Chapter 8 — Gravitation

Overview: Kepler's laws describe the motions of planets (and other bodies) around the Sun based on observations. They apply to two-body gravitational systems and are a foundation for celestial mechanics.

1. Kepler's First Law (Law of Ellipses): The orbit of every planet is an ellipse with the Sun at one focus.

Ellipse equation (Cartesian): x²/a² + y²/b² = 1, where a = semimajor axis, b = semiminor axis. Eccentricity e = c/a where c is the focal distance; b² = a²(1 - e²). Perihelion and aphelion are the closest and farthest points from the Sun.

2. Kepler's Second Law (Law of Areas): A line joining a planet and the Sun sweeps out equal areas in equal times.

Mathematically: dA/dt = constant. This implies conservation of angular momentum for the planet: L = m r² dθ/dt = constant, so r² dθ/dt = const. Consequence: planets move faster at perihelion and slower at aphelion.

3. Kepler's Third Law (Law of Periods): The square of the orbital period T of a planet is proportional to the cube of the semimajor axis a of its orbit.

Proportional form: T² ∝ a³. For a two-body system (star mass M and planet mass m) the exact relation from Newtonian gravity is:
T² = (4π²/(G (M + m))) a³. For M >> m (planet around the Sun), this simplifies to T² = (4π²/GM) a³.

Connection with Newton's Law: Using F = G M m / r² and centripetal approximation for circular orbits (m v²/r = G M m / r²) and v = 2πr/T leads directly to T² = (4π²/GM) r³, which is Kepler's third law (with r = a for circular motion).

Key consequences & limitations: Kepler's laws describe ideal two-body motion. For systems with more bodies, perturbations alter orbits slightly. Kepler's laws are exact in Newtonian gravity for two-body motion and are generalized in celestial mechanics (orbital elements, perturbation theory) and relativistic corrections (e.g., Mercury's perihelion precession).

Practical importance: orbit prediction, satellite design (orbital elements use Keplerian parameters), mission planning (Hohmann transfers), and measuring masses of stars/planets from orbital periods and semimajor axes.

Statement: Every particle of matter in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. In scalar form: F = G m1 m2 / r^2, directed along the line joining the two masses.

Vector form: F_{12} = -G (m1 m2 / r^2) r̂_{12}, where r̂_{12} is the unit vector from mass 1 to mass 2. The minus sign shows the force is attractive.

Universal gravitational constant: G = 6.67430 × 10^{-11} N·m^2/kg^2 (measured by the Cavendish experiment).

Important derived quantities: gravitational field (or acceleration due to gravity) at distance r from a mass M is g(r) = GM / r^2 (vector g points toward M). Gravitational potential (per unit mass) is V(r) = -GM / r and gravitational potential energy of masses M and m is U(r) = -GMm / r.

Key features:

• Universal: applies to all masses anywhere in the universe.
• Always attractive.
• Long-range and follows an inverse-square law (intensity falls as 1/r^2).
• Superposition principle: total gravitational force on a mass is the vector sum of forces from all other masses.

Applications and consequences: explains planetary orbits and Kepler's laws (for circular orbits equating centripetal force to gravity gives orbital speed v = sqrt(GM/r) and period T satisfies T^2 = (4π^2/GM) r^3), escape velocity v_{esc} = sqrt(2GM / R), tides (differential gravity of Moon and Sun), behaviour of satellites, variation of weight with altitude, and allows determination of Earth’s mass and density from G and g.

Limitations: Newton's law is extremely accurate for most everyday and astronomical problems but is replaced by General Relativity when very strong gravitational fields, high precision or relativistic speeds are involved.

Gravitational force between extended bodies is obtained by applying Newton's law of gravitation to every mass element and using superposition (integration). For point masses Newton's law is F = G m1 m2 / r². For extended bodies we sum (integrate) the contributions of all infinitesimal elements dm of the two bodies.

General expression (vector form): If body 1 has mass distribution ρ₁(r₁) and body 2 has ρ₂(r₂), the gravitational force on body 2 by body 1 is

F = G ∬_V1,V2 (ρ₁(r₁) ρ₂(r₂) (r₂ − r₁) / |r₂ − r₁|³) dV₁ dV₂

This double volume integral is often simplified using symmetry.

