Question 305: Gravitation

Question

The period of a planet around the sun is 8 times that of earth. The ratio of radius of planet's orbit to the radius of the earth's orbit is

Accepted Answer

Kepler's Third Law of Planetary Motion states that the square of the period of orbit ($T$) of a planet is directly proportional to the cube of the semi-major axis of its orbit ($r$). Mathematically, this is given by:
$$ T^2 \propto r^3 $$
For the Earth, we have:
$$ T_{	ext{earth}}^2 = k \cdot r_{	ext{earth}}^3 $$
For the planet with a period 8 times that of the Earth, we have:
$$ (8T_{	ext{earth}})^2 = k \cdot r_{	ext{planet}}^3 $$
Simplifying this expression:
$$ 64T_{	ext{earth}}^2 = k \cdot r_{	ext{planet}}^3 $$
Using the equation for Earth, $T_{	ext{earth}}^2 = k \cdot r_{	ext{earth}}^3$, we substitute:
$$ 64(k \cdot r_{	ext{earth}}^3) = k \cdot r_{	ext{planet}}^3 $$
The equation simplifies to:
$$ 64r_{	ext{earth}}^3 = r_{	ext{planet}}^3 $$
Taking the cube root on both sides:
$$ 4r_{	ext{earth}} = r_{	ext{planet}} $$
Therefore, the ratio of the radius of the planet's orbit to the radius of the Earth's orbit is 4.
Option A: 4

Answer

4

Answer

8

Answer

16

Answer

64

Gravitation

The period of a planet around the sun is 8 times that of earth. The ratio of radius of planet's orbit to the radius of the earth's orbit is

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