AP Physics C: Mechanics · Topic 2.6

Topic 2.6: Gravitational Force

Unit 2: Force and Translational Dynamics20-25% of the multiple-choice section

Topic 2.6 covers universal gravitation, the gravitational field, weight, apparent weight and the two kinds of mass. AP Physics C adds a fifth objective AP Physics 1 lacks: the force from an extended sphere, including the result that inside it grows linearly with distance from the centre.

AP Physics: Unit 2 (topics 2.6 Gravitational Force). AP Physics C: Mechanics Unit 2, Topic 2.6. FIVE learning objectives, more than any other topic in the unit, which has nineteen in total. 2.6.A (gravitational interaction between two objects with mass), supported by 2.6.A.1 with the printed |F_g| = G m1 m2 / r^2 and sub-statements i, ii, iii, plus 2.6.A.2 (a field models the effects of a noncontact force at various positions in space), 2.6.A.2.i (DERIVED: |g| = |F_g|/m = GM/r^2), 2.6.A.2.ii (with gravity the only force, the acceleration in m/s^2 numerically equals the field strength in N/kg) and 2.6.A.3 (DERIVED: Weight = F_g = mg). 2.6.B (situations where the gravitational force can be considered constant), with 2.6.B.1 and 2.6.B.2 (near Earth's surface g is approximately 10 N/kg). 2.6.C (apparent weight), with 2.6.C.1 (apparent weight is the magnitude of the normal force), 2.6.C.2, 2.6.C.3 (a system appears weightless when there are no forces on it OR when gravity is the only force on it, both clauses) and 2.6.C.4 (the equivalence principle). 2.6.D (inertial and gravitational mass), with 2.6.D.1 to 2.6.D.3. 2.6.E, describe the gravitational force exerted on an object by a uniform spherical distribution of mass, has NO counterpart in AP Physics 1, whose Topic 2.6 ends at 2.6.D. It carries 2.6.E.1 (the net force is the sum of the individual forces from small differential masses comprising the distribution), 2.6.E.2 (Newton's shell theorem) with 2.6.E.2.i (zero net force inside a thin spherical shell), 2.6.E.2.ii (outside a shell, treat it as a point mass at the centre), 2.6.E.2.iii (inside a uniform sphere only a partial mass contributes), 2.6.E.2.iv (DERIVED: m_partial = rho (4/3) pi r_partial^3) and 2.6.E.3 (DERIVED: F_g,partial = -k r_partial). Boundary statement, in full: AP Physics C: Mechanics does not expect students to mathematically prove or derive Newton's shell theorem. Objective 2.6.E is scoped to uniform SPHERICAL distributions only; the CED does not extend it to rods, discs or rings. Four of the unit's eleven derived equations are in this topic and none of them is printed on the equation sheet; what is printed is |F_g| = G m1 m2 / r^2. The constants box prints g = 9.8 m/s^2 and g = 9.8 N/kg while 2.6.B.2 gives g approximately 10 N/kg; both are in the same document. Suggested skills are 1.C, 2.A, 2.D and 3.B; AP Physics 1 lists 1.A, 2.A, 2.C, 2.D and 3.B for its version.

One whole learning objective that AP Physics 1 does not have

Both courses run Topic 2.6 through objectives 2.6.A, 2.6.B, 2.6.C and 2.6.D with the same statements. AP Physics C: Mechanics then adds 2.6.E: describe the gravitational force exerted on an object by a uniform spherical distribution of mass. There is no 2.6.E in AP Physics 1; its Topic 2.6 ends at 2.6.D.

That single extra objective makes Topic 2.6 the largest topic in Unit 2 by objective count, with five of the unit's nineteen, and it is the only place in the unit where you treat a source of force as an extended body rather than a point.

Statement 2.6.E.1 sets the frame: the net gravitational force exerted on an object by a uniform spherical distribution of mass is the sum of the individual forces from small differential masses that comprise the distribution. That is a summation over $dm$ , which is the same move Topic 2.1 makes for the centre of mass, applied here to a force instead of a position.