Shell theorem (useful results for spherical symmetry):

• A spherically symmetric shell of mass attracts a particle outside as if all mass were concentrated at the center (force ∝ 1/r² for r > R).
• Inside a thin spherical shell (r < R) the net gravitational force is zero.
• For a uniform solid sphere of total mass M and radius R: for r ≥ R the field is identical to a point mass, g(r) = GM/r²; for r ≤ R the field varies linearly with r, g(r) = GMr/R³ (i.e. F ∝ r).

Practical approach: If the separation between two extended bodies is large compared with their sizes and they are approximately spherically symmetric, treat each as a point mass at its center of mass. If not, set up and evaluate the necessary integrals or use numerical methods.

Important physical consequences:

• Weight variation with depth inside Earth: roughly linear decrease of g with depth (ignoring density variations).
• Spherical shells/hollow planets produce zero gravity inside (thought experiments and engineering implications for cavities).
• Gravitational interaction of non-spherical objects (rod, disk, ring) requires integration and gives characteristic field patterns (e.g., field on the axis of a ring).

Definition: The gravitational field (or gravitational field intensity) at a point in space is defined as the gravitational force experienced by a unit test mass placed at that point. If a mass m experiences a gravitational force F, the field intensity g is

g = F / m (direction: toward the attracting mass). Units: N kg⁻¹ (same as m s⁻²).

Gravitational field of a point mass: For a point mass M located at the origin, Newton's law of gravitation gives the magnitude

g(r) = GM / r²,

directed radially inward. Vector form: g = - GM r̂ / r², where r̂ is the unit vector from the source mass to the field point. G is the universal gravitational constant.

Field near Earth's surface: Treating Earth as a sphere of mass M_E and radius R_E, at the surface

g = GM_E / R_E² ≈ 9.8 m s⁻².

For small heights h above the surface (h << R_E),

g(h) = GM_E / (R_E + h)² ≈ g(1 - 2h / R_E) (first-order approximation).

Superposition: Gravitational fields from multiple masses add vectorially. For discrete masses M_i at positions r_i,

g_total(r) = Σ (-G M_i (r - r_i) / |r - r_i|³).

Special results for spherically symmetric mass distributions (important CBSE results):

• Outside a spherically symmetric mass distribution (radius R), the field is the same as if all mass were concentrated at the center: g = GM / r² for r ≥ R.
• Inside a thin spherical shell of mass, the gravitational field is zero everywhere inside the shell.
• Inside a uniform solid sphere of total mass M and radius R, at distance r from center (r ≤ R): g(r) = G M r / R³. Thus g increases linearly with r inside and reaches maximum at the surface, then falls as 1/r² outside.

Relation with gravitational potential V: The gravitational field is the negative gradient of the gravitational potential V: g = -∇V. For a point mass V(r) = -GM / r, so g = -dV/dr = GM / r² (directed inward).

Direction and sign: Gravitational field intensity is a vector that points toward masses (attractive). When using scalar magnitude g we usually quote a positive number and state the inward direction explicitly.

Properties (summary):

• Vector quantity, units N kg⁻¹ or m s⁻².
• Obeys superposition.
• For spherically symmetric mass: outside behaves as point mass; inside a shell field = 0; inside solid sphere g ∝ r.
• Determines weight: weight W = m g (local g).

Definition: Gravitational potential (V) at a point in a gravitational field is the work done by an external agent per unit mass in bringing a small test mass from infinity (where potential is taken as zero) to that point, without acceleration. It is a scalar quantity.

Mathematical meaning:

• For a point mass M at distance r, V(r) = -GM/r. (Convention: V(∞) = 0.)
• Gravitational potential energy (U) of a mass m placed at that point: U = mV = -GmM/r.
• For a system of point masses: V = -G Σ (mi/ri), where ri is distance from the i-th mass to the point.

Relation to gravitational field:

• The gravitational field (vector) g is the negative gradient of potential: g = -∇V. For radial symmetry, g(r) = -dV/dr (directed inward).
• Because V is scalar, potentials from different sources add algebraically (superposition), which simplifies calculations.

