Newton's Law of Universal Gravitation states that every particle in the universe attracts every other particle with a force (F) that is directly proportional to the product of their masses (m₁ and m₂) and inversely proportional to the square of the distance (r) between their centers.
Formula (Magnitude): F = G * (m₁m₂) / r²
Where G is the universal gravitational constant (G ≈ 6.674 × 10-11 N·m²/kg²).
In vector form, the force F₁₂ (force on m₁ due to m₂) is:
Formula (Vector): F₁₂ = -G * (m₁m₂) / r² * r̂₁₂
Where r̂₁₂ is the unit vector pointing from m₁ to m₂. The negative sign indicates that the force is attractive.
A central force is a force F(r) that is always directed along the line connecting the particle to a fixed center (the origin) and whose magnitude depends only on the distance |r| from the center.
Formula: F(r) = f(r) r̂
Gravitation is a perfect example of a central force, where f(r) = -G(Mm)/r².
The torque (τ) on the particle (mass m) about the force center is:
τ = r × F = r × (f(r) r̂)
Since r and r̂ are parallel, r × r̂ = 0. Therefore, the net torque is zero.
From Newton's second law for rotation, τ = dL/dt, where L is the angular momentum.
If τ = 0, then dL/dt = 0, which means L is constant.
L = r × p = constant. By definition, the vector L is perpendicular to both r and p (velocity v). Since L is a constant vector (it doesn't change direction), the position r and velocity v must *always* remain in the plane that is perpendicular to L.
Conclusion: All central force motion is confined to a plane.
As shown above, the torque τ is always zero for a central force. This directly implies that the total angular momentum L of the particle is conserved (it is constant in both magnitude and direction).
Areal velocity (dA/dt) is the rate at which the position vector r "sweeps out" an area as the particle moves.
In a small time dt, the particle moves by dr = v dt. The small area (dA) of the triangle swept out is:
dA = (1/2) |r × dr| = (1/2) |r × v dt|
So, the areal velocity is:
dA/dt = (1/2) |r × v| = (1/2) |r × (p/m)| = (1/2m) |r × p|
Since L = r × p, we get:
Formula: dA/dt = L / (2m)
Because angular momentum (L) and mass (m) are both constant in central force motion, the areal velocity (dA/dt) is also constant. This is the content of Kepler's Second Law.
Kepler's three laws describe the motion of planets around the Sun.
Statement: All planets move in elliptical orbits with the Sun at one of the two foci.
A circle is just a special case of an ellipse where the two foci coincide. The shape of the orbit (circle, ellipse, parabola, hyperbola) is determined by the total energy of the planet.
Statement: A line segment joining a planet and the Sun sweeps out equal areas in equal intervals of time.
This is a direct consequence of the conservation of angular momentum (areal velocity is constant), as proved in the section above. This means the planet moves *faster* when it is closer to the Sun (at perihelion) and *slower* when it is farther away (at aphelion).
Statement: 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.
Formula: T² ∝ a³ or T²/a³ = constant
For a simple circular orbit of radius 'r' (where a = r) around a large mass M, we can derive this:
Gravitational Force = Centripetal Force
G(Mm) / r² = m(v²) / r
G M / r = v²
Since v = (circumference) / (period) = (2πr) / T, we have v² = 4π²r² / T²
G M / r = 4π²r² / T²
Rearranging for T²: T² = (4π² / GM) r³
This shows T² ∝ r³, and the constant of proportionality is 4π²/GM.
For a satellite of mass 'm' in a circular orbit of radius 'r' (from Earth's center) around Earth (mass M):
As derived above (G M / r = v²):
Formula: vo = √(GM / r)
Note that orbital velocity *decreases* as the orbital radius 'r' *increases*.
A geosynchronous orbit is any orbit around the Earth with an orbital period (T) of exactly 24 hours (one sidereal day, to be precise: ~23h 56m).
This means the satellite returns to the exact same point in the sky at the same time each day.
A geostationary orbit is a *special type* of geosynchronous orbit.
Conditions:
Result: A satellite in this orbit appears to be fixed at a single point in the sky. All three conditions must be met.
Applications: This is extremely useful for communications satellites and weather satellites. You can point a stationary satellite dish on Earth at the satellite, and it will never move.
Weight is the force you feel (e.g., the normal force from the floor pushing up on you). "Weightlessness" is the sensation of having no weight. It is not the absence of gravity!
True weightlessness occurs during free-fall. An astronaut in a satellite orbiting the Earth is in a constant state of free-fall. The satellite (and everything in it) is "falling" around the Earth. The gravitational force (Fg) is providing the *exact* centripetal force (Fc) needed to stay in orbit.
Since the astronaut and the satellite are "falling" together, the astronaut does not press against the floor. The normal force is zero, and they feel "weightless."
The Global Positioning System (GPS) is a satellite-based navigation system.
Basic Idea: It consists of a constellation of ~30 satellites orbiting the Earth. Each satellite continuously broadcasts a signal containing its precise location and the precise time (from an onboard atomic clock).
A GPS receiver on Earth (like in your phone) receives these signals.