Topic 2.6 Gravitational Force: apply Newton's law of universal gravitation, define the gravitational field strength, relate it to weight, and treat gravity inside and outside a spherical mass.

What this topic is asking

The College Board (Topic 2.6) wants you to apply Newton's law of universal gravitation, recognize its inverse-square character, define the gravitational field strength $g$, relate it to weight, and handle gravity both outside and inside a spherical mass. The 2024-25 revision sharpened the expectation that you can treat the field within an extended body, which is a distinctive calculus-flavoured part of AP Physics C.

Newton's law of universal gravitation

The defining feature is the inverse-square dependence: tripling the separation cuts the force to one ninth. The force is always attractive and acts along the line of centers. A useful theorem (which Newton proved with calculus) is that a uniform spherical mass attracts external objects exactly as if all its mass were concentrated at its center, so you use the distance to the center, not to the surface.

The gravitational field and weight

It is often cleaner to think of one mass creating a gravitational field that the other responds to. The field strength at distance $r$ from a mass $M$ is the force per unit test mass:

Near a planet's surface ($r = R$) this gives the familiar $g = GM/R^2$, about $9.8$ m/s squared on Earth, and the weight of a mass $m$ is $W = mg$. Because $g$ depends on $M$ and $r$, weight varies from planet to planet and falls with altitude, while the mass itself is unchanged. The field is a vector pointing toward the attracting mass, and its units (m/s squared, equivalently N/kg) match an acceleration: a freely falling object accelerates at exactly $g$.

Gravity inside a uniform sphere

A hallmark AP Physics C result concerns the field inside a uniform sphere. By the shell theorem, the mass in shells outside your radius $r$ contributes nothing; only the mass enclosed within radius $r$ pulls you. For uniform density the enclosed mass is $M_{enc} = M\dfrac{r^3}{R^3}$, so the field inside is

which grows linearly from zero at the center to $GM/R^2$ at the surface. Outside, it falls off as $1/r^2$. The full graph of $g$ versus $r$ therefore rises as a straight line up to the surface and then decays as an inverse square, a shape the exam asks you to sketch.

Try this

Q1. The Moon's mass is about $1/81$ of Earth's and its radius about $1/3.7$. Estimate the Moon's surface $g$ as a fraction of Earth's. [3 points]

• Cue. $g \propto M/R^2$, so $\dfrac{g_{Moon}}{g_{Earth}} = \dfrac{1/81}{(1/3.7)^2} = \dfrac{1/81}{1/13.7} = \dfrac{13.7}{81} \approx 0.17$, about one sixth.

Q2. Explain why an astronaut in a low orbit feels weightless even though gravity there is nearly as strong as at the surface. [2 points]

• Cue. The astronaut and station are in free fall together, accelerating at the local $g$, so there is no normal force; gravity still acts strongly.