Special cases for spherical mass distributions (total mass M, radius R):

• Outside (r ≥ R): V(r) = -GM/r (same as point mass located at centre).
• Hollow thin spherical shell: inside (r < R) potential is constant: V = -GM/R. The field inside is zero.
• Uniform solid sphere: for r ≤ R, V(r) = -G M (3R^2 - r^2) / (2 R^3). At centre V(0) = -3GM/(2R); at surface r = R gives -GM/R.

Properties and insights:

• V is always negative (for attractive gravity with zero at infinity) for isolated masses; its magnitude increases as you move closer.
• Equipotential surfaces are surfaces of constant V; the gravitational field is always perpendicular to these surfaces. No work is done by gravity when moving along an equipotential.
• Potential difference ΔV between two points is the work done per unit mass in moving between them: ΔV = V(B) - V(A) = -∫_A^B g·dr.

Why negative? Because gravitational force is attractive, bringing a mass from infinity to a finite r releases energy (work is done by the field), so potential (work done by external agent) is negative relative to zero at infinity.

Definition: Gravitational potential energy (GPE) of a system of two masses is the energy associated with their relative position in a gravitational field. For a test mass m in the gravitational field of a mass M, the gravitational potential energy at distance r (measured from the centre of M) is defined with the zero of potential energy at infinity:

U(r) = -G M m / r.

Why the negative sign? The negative sign indicates that bound configurations (finite r) have less energy than the separated state (r → ∞, U = 0). Work must be done against gravity to move the masses apart to infinity.

Relation to work and conservative force: Gravity is a conservative force, so the change in gravitational potential energy when moving from r1 to r2 is the negative of the work done by the gravitational force:

ΔU = U(r2) − U(r1) = − ∫_{r1}^{r2} F · dr.

For radial motion under gravity (magnitude of force F = GMm/r^2 directed inward), the work done by gravity when moving from r1 to r2 is W_by = GMm(1/r2 − 1/r1), and hence U(r) = −GMm/r satisfies ΔU = −W_by.

Approximation near Earth's surface: If the mass m is at height h above Earth of radius R (h << R), expand U(R + h) to get

U(R + h) ≈ −GMm/R + m g h

Because the constant term −GMm/R is the same for all heights measured from the surface, changes in GPE near the surface are given by the familiar expression

ΔU ≈ m g h.

Key physical points:

• The zero of gravitational potential energy is chosen by convention (commonly U = 0 at r = ∞). Only differences in U are physically measurable.
• GPE is negative for bound two-body systems when zero is set at infinity.
• The spatial derivative of U gives the force: F_r = −dU/dr. For U = −GMm/r, this yields the inverse-square law magnitude GMm/r^2 directed inward.
• Units: joule (J).

What is g?
Acceleration due to gravity (g) at a point near Earth is the acceleration that a small test mass would experience solely due to Earth's gravitational pull. Its SI unit is m/s². Near the Earth's surface the average value is g ≈ 9.8 m/s².

Origin (Newton’s law)
From Newton's law of gravitation, the gravitational force on a mass m due to Earth of mass M and centre-to-point distance r is F = G M m / r². The acceleration (force per unit mass) is therefore

g(r) = G M / r².

At the Earth's surface (r = R), g = G M / R². Using M ≈ 5.972×10²⁴ kg, R ≈ 6.371×10⁶ m and G ≈ 6.674×10⁻¹¹ N·m²/kg² gives g ≈ 9.8 m/s².

Variation with height (h above surface)
At altitude h above Earth’s surface the distance is r = R + h, so

g(h) = G M / (R + h)² = g₀ (R²/(R + h)²).

For small heights (h << R) a binomial approximation gives

g(h) ≈ g₀ (1 − 2h/R).

Variation with depth (d below surface)
If Earth is approximated as a uniform sphere, the gravitational acceleration at depth d (distance r = R − d from centre) comes only from the mass enclosed within radius r, giving

g(d) = g₀ (1 − d/R).

Thus g decreases approximately linearly with depth and goes to zero at the centre (d = R).

