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35. Kepler's third law says that the square of period of revolution (T) of a planet around the sun is proportional to third power of average distance r between the sun and the planet i.e. $T^2=K{r}^3.$ Here K is constant

If the mass of sun and planet are M and m respectively then as per Newton's law of gravitation force of attraction between them is
$$\begin{gathered} F=\frac{GMm}{r^2},hereGisgravitationalcons\tan t. \\ TherelationbetweenGandKisdescribedas \\ a)GMK=4{\mathrm{\pi }}^2\mathrm{b})\mathrm{K}=\mathrm{G} \\ \mathrm{c})\mathrm{K}=\frac{1}{\mathrm{G}}\mathrm{d})\mathrm{GK}=4{\mathrm{\pi }}^2 \end{gathered}$$

Answer 1

Dear Student
$$\begin{gathered} T^2=k{r}^{3}andF=\frac{GmM}{r^2}\left(given\right) \\ \frac{m{v}^2}{r}=\frac{GmM}{r^2} \\ v=\sqrt{\frac{GM}{r}}......1> \\ now \\ {T}^2=k{r}^3 \\ {\left(\frac{2\pi r}{v}\right)}^2=k{r}^3 \\ {\left(\frac{2\pi r}{\sqrt{\frac{GM}{r}}}\right)}^2=k{r}^3\{fromeq1\} \\ \frac{4\pi r^3}{GM}=k{r}^3 \\ GMk=4\pi \end{gathered}$$
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