Statement 2.6.E.2 then introduces Newton's shell theorem, which describes the net gravitational force exerted on an object by a uniform spherical shell of mass, and gives three results you may use:

2.6.E.2.i: the net gravitational force exerted on an object inside a thin spherical shell is zero.
2.6.E.2.ii: the net gravitational force exerted on an object outside a thin spherical shell can be determined by treating the shell as a single massive object located at the centre of the shell.
2.6.E.2.iii: an object inside a sphere of uniform density experiences a net gravitational force from only a partial mass of the sphere.

Statement 2.6.E.2.iv makes the partial mass explicit. It is the portion of the sphere's mass located a distance less than or equal to the object's distance from the centre, and it can be calculated using the density of the sphere. Its derived equation is

m_{\text{partial}} = \rho \frac{4}{3}\pi \left(r_{\text{partial}}\right)^3

And 2.6.E.3 delivers the payoff: the gravitational force exerted on an object within a uniform sphere can be shown to be proportional to the object's distance from the sphere's centre, with the derived equation

F_{g,\text{partial}} = -k r_{\text{partial}}

A linear restoring force, with the same form as Hooke's law. That is an oscillation from Unit 7 hiding inside a Unit 2 gravitation topic, and the first worked example follows it through.

The boundary statement, and exactly what it fences off

Topic 2.6 carries one boundary statement, printed under objective 2.6.E:

"AP Physics C: Mechanics does not expect students to mathematically prove or derive Newton's shell theorem."

Read the verbs. You are not expected to prove or derive the theorem, which would mean carrying out the integral over the shell that shows the inside contributions cancel. You are entirely expected to apply it, because 2.6.E.2.i, 2.6.E.2.ii and 2.6.E.2.iii are required content and each of them is a usable result.

So the division of labour is clean:

You may use	You need not produce
zero force inside a thin shell	the integral that shows it
a shell acting as a point mass from outside	the integral that shows it
only the enclosed mass counting inside a solid sphere	a proof of the shell theorem
$m_{\text{partial}} = \rho\frac{4}{3}\pi r_{\text{partial}}^3$
$F_{g,\text{partial}} = -kr_{\text{partial}}$

Statement 2.6.E.3's own phrasing supports this reading: the force "can be shown to be" proportional to distance. It is a result handed to you.

One more scope note, and this one is about what the CED does not say rather than what it does. Objective 2.6.E is specifically about a uniform spherical distribution of mass. The CED does not extend it to rods, discs, rings or arbitrary bodies. If you meet a problem asking for the field of a uniform rod, it is enrichment rather than required content for this topic, and the exam is entitled to keep the extended-body cases spherical.

What the CED requires across all five objectives

Suggested skills for the whole topic are 1.C, 2.A, 2.D and 3.B.

2.6.A: describe the gravitational interaction between two objects or systems with mass.

2.6.A.1: Newton's law of universal gravitation describes the gravitational force between two objects or systems as directly proportional to each of their masses and inversely proportional to the square of the distance between the systems' centres of mass. Relevant equation $\lvert\vec{F}_g\rvert = G\dfrac{m_1m_2}{r^2}$ .
2.6.A.1.i: the gravitational force is attractive.
2.6.A.1.ii: it is always exerted along the line connecting the centres of mass of the two interacting systems.
2.6.A.1.iii: the gravitational force on a system can be considered to be exerted on the system's centre of mass.
2.6.A.2: a field models the effects of a noncontact force exerted on an object at various positions in space.
2.6.A.2.i: the magnitude of the gravitational field created by a system of mass $M$ at a point in space equals the ratio of the gravitational force exerted by the system on a test object of mass $m$ to the mass of the test object. Derived equation $\lvert\vec{g}\rvert = \dfrac{\lvert\vec{F}_g\rvert}{m} = G\dfrac{M}{r^2}$ .
2.6.A.2.ii: if the gravitational force is the only force exerted on an object, the observed acceleration in $\mathrm{m/s^2}$ is numerically equal to the magnitude of the gravitational field strength in N/kg at that location.
2.6.A.3: the gravitational force exerted by an astronomical body on a relatively small nearby object is called weight. Derived equation $\text{Weight} = F_g = mg$ .

2.6.B: describe situations in which the gravitational force can be considered constant.