Effect of Earth’s rotation and latitude
Because Earth rotates, there is a centrifugal acceleration that reduces the effective gravity. Its vertical component at latitude φ is ω² R cos²φ, so the effective gravity is approximately

g_eff(φ) = g(φ) − ω² R cos²φ,

where ω ≈ 7.2921×10⁻5 rad/s. This makes g slightly smaller at the equator (~9.780 m/s²) than at the poles (~9.832 m/s²).

Measurement
g can be measured by timing free fall (s = ½ g t²), by a simple pendulum (T = 2π √(L/g)), or by gravimeters which detect tiny changes in g.

Important distinctions
g is the gravitational field (acceleration) near Earth. It depends on location (height, depth, latitude) and not on the test mass. It should not be confused with G (universal gravitational constant) or with weight W = m g (which depends on the test mass m).

Overview

Gravitational acceleration g at a point near Earth depends on the distance from Earth's centre. As you go up (altitude) g decreases because you are farther from the mass producing the field. As you go down into Earth (depth) the effective g is due only to the mass enclosed within that radius, so g decreases toward the centre and becomes zero at the centre (for a spherically symmetric Earth).

Derivation for altitude (above surface)

Let Earth's mass = M and radius = R. At the surface g = GM/R². At height h above surface (distance from centre = R + h):

g(h) = G M / (R + h)² = g × (R/(R + h))².

For small heights h << R, use binomial expansion:

g(h) ≈ g × (1 - 2h/R). (first-order approximation)

Derivation for depth (below surface)

Assume Earth is spherically symmetric. By Gauss's law for gravity (or Newton's shell theorem), only the mass inside radius r contributes to g at distance r from centre. Let depth d be measured below surface so r = R - d.

If density is uniform, M_r = M × (r³/R³). Then

g(r) = G M_r / r² = G M r / R³ = g × (r/R) = g × (1 - d/R).

Thus, for a uniform Earth g decreases linearly with depth and becomes zero at the centre (d = R).

More general form

If density varies with radius ρ(r), the gravitational acceleration at radius r is

g(r) = (G / r²) × ∫₀^r 4π r'² ρ(r') dr'.

Real Earth is not uniform: inner regions are denser, so g actually increases slightly below the surface to a maximum at some depth (~about 3000 km), then decreases to zero at the centre.

Notes and limitations

• Binomial approximation g(h) ≈ g(1 - 2h/R) is valid only for h << R.
• Uniform density assumption is idealized. Real Earth’s density increases with depth.
• Measured apparent gravity at a point also depends on Earth's rotation (centrifugal effect) and latitude; effective g = gravitational g minus centrifugal component (≈ ω² R cos²φ).

Physical consequences

• At mountain tops you weigh slightly less than at sea level; in deep mines you weigh slightly less than at the surface (for a uniform Earth) but the real effect depends on internal density.
• Satellites in low Earth orbit still experience significant g (not zero); they are in free fall, producing weightlessness for occupants.
• At Earth's centre gravitational acceleration is zero (net gravitational forces cancel).

Apparent weight (reading on a scale) is the normal reaction force N that a supporting surface exerts on a body. It is what a weighing scale measures, and need not equal the gravitational force mg. Apparent weight depends on the acceleration of the body (or the frame) relative to an inertial frame.

Basic idea and free-body analysis: For a body of mass m in a noninertial (accelerating) frame or attached to an accelerating support, draw forces: weight mg (down) and normal reaction N (up). Using Newton’s 2nd law in an inertial frame (taking upward as positive):

N − mg = m a ⇒ N = m(g + a)

Here a is the acceleration of the body (upward positive). Special cases:

• At rest or uniform velocity (a = 0): N = mg (true weight = apparent weight).
• Accelerating upward (a > 0): N = m(g + a) → apparent weight increases.
• Accelerating downward (a > 0 downward, i.e. a negative in upward sign convention): N = m(g − |a|) → apparent weight decreases.
• Free fall (a = g downward): N = 0 → weightlessness (scale reads zero).

Weightlessness: Occurs when the support does not exert a normal force (N = 0). That happens in free fall or in orbit when the centripetal acceleration of the body equals gravitational acceleration (objects are in continuous free fall around Earth). Important: gravity is not zero in orbit; the support force is zero, so occupants feel weightless.

