Question

The planet Jupiter revolves around the Sun in a period of about 12 years (3.79 × 108 seconds). What is its mean distance from the center of the Sun? The mass of the Sun is 1.99 × 1030 kilograms.
A. 1.1 × 10^11 meters
B. 1.5 × 10^11 meters
C. 2.3 × 10^11 meters
D. 5.8 × 10^11 meters
E. 7.8 × 10^11 meters

Answer 1

As we know that gravitational force of Sun will provide the centripetal force to all planets

so here we will have

$F_g = F_c$

$\frac{GMm}{r^2} = m\omega^2 r$

here we know that

$\omega = \frac{2\pi}{T}$

$\omega = \frac{2\pi}{3.79 \times 10^8}$

$\omega = 1.66 \times 10^{-8}$

now from above equation

$\frac{GM}{r^2} = \omega^2 r$

$\frac{GM}{\omega^2} = r^3$

$\frac{(6.67 \times 10^{-11})(1.99\times 10^{30})}{(1.66\times 10^{-8})^2} = r^3$

$r = 7.8 \times 10^{11} m$