2.6.B.1: if the gravitational force between two systems' centres of mass has a negligible change as their relative position changes, the force can be considered constant at all points between the initial and final positions.
2.6.B.2: near the surface of Earth, the strength of the gravitational field is $g \approx 10\ \mathrm{N/kg}$ .

2.6.C: describe the conditions under which the magnitude of a system's apparent weight differs from the magnitude of the gravitational force exerted on it.

2.6.C.1: the magnitude of the apparent weight of a system is the magnitude of the normal force exerted on the system.
2.6.C.2: if the system is accelerating, the apparent weight is not equal to the magnitude of the gravitational force exerted on it.
2.6.C.3: a system appears weightless when there are no forces exerted on the system or when the force of gravity is the only force exerted on the system.
2.6.C.4: the equivalence principle states that an observer in a noninertial reference frame is unable to distinguish between an object's apparent weight and the gravitational force exerted on the object by a gravitational field.

2.6.D: describe inertial and gravitational mass.

2.6.D.1: objects have inertial mass, or inertia, a property that determines how much an object's motion resists changes when interacting with another object.
2.6.D.2: gravitational mass is related to the force of attraction between two systems with mass.
2.6.D.3: inertial mass and gravitational mass have been experimentally verified to be equivalent.

2.6.E is the fifth, set out in the first section above.

The two values of g, and why both are correct

Statement 2.6.B.2 says that near the surface of Earth the strength of the gravitational field is $g \approx 10\ \mathrm{N/kg}$ . The constants box on the AP Physics C: Mechanics Table of Information prints two lines:

magnitude of the acceleration due to gravity at Earth's surface, $g = 9.8\ \mathrm{m/s^2}$
magnitude of the gravitational field strength at Earth's surface, $g = 9.8\ \mathrm{N/kg}$

Both of those facts are in the same document. The framework rounds to 10 for estimation, and the booklet you are handed in the exam room prints 9.8. Neither is a misprint, and this site uses 9.8 throughout because that is the value printed on the sheet.

What matters more than the number is the pair of lines in the constants box. The same symbol $g$ carries two units, $\mathrm{m/s^2}$ and N/kg, and statement 2.6.A.2.ii explains why they are numerically equal: if the gravitational force is the only force on an object, the observed acceleration in $\mathrm{m/s^2}$ equals the field strength in N/kg. Those are genuinely different quantities. The field exists at a point in space whether or not anything is there; the acceleration is a property of an object in free fall. They coincide because dividing $F_g = mg$ by $m$ gives back $g$ , which is 2.6.D.3 doing quiet work in the background: the $m$ that appears in the gravitational force and the $m$ that appears in $F = ma$ are experimentally the same number.

The is g 9.8 or 10 guide works through the discrepancy across all four AP Physics courses. For the number itself, use 9.8 unless a question tells you otherwise, and never lose marks arguing about the third significant figure.

Field, weight and apparent weight, kept separate

Three ideas get conflated here, and the CED distinguishes them precisely.

The field is defined at a point in space. Statement 2.6.A.2 says a field models the effects of a noncontact force exerted on an object at various positions in space, and 2.6.A.2.i defines its magnitude as the force per unit mass on a test object, $\lvert\vec{g}\rvert = GM/r^2$ . The test mass cancels out, which is the whole point: the field belongs to the source, not to the object feeling it.

Weight is a force. Statement 2.6.A.3 defines it as the gravitational force exerted by an astronomical body on a relatively small nearby object, with $\text{Weight} = F_g = mg$ . Note that both 2.6.A.2.i and 2.6.A.3 are labelled derived equations, so neither is printed on the equation sheet. What is printed is $\lvert\vec{F}_g\rvert = Gm_1m_2/r^2$ , and you get the other two from it.

Apparent weight is a different force again. Statement 2.6.C.1 says the magnitude of the apparent weight of a system is the magnitude of the normal force exerted on the system. It is what a bathroom scale reads, and it equals the true weight only when the system is not accelerating vertically. Statement 2.6.C.2 says so directly.