Accelerating (non-inertial) frames and pseudo (inertial) forces: When working entirely inside a noninertial frame that accelerates with acceleration a_frame, introduce a pseudo force F_pseudo = −m a_frame acting on each mass, opposite to the frame acceleration. Then static balance in the accelerating frame gives:

N + (−m a_frame) − m g = 0 ⇒ N = m(g + a_frame) (signs depend on chosen axis)

Using consistent sign conventions this yields the same expressions as above. The pseudo force idea helps analyze more complex accelerating systems (e.g., accelerating vehicles, rotating frames).

Rotating frames and centrifugal effect: In a frame rotating with angular speed ω, an additional outward (centrifugal) pseudo acceleration ω²r acts on a mass at radius r (radially outward). The effective gravity (vector sum of real gravity and centrifugal acceleration) determines apparent weight. For a horizontal rotating platform the apparent normal force may increase or decrease depending on orientation of g and centrifugal acceleration.

Key physical points:

• Scale reading = normal reaction N = apparent weight, not necessarily mg.
• Weightlessness means N = 0, not absence of gravity.
• Pseudo forces must be used when applying Newton’s laws inside accelerating frames.
• Apparent weight can be changed deliberately (elevators, rockets, centrifuges) or happen naturally (free fall, orbit).

Overview
Motion under gravity for bodies in space (planets, moons, satellites) is governed by the gravitational force which provides the required centripetal force for circular orbits and determines orbital parameters for more general (elliptical) motion. Key ideas: gravity supplies centripetal acceleration, orbital speed depends on central mass and orbital radius, period follows Kepler's third law, and escape velocity sets the threshold to leave a bound orbit.

Gravitational force as centripetal force
For a small mass m orbiting a large mass M at distance r (center to center), Newton's law of gravitation gives the inward force F = GMm/r². If the orbit is circular with speed v, centripetal force required is m v²/r. Equating them:

• GMm/r² = m v²/r

Thus, the circular orbital speed is v = sqrt(GM/r). Note that v is independent of the orbiting mass m.

Orbital period
The time to complete one revolution (period T) for a circular orbit of radius r is T = 2πr/v. Using v = sqrt(GM/r) gives Kepler's third-law form:

• T = 2π sqrt(r³ / GM)

This shows T ∝ r^3/2 for orbits around the same central mass.

Energy of a circular orbit
Total mechanical energy (kinetic + potential) of mass m in a circular orbit:

• Kinetic energy: K = 1/2 m v² = GMm/(2r)
• Gravitational potential energy: U = - GMm/r
• Total energy: E = K + U = - GMm/(2r) (negative for bound orbits)

Escape velocity
Escape velocity from distance r is found from energy conservation (set final total energy → 0):

• v_esc = sqrt(2GM/r)

Variation of g with altitude
Acceleration due to gravity at distance r from Earth's center is g(r) = GM/r². For height h above Earth's surface (r = R + h):

• g(h) = g_0 (R / (R + h))², where g_0 is surface gravity and R is Earth's radius.

Elliptical orbits and Kepler's laws (brief)
General orbits under inverse-square gravity are conic sections. For bound motion, orbits are ellipses with the central mass at one focus. Kepler's laws summarize the motion: (1) Planets move in ellipses; (2) Equal areas are swept in equal times (angular momentum conservation); (3) T² ∝ a³ (a = semi-major axis).

Apparent weightlessness
Objects in orbit (e.g., astronauts) experience apparent weightlessness because they are in continuous free fall around Earth — gravity acts but produces centripetal acceleration instead of normal force.

Important notes
- Orbital speed decreases with increasing radius (v ∝ 1/√r).
- Geostationary orbit: a circular orbit in Earth's equatorial plane with period 24 h; altitude ≈ 35,786 km above Earth’s surface. Satellite mass does not enter v or T formulas (when M ≫ m).

Definition: The orbital period (T) is the time a body takes to complete one full orbit about another body. Kepler's third law (empirical) states that for planets orbiting the Sun, the square of the orbital period is proportional to the cube of the semi-major axis: T^2 \propto a^3. Newtonian gravitation provides the physical basis and the exact constant of proportionality.

