State Newton's law of universal gravitation, apply $F_g = Gm_1m_2/r^2$ to calculate the gravitational force, and use the inverse-square relationship to reason about how the force changes with distance.

What this topic is asking

Universal gravitation is Newton's account of the attractive force between any two masses, from a falling apple to the orbit of a planet. The Physical Setting/Physics course asks you to state the law, calculate the force with $F_g = \dfrac{Gm_1 m_2}{r^2}$, and reason with the inverse-square relationship: how the force changes when the masses or the distance change. The Regents tests both direct calculation and the proportional reasoning the inverse-square law invites.

Newton's law of universal gravitation

The force acts on both masses equally and oppositely (Newton's third law), no matter how different their sizes: Earth pulls a satellite with the same force the satellite pulls Earth. Gravity is always attractive, never repulsive, and it acts at a distance, with no contact needed. Because $G$ is tiny, the force is negligible between everyday objects and significant only when at least one mass is astronomical.

The inverse-square law

The inverse-square reasoning is a favorite Regents item, often without numbers. The procedure is to square the distance factor and invert it. The masses, by contrast, enter linearly: doubling one mass doubles the force, and doubling both quadruples it. Keeping the mass dependence (linear) separate from the distance dependence (inverse-square) avoids confusion.

Gravitational field strength and weight

Near a planet of mass $M$ and radius $r$, the gravitational force on a small mass $m$ is its weight, and equating $F_g = mg$ with the gravitation law gives the gravitational field strength:

This explains why $g$ differs between planets (the Moon's smaller mass gives a smaller $g$, about $1.6$ m/s squared) and why $g$ decreases with altitude (larger $r$). It also shows that the weight $F_g = mg$ near a surface is just the universal law evaluated at that surface. The gravitational field is the region around a mass where another mass feels a force, and $g$ measures its strength in newtons per kilogram (equivalently m/s squared).

Reference Tables note

The Reference Tables print $F_g = \dfrac{Gm_1 m_2}{r^2}$ and $F_g = mg$ in the Mechanics section, and the constant $G = 6.67 \times 10^{-11}$ N m squared per kg squared in the constants list, along with the masses and radii of Earth and other bodies on some editions. The field-strength form $g = \dfrac{GM}{r^2}$ is not printed separately but follows by equating the two printed expressions. Gravity supplies the centripetal force for orbits, linking this topic to uniform circular motion.

Try this

Q1. State how the gravitational force between two masses changes if the distance between them is tripled. [1 point]

• Cue. It becomes one ninth as large (inverse-square: $3^2 = 9$).

Q2. State two factors that determine the gravitational force between two objects. [2 points]

• Cue. The product of their masses, and the (inverse square of the) distance between their centers.