Unit 3: Gravitation

Table of Contents

Newton's Law of Gravitation

Newton's Law of Universal Gravitation states that every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The magnitude of the force Fg is:

Fg = G (m₁m₂) / r²

In vector form, the force on m₂ due to m₁ is:

F21 = -G (m₁m₂) / r² r̂12

Where r̂12 is the unit vector pointing from m₁ to m₂. The negative sign is crucial, as it indicates the force is attractive (it pulls m₂ *back* towards m₁).

Motion in a Central Force Field

Definition of a Central Force

A central force is a force on a particle that is always directed along the line connecting the particle and a fixed center point (the origin).

Its magnitude depends only on the distance `r` from the center, not on the angle.

F(r) = f(r) r̂

Gravity (f(r) = -GMm/r²) and the electrostatic force (f(r) = kq₁q₂/r²) are the two most important examples.

Key Properties of Central Force Motion

  1. The force is conservative. A force `F(r) = f(r)r̂` is always conservative, which means a potential energy function `U(r)` exists (where F = -∇U) and total mechanical energy (E = K + U) is conserved.
  2. Torque is always Zero. The torque `τ` about the center point is defined as `τ = r × F`.
    Since F is parallel to r (or anti-parallel, like gravity), their cross product is zero.
    τ = r × (f(r)r̂) = 0
  3. Angular Momentum (L) is Conserved.
    From Unit 2, we know `τ = dL/dt`.
    Since `τ = 0` for a central force, we must have `dL/dt = 0`, which means:
    L = constant
    This is the *most important characteristic* of central force motion.
  4. The motion is confined to a plane.
    The angular momentum vector L = r × p is constant. This means L always points in the same fixed direction.
    By the definition of the cross product, both the position vector r and the momentum vector p must always be perpendicular to L.
    Therefore, the entire motion (the path of the particle) must lie in a single, fixed plane that is perpendicular to the constant angular momentum vector L.

Kepler’s Laws

Kepler's laws are three empirical laws that describe the motion of planets around the Sun. They are all direct consequences of the Sun exerting an inverse-square (1/r²) central force (gravity) on the planets.

Kepler's First Law (Law of Orbits)

Law: "All planets move in elliptical orbits with the Sun at one of the two foci."

Reason: This is the specific mathematical solution to the equation of motion for an attractive 1/r² force, for a bound system (one with negative total energy).

Kepler's Second Law (Law of Areas)

Law: "A line segment joining a planet and the Sun sweeps out equal areas in equal intervals of time."

Derivation (Areal Velocity is Constant):

  1. This law is a direct consequence of the conservation of angular momentum (L), which is true for *any* central force.
  2. The small area `dA` swept out by the position vector `r` in a time `dt` is the area of a small triangle: `dA = (1/2) |r × dr|`.
  3. Since `dr = v dt`, we have `dA = (1/2) |r × v| dt`.
  4. We know `L = r × p = m(r × v)`. So, `|r × v| = L/m` (where L = |L|).
  5. Substitute this in: `dA = (1/2) (L/m) dt`.
  6. Rearranging gives the areal velocity:
    dA/dt = L / (2m)
  7. Since the gravitational force is central, `L` is constant. The planet's mass `m` is also constant.
  8. Therefore, dA/dt = constant. This is Kepler's Second Law.
  9. Implication: A planet must move *faster* when it is closer to the Sun (perihelion) and *slower* when it is farther away (aphelion).

Kepler's Third Law (Law of Periods)

Law: "The square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (a) of its orbit."

T² ∝ a³

Derivation (for a simple circular orbit of radius r):

  1. For a circular orbit, the gravitational force provides the exact centripetal force required to keep the planet (mass m) in orbit around the Sun (mass M).
  2. Fgravity = Fcentripetal
  3. G(Mm) / r² = m v² / r
  4. Simplify: G M / r = v²
  5. The orbital speed `v` is the circumference (2πr) divided by the period (T): `v = 2πr / T`.
  6. Substitute `v`: G M / r = (2πr / T)² = 4π²r² / T²
  7. Rearrange to solve for T²:
    T² G M = 4π²r³
    T² = [ 4π² / (GM) ] r³
  8. Since `(4π² / GM)` is a constant for all planets in the solar system, we have T² ∝ r³. (For a full ellipse, `r` is replaced by the semi-major axis `a`).

Motion of Satellite

The motion of an artificial satellite around a planet (like Earth) follows the same laws as a planet around the Sun.

Orbital Velocity

This is the speed `v_orb` required to maintain a stable circular orbit at a radius `r` (measured from the planet's center).

From the derivation of Kepler's 3rd Law (step 4): G M / r = v²

vorb = √(GM / r)

Where `M` is the mass of the planet (e.g., Earth).

Escape Velocity

This is the minimum speed `v_esc` an object needs at a given distance `R` (e.g., the planet's surface) to escape the planet's gravitational pull completely (i.e., to reach `r = ∞` with zero kinetic energy).

We use the conservation of energy (E = K + U):

Einitial (at surface) = Efinal (at infinity)

(Ki + Ui) = (Kf + Uf)

(1/2)mvesc² + (-GMm/R) = 0 + 0

(1/2)mvesc² = GMm/R

vesc² = 2GM/R

vesc = √(2GM / R)
Key Relationship:
Compare the escape velocity from the surface (R) with the orbital velocity *at* the surface (r=R): Therefore: vesc = √2 × vorb
Escape velocity (≈ 11.2 km/s for Earth) is about 41.4% faster than the low-Earth-orbit velocity (≈ 7.9 km/s).

Weightlessness

Weightlessness is the sensation (or state) of having no apparent weight. It is NOT the absence of gravity.

Basic idea of global positioning system (GPS)

The Global Positioning System (GPS) is a utility that provides users with positioning, navigation, and timing (PNT) services. It is a satellite-based navigation system.