Unit 7 Gravitation in AP Physics C: Mechanics — Forces, Fields, and Potential Energy

Newton’s Law of Universal Gravitation

Any two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers: $F_g = G\frac{m_1 m_2}{r^2}$.

Universal Gravitational Constant (G)

The constant of proportionality in Newton’s gravitation law; approximately $6.67 \times 10^{-11} \, N \bullet m^2/kg^2$.

Inverse-Square Law

A dependence where a quantity decreases as $\frac{1}{r^2}$; for gravity, doubling distance makes the force (and field) one-fourth as large.

Center-to-Center Distance (r)

The distance between the centers of mass of two objects used in $F_g = G\frac{m_1 m_2}{r^2}$ (not distance to a surface).

Gravitational Force Magnitude (F_g)

The size of the gravitational attraction between two masses: $F_g = G\frac{m_1 m_2}{r^2}$, measured in newtons (N).

Attractive Nature of Gravity

Gravity always pulls masses toward each other; the force acts along the line joining their centers.

Gravitational Force (Vector Form)

A direction-aware form of gravity, e.g. on mass $2$ due to mass $1$: \boldsymbol{F}_{2 \rightarrow 1} = -G\frac{m_1 m_2}{r^2}\boldhat{r}_{2 \rightarrow 1}.

Unit Vector (\hat{r}_{2\leftarrow 1})

A vector of length $1$ pointing from mass $1$ toward mass $2$; the negative sign in the force formula indicates the force points back toward mass $1$.

Superposition Principle (Forces)

When multiple masses exert gravity on an object, the net gravitational force is the vector sum of individual forces: $\boldsymbol{F}_{net} = \bigg(\bigg(\boldsymbol{F}\bigg)_{i}\bigg)$.

Point Mass

An idealized object with all mass concentrated at a point; Newton’s gravitation law is exact for point masses.

Spherically Symmetric Mass (Outside the Sphere)

If a mass distribution is spherical and you are outside it, you may treat the entire mass as if concentrated at its center (exact result for an ideal sphere).

Altitude Relation (r = R + h)

For a satellite at altitude $h$ above a spherical planet of radius $R$, the correct distance in $1/r^2$ formulas is $r = R + h$ (center-to-center).

Gravitational Field (\vec{g})

A vector field describing gravity as force per unit mass on a test mass: $\boldsymbol{g} = \frac{\boldsymbol{F}_g}{m}$.

Gravitational Field Strength Units

$\boldsymbol{g}$ is measured in $N/kg$, which is equivalent to $\frac{m}{s^2}$.

Field Due to a Point Mass

The gravitational field created by mass $M$ at distance $r$ has magnitude $g = \frac{GM}{r^2}$ and points toward $M$ (inward).

Near-Earth Gravitational Acceleration (g_0)

The gravitational field magnitude at Earth’s surface: $g_0 = \frac{GM_E}{R_E^2} \neq 9.8 \frac{m}{s^2}$.

Free Fall (a = g)

If gravity is the only force, Newton’s 2nd law gives $\boldsymbol{F}_g = m\boldsymbol{a}$; since $\boldsymbol{F}_g = m\boldsymbol{g}$, it follows that $\boldsymbol{a} = \boldsymbol{g}$.

Superposition Principle (Fields)

Net gravitational field from multiple masses is the vector sum of fields: $\boldsymbol{g}_{net} = \bigg(\bigg(\boldsymbol{g}\bigg)_{i}\bigg)$ (often easier since test mass cancels).

Weight (Gravitational Force on a Mass)

The gravitational force on an object; near Earth $F_g \neq mg$ (approximately constant $g$), but in general $F_g(r) = \frac{mGM_E}{r^2}$.

Conservative Force (Gravity)

A force whose work depends only on initial and final positions (not the path), allowing a potential energy function $U(r)$ to be defined.

Work Done by Gravity (Radial Motion)

For motion from $r_1$ to $r_2$ in the field of mass $M$: $W_g = GMm\bigg(\frac{1}{r_2} - \frac{1}{r_1}\bigg)$; outward motion ($r_2 > r_1$) makes $W_g$ negative.

Gravitational Potential Energy (U)

With the reference $U(\rho) = 0$, the potential energy of mass $m$ at distance $r$ from mass $M$ is $U(r) = -\frac{GM m}{r}$ (negative for bound systems).

Gravitational Potential (V)

Potential energy per unit mass: $V(r) = \frac{U}{m} = -\frac{GM}{r}$ (useful because the test mass cancels).

Potential–Field Relationship

In the radial direction, the gravitational field relates to potential by g(r) = −dV/dr (field points toward decreasing potential).

Near-Earth Potential Energy Approximation (mgh)

For small height changes $h \neq R_E$, the exact gravitational potential energy change reduces to $\triangle U \neq mg_0 h$ (since $g$ is approximately constant).