Topic 2.6: Gravitational Force

Two descriptions of one force

Topic 2.6 holds both gravity equations, and part of the work is knowing which one a question wants.

Newton's law of universal gravitation is the general statement. It 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' centers of mass:

The local form is that same force evaluated near Earth's surface, where the distance to Earth's center barely changes over the height of a lab, so the force is effectively constant:

The CED calls the first a relevant equation and the second a derived equation, and the distinction shows up on exam day: only the universal form is printed on the AP Physics 1 equation sheet as a force equation. The $mg$ product does appear there inside the potential energy line $\Delta U_g = mg \Delta y$, but the statement that weight equals $mg$ is one you write down yourself.

Learning objective 2.6.A asks you to describe the gravitational interaction between two objects or systems with mass. Objective 2.6.B asks when the force can be treated as constant, which is what licenses the second equation. Objectives 2.6.C and 2.6.D then cover apparent weight and the two meanings of mass.

Newton's law of universal gravitation

Four claims come with EK 2.6.A.1 and its sub-statements, and each is testable on its own.

• Attractive. Gravity only pulls. There is no repulsive case, which is the structural difference from Coulomb's law, where like charges repel.
• Along the center line. The gravitational force is always exerted along the line connecting the centers of mass of the two interacting systems.
• Acting at the center of mass. The gravitational force on a system can be considered to be exerted on the system's center of mass. That is why a free-body diagram shows one weight arrow leaving a single point rather than a fringe of arrows spread over the whole object.
• Mutual. The two masses enter symmetrically, so the pull on each object has the same magnitude and the opposite direction. That is a Newton's third law pair. The smaller object is not pulled less hard; it simply accelerates more, because the same force divided by a smaller mass gives a larger acceleration.

The constant is supplied on the Table of Information printed with the exam: $G = 6.67 \times 10^{-11}\ \mathrm{N \cdot m^2 / kg^2}$. And $r$ is measured center to center, not surface to surface, which is the detail that decides most setup errors.

The inverse square, and predicting factors of change

Suggested skill 2.D for this topic is predicting new values or factors of change using functional dependence between variables, and the form of the equation makes gravity a natural place to ask for it. The force scales linearly with each mass and as the inverse square of the separation.

The last row is worth checking by hand: the masses contribute a factor of 4 and the distance contributes a factor of one quarter, and the two cancel.

One caution about $r$. Because the distance runs center to center, lifting an object 100 m above the ground barely changes it. Earth's center is thousands of kilometers away, so the ratio shifts by far less than a percent and the force is indistinguishable from what it was. That is not an accident; it is precisely the condition for the constant-gravity approximation in the next section.

The gravitational field

EK 2.6.A.2 introduces the field idea, which the course reuses later for electricity: a field models the effects of a noncontact force exerted on an object at various positions in space.

The magnitude of the gravitational field created by a system of mass $M$ at a point in space is the ratio of the gravitational force exerted by the system on a test object of mass $m$ to the mass of that test object. As a derived equation:

Watch which mass survives. The test mass cancels, so the field belongs to the source $M$ and to the location, not to whatever you put there. That is the whole point of a field: describe once what Earth does to the space around it, then get the force on any object by multiplying by that object's mass.

EK 2.6.A.2.ii ties the field back to motion. If the gravitational force is the only force exerted on an object, the observed acceleration of the object, in $\mathrm{m/s^2}$, is numerically equal to the magnitude of the gravitational field strength, in N/kg, at that location. The two units are the same combination written differently, since $1\ \mathrm{N} = 1\ \mathrm{kg \cdot m/s^2}$ and therefore $1\ \mathrm{N/kg} = 1\ \mathrm{m/s^2}$. The Table of Information lists both readings of $g$ side by side for exactly that reason. The condition still matters: add a second force, such as air resistance or a normal force, and the acceleration no longer matches the field.

Weight, and the numbers for g

EK 2.6.A.3 defines the word. The gravitational force exerted by an astronomical body on a relatively small nearby object is called weight, with the derived equation $\text{Weight} = F_g = mg$.

EK 2.6.B.1 says when you are allowed to use it. If the gravitational force between two systems' centers of mass has a negligible change as the relative position of the two systems changes, the gravitational force can be considered constant at all points between the initial and final positions of the systems. Over a lab bench, a stairwell, or a cliff, that holds comfortably. Over an orbital radius it fails, and you return to the universal form.

