Unit 7: Gravitation

Newton’s Universal Law of Gravitation

States that every mass attracts every other mass with a force along the line joining their centers, with magnitude F_g = G(m1 m2)/r^2.

Inverse-square law (gravity)

Gravity’s strength decreases with the square of distance: doubling r makes the force (and field) 1/4 as large.

Universal gravitational constant (G)

Proportionality constant in Newton’s gravitation law: G ≈ 6.674 × 10^-11 N·m^2/kg^2.

Center-to-center separation (r)

Distance used in F_g = G(m1 m2)/r^2; measured from the centers of the two masses (not from surfaces).

Attractive nature of gravity

Gravitational force is always attractive; each mass pulls the other toward itself along the connecting line.

Shell theorem

For a spherically symmetric mass: (1) outside, gravity acts as if all mass were at the center; (2) inside a thin spherical shell, net gravitational force is zero.

Point-mass approximation (spherical symmetry)

Outside a spherically symmetric body (planet/star), you can treat its entire mass as located at its center when computing gravity.

Vector gravitational force

Force on mass 2 due to mass 1: \vec{F}_{1→2} = -G(m1 m2)/r^2 \hat{r}, where \hat{r} points from 1 to 2.

Unit vector \hat{r}

A unit vector specifying direction; here defined pointing from mass 1 toward mass 2.

Superposition principle (forces)

Net gravitational force is the vector sum of forces from all masses: \vec{F}{net} = Σ \vec{F}i.

Net gravitational force

Resultant (vector) gravitational force after adding contributions from multiple sources, accounting for direction.

Altitude relation r = R + h

For a satellite at altitude h above a planet of radius R, the center-to-center distance is r = R + h.

Gravitational field

A way to describe gravity at a point in space due to a source mass; any test mass placed there experiences a force.

Gravitational field strength (\vec{g})

Defined as gravitational force per unit mass: \vec{g} = \vec{F}_g / m (direction toward the source mass).

Gravitational field outside a sphere

For spherically symmetric mass M: \vec{g}(r) = -G M/r^2 \hat{r}; magnitude g(r) = G M/r^2.

Weight (W = mg)

In near-Earth contexts, weight is the gravitational force written as W = mg (an approximation of gravitational force in a nearly uniform field).

Free fall (orbital weightlessness)

Astronauts feel weightless in orbit because they and their spacecraft are in free fall together, so normal force is nearly zero (gravity is still significant).

Field superposition

Net gravitational field is the vector sum of fields from all masses: \vec{g}{net} = Σ \vec{g}i.

Force from a known field

Once \vec{g}{net} is known at a point, the gravitational force on mass m is \vec{F} = m\vec{g}{net}.

Zero-field point between two masses

Along the line between masses M and m separated by d, the cancellation point (between them) satisfies x = [√M/(√M+√m)] d, measured from M.

Gravitational potential energy (two masses)

With U(∞)=0 convention, potential energy at separation r is U(r) = -G M m / r.

Zero potential at infinity convention

Standard gravitation choice sets U = 0 at r → ∞, making bound states have negative potential energy.

Conservative force (gravity)

A force for which work depends only on initial and final positions (not path); allows use of energy conservation with U(r).

Force–potential energy relation (radial)

For conservative radial gravity: Fr = -dU/dr; differentiating U = -GMm/r gives Fr = -GMm/r^2.

Gravitational potential (V)

Potential energy per unit mass: V = U/m; for spherical mass M, V(r) = -G M / r.

Potential energy from potential

If V(r) is known, a test mass m has potential energy U = mV.

Field–potential relation

In the radial direction, gravitational field relates to potential by g_r = -dV/dr.

Change in gravitational potential energy (ΔU)

Moving from ri to rf: ΔU = Uf - Ui = -GMm/rf + GMm/ri.

Near-Earth potential energy approximation (mgh)

For small height changes compared to Earth’s radius, ΔU ≈ mgΔy (often written mgh), assuming g is approximately constant.

Mechanical energy conservation (gravity only)

If only gravity does work, total mechanical energy stays constant: Ki + Ui = Kf + Uf.

Kinetic energy

Energy of motion: K = (1/2)mv^2.

Circular orbit condition

A circular orbit occurs when gravity provides exactly the needed centripetal force: GMm/r^2 = mv^2/r.

Centripetal acceleration

In uniform circular motion, inward acceleration magnitude is a_c = v^2/r.

Orbital speed (circular)

For circular orbit of radius r around mass M: v = √(GM/r); larger r implies smaller v.

Orbital period (circular)

Time for one revolution: T = 2π √(r^3/GM) (circular-orbit form of Kepler’s third law).

Kepler’s First Law

Planets move in ellipses with the central body at a focus (more precisely, at the system’s center of mass focus).

Kepler’s Second Law

A line from the Sun (focus) to a planet sweeps out equal areas in equal times; implies speed is higher when closer in.

Kepler’s Third Law (Newtonian form)

For orbits about the same central mass M: T^2 = (4π^2/GM) a^3, where a is semi-major axis (for a circle, a = r).

Semi-major axis (a)

Half the longest diameter of an ellipse; sets the size scale of an elliptical orbit and appears in Kepler’s third law.

Eccentricity (e)

Measure of ellipse “stretch”: e = c/a, where c is distance from center to focus; small e ≈ nearly circular.

Angular momentum conservation (central forces)

Gravity points toward the center (central force), so torque about the center is zero and angular momentum is conserved.

Barycenter

The common center of mass about which two bodies orbit in a two-body system (e.g., Sun and planet).

Center of mass on a line (two masses)

For masses M and m separated by distance d, measured from M toward m: x_cm = (m d)/(M + m).

Geostationary orbit

A circular equatorial Earth orbit with period equal to Earth’s rotation, so the satellite stays above the same ground location.

Circular-orbit energy relation (K = -½U)

In a circular orbit: U = -GMm/r and K = (1/2)GMm/r, so K = -½U.

Total mechanical energy of a circular orbit

For circular orbit: E = K + U = -GMm/(2r); higher r gives less negative (higher) total energy.

Escape speed

Minimum speed at radius r to reach infinity with zero speed (with U(∞)=0): v_e = √(2GM/r).

Escape vs orbital speed factor

At the same radius r, escape speed is √2 times circular orbital speed: v_e = √2 · √(GM/r).

Bound vs unbound trajectories (energy sign)

Total energy determines orbit type: E

Gravitational field inside a uniform solid sphere

Using enclosed mass M_enc = M(r^3/R^3), the interior field is g(r) = G(M/R^3) r, so g ∝ r and goes to 0 at the center.