Derivation for circular orbits (Newtonian):

For a small mass m orbiting a large mass M at radius r (circular orbit):
Centripetal force = Gravitational force
m v^2 / r = G M m / r^2
=> v = sqrt(G M / r)
Orbital period T = circumference / speed = 2 π r / v
=> T = 2 π r / sqrt(G M / r) = 2 π sqrt(r^3 / (G M))

Kepler's third law (Newtonian form):

T^2 = (4 π^2 / (G (M + m))) a^3

Here a is the semi-major axis of the relative orbit. For M >> m (e.g., planet around Sun, satellite around Earth) this simplifies to

T^2 = (4 π^2 / (G M)) a^3,

Elliptical orbits: The same relation holds if r (circular radius) is replaced by the semi-major axis a of the ellipse. The period depends only on a (and central mass), not on eccentricity.

Physical meaning and assumptions:

• Assumes inverse-square gravity and two-body motion (point masses or spherically symmetric masses).
• Neglects perturbations from other bodies, atmospheric drag, relativity (important only for very precise cases).
• Shows why more distant objects have much longer periods (T ∝ a^{3/2}).

Using the law: Given a and M, you can compute T. Conversely, measuring T and a for two orbiting objects lets you find the total mass M + m (useful in astronomy to find star masses from binary orbits).

Definition: Escape velocity is the minimum speed that an object must have at the surface (or at a given distance) of a gravitating body so that it can move away indefinitely and reach infinity with zero remaining speed, i.e., just escape the gravitational field without further propulsion.

Derivation (energy conservation): Take gravitational potential energy U = -GMm/r (zero at r → ∞). For a mass m at radius r = R with initial speed v, total energy is E = (1/2)mv^2 - GMm/R. To just escape, the final speed at infinity is zero, so final total energy = 0. Thus

(1/2)mv_esc^2 - GMm/R = 0

Solving gives v_esc = sqrt(2GM/R). Important points:

• Escape velocity is independent of the mass m of the escaping object.
• For escape from altitude h above the surface, use radius r = R + h: v_esc = sqrt(2GM/(R+h)).
• Using surface gravity g = GM/R^2, v_esc can also be written v_esc = sqrt(2gR).

Minimum energy required: The minimum kinetic energy required (at the starting point) for escape equals the magnitude of the gravitational potential energy at that point. From the energy equation,

E_min = (1/2) m v_esc^2 = GMm/R = m g R.

This is the least amount of kinetic energy that must be supplied (ignoring atmosphere, drag, and rotation effects). If the object has more energy than this, it will escape with some residual speed; if less, it will remain bound and follow a closed or elliptical orbit or fall back.

Relationship with orbital speed: Circular orbital speed at the surface (idealized) is v_orb = sqrt(GM/R). Therefore v_esc = sqrt(2) v_orb (escape speed is √2 times the circular speed).

Physical remarks: - In presence of an atmosphere, drag increases required energy. - Planetary rotation helps if the launch is eastward at the equator (reduces required rocket delta-v). - If total energy > 0, the trajectory is unbound (parabolic for exactly zero total energy, hyperbolic for positive energy).

Overview: The gravitational field produced by a mass M exerts a central force on another mass m. This force is conservative, so work done by gravity depends only on initial and final positions (not path) and mechanical energy (kinetic + potential) is conserved when only gravity does work.

Gravitational force (point mass): Two point masses M and m attract each other with

F(r) = G M m / r² directed toward M (radial). In vector form F = -G M m / r² r̂.

Work done by gravity: If the mass m moves quasi-statically from r = r1 to r = r2 (radial motion), the work done by the gravitational force is

W = ∫(r1→r2) F · dr = -G M m (1/r2 - 1/r1) = G M m (1/r1 - 1/r2).

Note sign: gravity does positive work when the object moves closer (r2 < r1) and negative work when moving away.

Gravitational potential energy (GPE): Choose zero of potential energy at infinity. The potential energy of m at distance r from M is

U(r) = - G M m / r.

Change in potential energy from r1 to r2 is ΔU = U(r2) - U(r1) = -G M m (1/r2 - 1/r1). This gives W = -ΔU (work done by gravity equals negative change in potential energy).