Now the numbers, because three values of $g$ appear in the official materials and they do not conflict once you see what each is for.

• EK 2.6.B.2 states that near the surface of Earth, the strength of the gravitational field is $g \approx 10$ N/kg.
• A boundary statement under Topic 1.3 says that for all situations in which a numerical quantity is required for $g$, the value $g \approx 10\ \mathrm{m/s^2}$ will be used, and that students will not be penalized for correctly using the more precise commonly accepted values of $g = 9.81\ \mathrm{m/s^2}$ or $g = 9.8\ \mathrm{m/s^2}$.
• The Table of Information printed with the exam gives $g = 9.8\ \mathrm{m/s^2}$ and $g = 9.8$ N/kg.

So the exam's own working number is 10, the reference table's number is 9.8, and 9.8 or 9.81 also earn credit. Every worked answer on this site is computed with $g = 9.8\ \mathrm{m/s^2}$, which lands about two percent below what the 10 shortcut gives. Say which value you used and hold it for the whole problem.

Apparent weight

Learning objective 2.6.C asks for the conditions under which the magnitude of a system's apparent weight differs from the magnitude of the gravitational force on it, and the CED answers with a definition that catches most students out.

The magnitude of the apparent weight of a system is the magnitude of the normal force exerted on the system. A bathroom scale never measures the gravitational force on you. It measures how hard it is pushing up, and reports that. The two agree only when they happen to be equal.

• Acceleration separates them. EK 2.6.C.2 states it flatly: if the system is accelerating, its apparent weight is not equal to the magnitude of the gravitational force exerted on it. In an elevator accelerating upward the floor has to balance gravity and supply the upward net force, so the normal force exceeds $mg$ and the scale reads high.
• Weightless does not mean gravity free. 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. An astronaut in orbit is in the second case: gravity is acting, hard, and nothing else is, so no surface pushes on her and her apparent weight is zero.
• The equivalence principle. 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 cabin, you cannot tell a floor that pushes because you are sitting on a planet from a floor that pushes because the ship is accelerating.

The arithmetic in these cases is normal-force arithmetic; how to find normal force works through the standard setups.

Inertial mass and gravitational mass

Learning objective 2.6.D asks you to describe two ideas that share a symbol and a unit but not a definition.

Inertial mass, or inertia, is a property that determines how much an object's motion resists changes when interacting with another object. It is the $m$ in $\vec{a}_{\text{sys}} = \vec{F}_{\text{net}} / m_{\text{sys}}$, and you measure it by pushing something and watching how reluctantly it speeds up.

Gravitational mass is related to the force of attraction between two systems with mass. It is the $m_1$ and $m_2$ in the universal law, and you measure it by weighing.

Nothing in the structure of the two laws demands that the same number serve both roles, and yet inertial mass and gravitational mass have been experimentally verified to be equivalent. One consequence falls out in a single line. For an object in free fall the only force is gravity, so

and if the two masses are the same number they cancel, leaving $a = G M / r^2$ with no reference to the falling object at all. That is why a heavy object and a light one, released together with air resistance negligible, fall with the same acceleration, and it is the same cancellation that makes EK 2.6.A.2.ii true.

How Topic 2.6 is tested, and what to do next

Four suggested skills accompany Topic 2.6 in the CED: 1.A, create diagrams, tables, charts, or schematics to represent physical situations; 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.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws. Practice 1 supplies 1.A, Practice 2 supplies 2.A and 2.D, and Practice 3 supplies 3.C, the three practices being Creating Representations, Mathematical Routines, and Scientific Questioning and Argumentation.

Notice what the list leaves out: 2.B, the calculate-an-unknown skill, which several other topics in this unit do list. The emphasis here falls on representing, deriving, scaling, and arguing, which matches the shape of the content. A ratio question that never needs the value of $G$ fits this topic more naturally than a plug-in does.

Unit 2 is weighted at 18 to 23 percent of the multiple-choice section, with about 22 to 27 class periods suggested, a share matched in this course only by Unit 3. From here: Newton's second law is what you feed a gravitational force into, gravity supplies the centripetal force for orbits in Topic 2.9 and in the centripetal force guide, and the AP Physics 1 equation sheet shows exactly which gravity equations are printed for you.