Statement 2.6.C.3 is worth quoting carefully because it has two clauses that get truncated: a system appears weightless when there are no forces exerted on the system or when the force of gravity is the only force exerted on the system. The second clause is the one that explains orbit. An astronaut on the International Space Station is not beyond Earth's gravity; the gravitational field there is most of its surface value. They are in free fall, which is to say gravity is the only force acting, so there is no normal force and the apparent weight is zero.

Statement 2.6.C.4 generalises it: the equivalence principle states that an observer in a noninertial reference frame is unable to distinguish between an object's apparent weight and the gravitational force exerted on the object by a gravitational field. Sealed in a windowless lift, no experiment tells you whether you are accelerating or sitting in a stronger field. The weight versus normal force comparison and the apparent weight glossary entry cover the practical side.

And 2.6.D closes the loop on mass. Inertial mass, 2.6.D.1, is resistance to changes in motion. Gravitational mass, 2.6.D.2, is what enters the force of attraction. Statement 2.6.D.3 says they have been experimentally verified to be equivalent, which is not obvious and is the empirical fact the equivalence principle rests on. The mass versus weight comparison is the plain-language version.

Traps this topic sets

Measuring $r$ from the surface. In $\lvert\vec{F}_g\rvert = Gm_1m_2/r^2$ , statement 2.6.A.1 specifies the distance between the systems' centres of mass. A satellite at an altitude of 400 km above Earth is not at $r = 400$ km; it is at Earth's radius plus 400 km.

Applying the inverse square inside a body. The inverse square is for the region outside the mass. Inside a uniform sphere, 2.6.E.3 says the force is proportional to $r$ , not to $1/r^2$ . The two agree at the surface and diverge in opposite directions on either side of it: going up, the field falls off as $1/r^2$ ; going down, it falls off linearly to zero at the centre.

Confusing weight with apparent weight. The scale in a lift reads the normal force. The gravitational force has not changed.

Thinking weightless means no gravity. Statement 2.6.C.3's second clause says a system also appears weightless when gravity is the only force acting on it. That is orbit.

Assuming the field is the acceleration. They are numerically equal only in the case 2.6.A.2.ii names, where the gravitational force is the only force exerted on the object. A block on a table sits in a field of 9.8 N/kg and has zero acceleration.

Trying to prove the shell theorem. The boundary statement says the course does not expect you to mathematically prove or derive it. Time spent doing so is time not spent on the results you actually need.

If you want the algebra-based version of this topic

The first four objectives are shared, so if you want a slower treatment of the field, weight, apparent weight and the two kinds of mass, AP Physics 1 Topic 2.6: Gravitational Force covers exactly that material with algebra-based examples. If you are taking AP Physics C: Mechanics, the objective that is only here, 2.6.E, is the one worth your time on this page.

	AP Physics 1 Topic 2.6	AP Physics C Topic 2.6
Objectives	2.6.A to 2.6.D	2.6.A to 2.6.E
Extended spherical bodies	not in the course	objective 2.6.E
Shell theorem	not in the course	2.6.E.2, apply but do not prove
Inside a uniform sphere	not in the course	2.6.E.3, $F = -kr$
Boundary statements	none on this topic	one, on the shell theorem
Suggested skills	1.A, 2.A, 2.C, 2.D, 3.B	1.C, 2.A, 2.D, 3.B

The algebra-based course lists five suggested skills here to the C course's four, which is a reminder that skill count is not difficulty. What the C course lists is 1.C, create qualitative sketches of graphs, where AP Physics 1 lists 1.A, create diagrams. Sketching the field of a sphere against distance from its centre, linear inside and inverse square outside, is precisely a 1.C task and precisely a 2.6.E question.

Related on this site: the gravitational field glossary entry, the universal gravitation glossary entry, and Topic 2.10, where gravitation supplies the net force producing centripetal acceleration in orbit.

How Topic 2.6 is tested

Unit 2 carries 20 to 25 percent of the multiple-choice section across about 15 to 25 class periods, the highest minimum weighting of any unit in AP Physics C: Mechanics (only Unit 3, at 15 to 25 percent, reaches the same 25 percent ceiling). Topic 2.6 holds five of the unit's nineteen learning objectives, more than any other topic in it.

