Question

Suppose a new planet is discovered between Uranus and Neptune. Its time period would be

• A. Less than that of Neptune.
• B. More than that of Neptune
• C. Equal to that of Neptune or Uranus.
• D. Less than that of Uranus.

Answer 1

The correct option is A

Less than that of Neptune.

Explanation for correct option

1. Kepler's third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit.
2. So for given $r$ the radius of the circular orbit the time period $T$ is $T^2\propto r^3$
3. As the distance of the new planet from the Sun is less than the distance between the Neptune and the Sun.
4. If $r_1$ is the distance between Sun and Uranus, $r_2$ the distance between the new planet and the sun and $r_3$ the distance between the Neptune and the Sun
5. $$\begin{gathered} \therefore r_1<r_2<r_2 \\ {r}_1^3<r_2^3<r_3^3 \end{gathered}$$
6. As $T^2$ directly proportional to $r^3$
7. So the time period of the new planet will be less than the time period of the Neptune but greater than Uranus

Hence, the correct option is Option (A)