Question

The gravitational force between sun and jupiter is approximately $4\times {10}^{23}$N. If mass of sun = $1.99\times {10}^{30}\mathrm{kg}$, the mean distance of the jupiter from the sun is $7.8\times {10}^{11}\mathrm{m}$, find the mass of jupiter.

Answer 1

Step 1: Given data:

The gravitational force between Sun and Jupiter, $\mathrm{F}=4\times {10}^{23}\mathrm{N}$

Mass of sun = ${\mathrm{m}}_{\mathrm{s}}$= $1.99\times {10}^{30}\mathrm{kg}$

The mean distance of the Jupiter from the Sun is, $\mathrm{r}=$ $7.8\times {10}^{11}\mathrm{m}$

Step 2: Calculating the mass of Jupiter:

Let the mass of Jupiter = ${\mathrm{m}}_{\mathrm{j}}$

According to the law of gravitation, the force between Sun and Jupiter is given by the formula-

$$\begin{gathered} \mathrm{F}=\mathrm{G}\frac{{\mathrm{m}}_{\mathrm{s}}{\mathrm{m}}_{\mathrm{j}}}{{\mathrm{r}}^2}...........\left(1\right) \\ \mathrm{Value}\mathrm{of}\mathrm{Gravitational}\mathrm{constant},\mathrm{G}=6.67\times {10}^{-11}\mathrm{N}{\mathrm{m}}^2{\mathrm{kg}}^{-2} \\ \mathrm{Using}\mathrm{equation}\left(1\right), \\ {\mathrm{m}}_{\mathrm{j}}=\frac{\mathrm{F}\times {\mathrm{r}}^2}{\mathrm{G}\times {\mathrm{m}}_{\mathrm{s}}} \\ \mathrm{Put}\mathrm{the}\mathrm{values}\mathrm{of}\mathrm{F},{\mathrm{r}}^2,\mathrm{G},{\mathrm{m}}_{\mathrm{s}}\mathrm{in}\mathrm{the}\mathrm{above}\mathrm{equation}\mathrm{we}\mathrm{get}, \\ {\mathrm{m}}_{\mathrm{j}}=\frac{4\times {10}^{23}\times {(7.8\times {10}^{11})}^2}{6.67\times {10}^{-11}\times 1.99\times {10}^{30}} \end{gathered}$$

$$\begin{gathered} {\mathrm{m}}_{\mathrm{j}}=\frac{4\times 7.8\times 7.8\times {10}^{26}}{6.67\times 1.99} \\ =\frac{243.36\times {10}^{26}}{13.27} \\ =18.34\times {10}^{26} \\ {\mathrm{m}}_{\mathrm{j}}=1.8\times {10}^{27}\mathrm{kg} \end{gathered}$$

Hence, the mass of Jupiter is $1.8\times {10}^{27}\mathrm{kg}$.