The suggested skills are 1.C, create qualitative sketches of graphs that represent features of a model or the behavior of the physical system; 2.A, derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway; 2.D, predict new values or factors of change of physical quantities using functional dependence between variables; and 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim.

Skill 2.A carries 25 to 30 percent of the multiple-choice section, the largest band the CED lists for any single skill, and the CED lists it for exactly four topics in this unit: 2.6, 2.7, 2.9 and 2.10. Those are the four topics with derived equations to produce, and 2.6 has four of the unit's eleven derived equations on its own.

Skill 2.D is the inverse-square skill. A question that asks what happens to the gravitational force if one mass triples and the separation doubles is answered from functional dependence alone: three over four of the original. No numbers needed, and no calculator.

Skill 1.C points at the graph this topic owns: gravitational field strength against distance from the centre of a uniform sphere. It rises linearly from zero at the centre to a maximum at the surface, then falls as $1/r^2$ outside. Being able to sketch that curve, with the kink at the surface and the correct shape on each side, is a compact test of 2.6.E.3 and 2.6.A.2.i together.

The Unit 2 hub sets this topic against the other nine, and the circular motion and gravitation practice set covers the surrounding material, though it is written for the algebra-based treatment.

Down a tunnel through a uniform planet

A planet of radius $R = 6.4 \times 10^6$ m has uniform density and a surface gravitational field strength of $9.8$ N/kg. A narrow tunnel is drilled straight through its centre and an object of mass $m$ is dropped in from the surface. Derive the force on the object at a distance $r$ from the centre, identify the constant $k$ in statement 2.6.E.3, and find the period of the resulting oscillation and the maximum speed.

Declare the convention: $r$ measured outward from the centre along the tunnel, positive away from the centre.
Apply the shell theorem, which 2.6.E.2 lets you use without proving. Every part of the planet farther out than the object lies in shells that enclose it, and 2.6.E.2.i says the net force from inside a thin shell is zero. So only the mass within radius $r$ contributes, which is 2.6.E.2.iii.
Write that partial mass using 2.6.E.2.iv: $m_{\text{partial}} = \rho\dfrac{4}{3}\pi r^3$ .
Treat the enclosed sphere as a point mass at the centre, which 2.6.E.2.ii licenses, and use universal gravitation: $F = -\dfrac{G m\, m_{\text{partial}}}{r^2} = -\dfrac{Gm\rho\frac{4}{3}\pi r^3}{r^2} = -\dfrac{4}{3}\pi G\rho m\, r$ . The minus sign says the force points back toward the centre.
That is 2.6.E.3's $F_{g,\text{partial}} = -kr_{\text{partial}}$ , with $k = \frac{4}{3}\pi G\rho m$ . The two powers of $r$ in the volume beat the two in the inverse square, leaving one.
Rewrite $k$ in terms of what the problem gives you. The planet's total mass is $M = \rho\frac{4}{3}\pi R^3$ , so $\frac{4}{3}\pi G\rho = \dfrac{GM}{R^3}$ , and the surface field is $g = GM/R^2$ by 2.6.A.2.i. So $\dfrac{GM}{R^3} = \dfrac{g}{R}$ and $k = \dfrac{mg}{R}$ .
Recognise the form. $F = -kr$ with a constant $k$ is Hooke's law, statement 2.8.A.2's $\vec{F}_s = -k\Delta\vec{x}$ with a different letter on the front. The object will oscillate, and the period comes from the printed line $T_s = 2\pi\sqrt{m/k}$ .
Substitute: $T = 2\pi\sqrt{\dfrac{m}{mg/R}} = 2\pi\sqrt{\dfrac{R}{g}}$ . The mass cancels, so every object dropped in takes the same time, which is the symbolic result worth writing down.
Numerically, $R/g = 6.4\times 10^6 / 9.8 = 6.531\times 10^5\ \mathrm{s^2}$ , whose square root is $808.1$ s. So $T = 2\pi(808.1) = 5078$ s, about 84.6 minutes.
Maximum speed, at the centre. For simple harmonic motion the maximum speed is $\omega A$ , with $\omega = 2\pi/T = 1.2374\times 10^{-3}\ \mathrm{rad/s}$ and amplitude $A = R$ : $v_{\max} = (1.2374\times 10^{-3})(6.4\times 10^6) = 7.92\times 10^3$ m/s.
Check that against something independent. Symbolically $v_{\max} = \omega R = \sqrt{g/R}\cdot R = \sqrt{gR}$ , and $\sqrt{(9.8)(6.4\times 10^6)} = \sqrt{6.272\times 10^7} = 7.92\times 10^3$ m/s. That is also the speed of a circular orbit skimming the surface, which is the kind of coincidence Topic 2.10 explains.
A scope note: the CED does not ask for a tunnel period under Topic 2.6. What it gives you is $F = -kr$ , and recognising that form is the transferable skill. The oscillation itself belongs to Unit 7.

