of average distance r between sun and planet i.e. $T squared equals K r cubed$ here K is constant. If the masses of sun and planet are M and m respectively then as for Newton's law of gravitation force of attraction between them is $F equals fraction numerator G M m over denominator r squared end fraction$ , here G is gravitational constant. The relation between G and K is described as

K=G
$K = \frac{1}{G}$
$G K = 4 π^{2}$
$G M K = 4 π^{2}$

The correct answer is: $G M K equals 4 pi squared$

$therefore fraction numerator G M m over denominator r squared end fraction equals fraction numerator m v squared over denominator r end fraction semicolon v squared equals fraction numerator G m over denominator r end fraction not stretchy rightwards arrow left parenthesis i right parenthesis T equals fraction numerator 2 pi r over denominator v end fraction comma T squared equals fraction numerator 4 pi squared r squared over denominator v squared end fraction equals fraction numerator 4 pi squared r squared over denominator open parentheses fraction numerator G M over denominator r end fraction close parentheses end fraction table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell T squared equals fraction numerator 4 pi squared r cubed over denominator G M end fraction not stretchy rightwards arrow left parenthesis i i right parenthesis end cell row cell K equals fraction numerator 4 pi squared over denominator G M end fraction therefore G M K equals 4 pi squared end cell end table$

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