Question

that piece of dust? The Earth is instantly replaced in its orbit by a speck of dust. Which statement best describes the subsequent orbital motion of that piece of dust? The dust particle will continue in the same orbit as the Earth did, orbiting the Sun in 1 year. The dust particle will move to a smaller orbit and orbit the Sun in less than 1 year. The dust particle will move to a larger orbit and orbit the Sun in more than 1 year. The dust particle will spiral into the Sun. The dust particle will be ejected from the Solar System.

Answer 1

The correct statement is:

The dust particle will move to a larger orbit and orbit the Sun in more than 1 year.

In fact, the dust particle has smaller mass than the Earth, therefore the gravitational attraction exerted by the Sun on the dust will be smaller, according to the formula of the gravitational force:

$F=G\frac{M_S m}{r^2}$

where G is the gravitational constant, MS is the Sun mass, m is the mass of the Earth (at first) and then of the dust particle (later), and r is the distance from the Sun. When we replace the Earth with the dust particle we see that m decreases, so the gravitational force F decreases, and the particle will move to a larger orbit. Then its orbital period will increase, according to the third law of Kepler, which states that the square of the orbital period is proportional to the cube of the orbital radius:

$T^2 \sim r^3$

therefore, since the orbit is larger, the orbital period is greater.