$F = -\dfrac{4}{3}\pi G\rho m r = -\dfrac{mg}{R}r$ , so $k = mg/R$ . The motion is simple harmonic with $T = 2\pi\sqrt{R/g} = 5.08\times 10^3$ s, about 84.6 minutes, independent of the object's mass, and the maximum speed at the centre is $\sqrt{gR} = 7.92\times 10^3$ m/s.

Field strength above and below the surface

For the same uniform planet, with surface field $9.8$ N/kg at radius $R$ , find the gravitational field strength at $r = 1.5R$ , at $r = 3R$ , and at a depth of $R/2$ below the surface. Then explain why two of those answers are close in size for completely different reasons, and sketch the field against $r$ .

Split the problem at the surface. Outside, 2.6.A.2.i gives $g(r) = GM/r^2$ , an inverse square in $r$ . Inside, 2.6.E.3 gives a force proportional to $r$ , so dividing by $m$ gives a field proportional to $r$ .
Write both in terms of the surface value $g_s = GM/R^2 = 9.8$ N/kg. Outside: $g(r) = g_s\left(\dfrac{R}{r}\right)^2$ . Inside: $g(r) = g_s\dfrac{r}{R}$ . Both give $g_s$ at $r = R$ , which is the continuity check.
At $r = 1.5R$ , outside: $g = 9.8/(1.5)^2 = 9.8/2.25 = 4.36$ N/kg.
At $r = 3R$ , outside: $g = 9.8/9 = 1.09$ N/kg.
At a depth of $R/2$ , so $r = 0.5R$ , inside: $g = 9.8(0.5) = 4.90$ N/kg.
Now the comparison. Going up to $1.5R$ gives 4.36 N/kg and going down to $0.5R$ gives 4.90 N/kg, which are within about 11 percent of each other, but the physics behind them is different. Above the surface the whole planet still pulls and the distance has grown, so the field falls as $1/r^2$ . Below the surface the distance has shrunk but most of the planet no longer counts: by 2.6.E.2.i the shells outside contribute nothing, and by 2.6.E.2.iv only $\rho\frac{4}{3}\pi(0.5R)^3$ , one eighth of the planet's mass, is doing anything.
Check that eighth explicitly. With one eighth the mass at half the distance, the inverse square gives $\frac{1}{8}\times\frac{1}{(1/2)^2} = \frac{1}{8}\times 4 = \frac{1}{2}$ of the surface field, so 4.90 N/kg. That agrees with the linear formula, as it must.
The sketch that skill 1.C wants: a straight line rising from the origin to the point $(R, 9.8)$ , then a curve falling away as $1/r^2$ , passing through $(1.5R, 4.36)$ and $(3R, 1.09)$ and approaching zero without reaching it. The maximum is at the surface, and the graph has a kink there, since the slope is positive just inside and negative just outside.
One prediction to take from the shape: the field is largest at the surface of a uniform planet, not at its centre. At the centre it is zero, because 2.6.E.2.i applies to every shell around you at once.

$g(1.5R) = 4.36$ N/kg, $g(3R) = 1.09$ N/kg, and $g(0.5R) = 4.90$ N/kg. Outside the planet the field falls as the inverse square of $r$ ; inside it falls linearly, because only the enclosed mass contributes. The field is largest at the surface and zero at the centre.

Frequently asked questions

What is Newton's shell theorem in AP Physics C?