Gravitational potential (per unit mass): Φ(r) = U/m = - G M / r. The field (acceleration) is g = -∇Φ, whose magnitude is g(r) = G M / r².

Near Earth's surface (approximation): For heights h small compared with Earth's radius R, expand U(R+h) ≈ -GMm/(R+h) ≈ -GMm/R + mgh, so only changes matter and we commonly use

ΔU ≈ m g Δh,

and take U = mgh (with appropriate zero reference) for practical problems.

Conservation of mechanical energy: If only gravity does work (no non-conservative forces), total mechanical energy E is conserved:

E = K + U = 1/2 m v² + U(r) = constant.

So as U decreases (object moves inward), kinetic energy K increases so that E stays fixed. For a free-falling object from rest at r1 to r2: 1/2 m v² = U(r1) - U(r2) = G M m (1/r2 - 1/r1).

Special results:

• Escape velocity from surface r = R: set final kinetic + potential at infinity to zero: 1/2 m v_e² - G M m / R = 0 ⇒ v_e = sqrt(2 G M / R). For Earth v_e ≈ 11.2 km/s.
• Circular orbital speed at radius r: centripetal requirement gives m v²/r = G M m / r² ⇒ v = sqrt(G M / r).
• Total mechanical energy of a circular orbit: E = - 1/2 G M m / r (negative, bound orbit). Kinetic energy equals half the magnitude of potential energy: K = + 1/2 G M m / r, U = - G M m / r.

Sign conventions and intuition: Gravitational potential energy is negative (with zero at infinity). Moving an object out (increasing r) increases U (less negative) and gravity does negative work; moving it in decreases U (more negative) and gravity does positive work. In closed loop motion under gravity only, net work = 0.

How to solve problems:

1. Decide whether to use exact expressions U = -GMm/r or near-surface approximation mgh.
2. Use energy conservation: K_i + U_i = K_f + U_f to find speeds or heights.
3. For work, evaluate W = ∫ F·dr or use W = -ΔU.

Principle: Gravitational interactions obey the superposition principle: the net gravitational field or potential at any point due to several masses is the algebraic (vector for field/force, scalar for potential) sum of the contributions from each mass, calculated as if the others were absent.

Gravitational field and force (N bodies): For a test mass m at position r, the net gravitational force is F_net = m g_net where

• g_net(r) = sum_{i=1}^{N} g_i(r) = -G sum_{i=1}^{N} M_i (r - r_i)/|r - r_i|^3.
• Equivalently, F_net = -G m sum_{i=1}^{N} M_i (r - r_i)/|r - r_i|^3.

Gravitational potential (scalar): The total potential at r is the sum of the individual potentials:

• Φ(r) = sum_{i=1}^{N} Φ_i(r) = -G sum_{i=1}^{N} M_i / |r - r_i|.
• Potential energy of test mass m: U(r) = m Φ(r) = -G m sum_{i=1}^{N} M_i / |r - r_i|.

Multi-body consequences and uses: Because forces add vectorially, you can compute the net pull on a body from multiple planets, stars, or particles by adding contributions. In many problems (e.g., orbital motion of two bodies), it is convenient to reduce the problem to the motion of the centre of mass (CM) plus relative motion. For an isolated system, the CM moves as if the total mass were concentrated at that point under external forces:

• R_CM = (1 / M_tot) sum_i m_i r_i, where M_tot = sum_i m_i.

Simple two-body line problem — equilibrium point between two masses: For two fixed masses M1 and M2 separated by distance d on a line, the point along the line between them where a test mass feels zero net gravitational force satisfies

• GM1/x^2 = GM2/(d - x)^2, giving x = d / (1 + sqrt(M2/M1)), measured from M1 toward M2.

Stability note: A point where net force is zero may be stable or unstable. For two masses the midpoint between equal masses is an equilibrium along the line, but in full 3D it is generally a saddle point (unstable to transverse displacements).

Astrophysical applications: superposition explains tidal forces (difference in gravitational pull across a body), spacecraft station-keeping at Lagrange points (special equilibrium points in the rotating two-body frame), and calculating net gravity in multi-planet systems and galactic dynamics.