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: Fg = G(m1 m_2)/r^2.

Universal Gravitational Constant (G)

The constant of proportionality in Newton’s gravitation law; approximately 6.67 × 10^(-11) N·m^2/kg^2.

Inverse-Square Law

A dependence where a quantity decreases as 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 Fg = G(m1 m_2)/r^2 (not distance to a surface).

Gravitational Force Magnitude (F_g)

The size of the gravitational attraction between two masses: Fg = G(m1 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: \vec{F}{2\leftarrow 1} = -G(m1 m2/r^2)\hat{r}{2\leftarrow 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: \vec{F}{net} = \sumi \vec{F}_i.

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: \vec{g} = \vec{F}_g/m.

Gravitational Field Strength Units

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

Field Due to a Point Mass

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

Near-Earth Gravitational Acceleration (g_0)

The gravitational field magnitude at Earth’s surface: g0 = GME/R_E^2 ≈ 9.8 m/s^2.

Free Fall (a = g)

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

Superposition Principle (Fields)

Net gravitational field from multiple masses is the vector sum of fields: \vec{g}{net} = \sumi \vec{g}_i (often easier since test mass cancels).

Weight (Gravitational Force on a Mass)

The gravitational force on an object; near Earth Fg ≈ mg (approximately constant g), but in general Fg(r) = 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 r1 to r2 in the field of mass M: Wg = GMm(1/r2 − 1/r1); outward motion (r2 > r1) makes Wg negative.

Gravitational Potential Energy (U)

With the reference U(∞)=0, the potential energy of mass m at distance r from mass M is U(r) = −GMm/r (negative for bound systems).

Gravitational Potential (V)

Potential energy per unit mass: V(r) = U/m = −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 ≪ RE, the exact gravitational potential energy change reduces to ΔU ≈ m g0 h (since g is approximately constant).