Essential knowledge 2.6.E.2 of the AP Physics C: Mechanics framework says Newton's shell theorem describes the net gravitational force exerted on an object by a uniform spherical shell of mass, and gives three results. Statement 2.6.E.2.i: the net gravitational force on an object inside a thin spherical shell is zero. Statement 2.6.E.2.ii: the force on an object outside a thin spherical shell can be determined by treating the shell as a single massive object located at the centre of the shell. Statement 2.6.E.2.iii: an object inside a sphere of uniform density experiences a net gravitational force from only a partial mass of the sphere. The boundary statement under this topic says the course does not expect students to mathematically prove or derive the theorem, so you apply these results rather than deriving them.

What is the gravitational force inside a uniform sphere?

It is proportional to your distance from the centre, not to the inverse square of it. Essential knowledge 2.6.E.3 states that the gravitational force exerted on an object within a uniform sphere can be shown to be proportional to the object's distance from the sphere's centre, with the derived equation that the partial gravitational force equals negative k times the partial radius. The reason is that only the enclosed mass contributes, and essential knowledge 2.6.E.2.iv gives that mass as the density times four thirds pi times the partial radius cubed. Three powers of r in the enclosed mass against two in the inverse square leaves one power of r.

Does AP Physics 1 cover the gravitational field inside a planet?

No. AP Physics 1's Topic 2.6 runs from learning objective 2.6.A through 2.6.D and stops there. AP Physics C: Mechanics adds a fifth learning objective, 2.6.E, describe the gravitational force exerted on an object by a uniform spherical distribution of mass, which has no counterpart anywhere in the AP Physics 1 framework. That objective is where the shell theorem, the partial-mass equation and the linear force inside a uniform sphere live. It is the single largest difference between the two versions of this topic, and it makes Topic 2.6 the largest topic in AP Physics C: Mechanics Unit 2 by objective count.

Is g 9.8 or 10 in AP Physics C Mechanics?

Both numbers appear in the same course and exam description and neither is a misprint. Essential knowledge 2.6.B.2 states that near the surface of Earth the strength of the gravitational field is approximately 10 newtons per kilogram, which is the framework's estimation value. The constants box on the AP Physics C: Mechanics Table of Information, the sheet handed to students in the exam room, prints the magnitude of the acceleration due to gravity at Earth's surface as 9.8 metres per second squared and the magnitude of the gravitational field strength at Earth's surface as 9.8 newtons per kilogram. This site uses 9.8 throughout, since that is the printed sheet value.

What is the difference between weight and apparent weight?

Weight is the gravitational force. Essential knowledge 2.6.A.3 defines it as the gravitational force exerted by an astronomical body on a relatively small nearby object, with the derived equation weight equals mg. Apparent weight is a different force entirely: essential knowledge 2.6.C.1 says the magnitude of a system's apparent weight is the magnitude of the normal force exerted on the system, which is what a scale reads. Statement 2.6.C.2 adds that if the system is accelerating, the apparent weight is not equal to the gravitational force. The two coincide only when there is no vertical acceleration.

Why do astronauts float if gravity still acts on them?

Because apparent weight is the normal force, and in free fall there is no normal force. Essential knowledge 2.6.C.3 of the AP Physics C: Mechanics framework says a system appears weightless when there are no forces exerted on the system or when the force of gravity is the only force exerted on the system. The second clause is the one that applies in orbit: the gravitational field a few hundred kilometres up is most of its surface value, and the station and everything in it are all accelerating together under it. With nothing pushing back, the normal force is zero and so is the apparent weight.

Are inertial mass and gravitational mass the same thing?

They are logically different properties that turn out to have the same value. Essential knowledge 2.6.D.1 defines inertial mass as the property that determines how much an object's motion resists changes when interacting with another object, which is the mass in Newton's second law. Essential knowledge 2.6.D.2 says gravitational mass is related to the force of attraction between two systems with mass, which is the mass in the law of universal gravitation. Statement 2.6.D.3 says the two have been experimentally verified to be equivalent. That equivalence is why all objects fall with the same acceleration in a given gravitational field, and it is the empirical basis of the equivalence principle stated at 2.6.C.4.