True or false on gravitation

Problem

True or false? (a) The gravitational force between two bodies is always attractive. (b) Kepler’s laws are consequences of the law of universal gravitation. (c) A satellite in orbit is not subject to gravity. (d) The period of a satellite depends only on the orbital radius and the mass of the central body.

Solution

(a) True. The law $F = Gm_1 m_2/r^2$ always gives attraction: there are no “negative masses”, and gravity is never repulsive. Masses simply attract each other.

(b) True. Newton showed that all three of Kepler’s laws (elliptical orbits, the law of areas, $T^2 \propto a^3$) follow from his laws of dynamics together with the law $F \propto 1/r^2$. Kepler had found them empirically; Newton provided the explanation.

(c) False. It is precisely Earth’s gravity that keeps the satellite in orbit, providing the centripetal force. The satellite is in free fall: it seems “weightless”, but gravity is very much at work.

(d) True. From Kepler’s third law $T = 2\pi\sqrt{r^3/(GM)}$: the period depends only on the orbital radius $r$ and the central mass $M$, not on the satellite’s mass.

$\ev{\text{(a) T}, \quad \text{(b) T}, \quad \text{(c) F}, \quad \text{(d) T}}$

Links

Topics: Gravitation Concepts: Law of universal gravitation · Kepler’s laws · Orbits · Weight force Objects: Satellite