Angular Momentum, Kepler’s Laws, and Gravity
Angular Momentum of a Particle
The angular momentum of a particle about a point O is the vector product of the position vector r, with respect to that point, and its momentum p: L = r x p, where p = mv, where m is the mass and v the velocity of the particle can be rewritten as L = mr x v.
The angular momentum is measured in kg⋅m²/s. L is a vector quantity, its magnitude is L = mrvsinθ, where θ is the angle between r and v; if they are parallel, then L = 0.
Angular momentum characterizes the rotation of a moving particle.
Change: dL/dt = dr/dt x p + r x dp/dt = v x p + r x F = M, where M is defined as the moment of force, the vector product of r and F. This result is fundamental to the study of rotations: its physical meaning is that M tends to change the direction of motion.
Theorem of Conservation: If the net force acting on a particle is zero, L is conserved. This occurs when the net force is 0 or when the force is parallel, as in the case of central forces.
Kepler’s Laws
Empirical laws enunciated by Kepler in the 17th century to describe the motion of planets around the sun.
1. Law of Orbits: The planets describe elliptical orbits, with the sun at one focus.
2. Law of Areas: The position vector from the sun to a planet sweeps out equal areas in equal times. This means the velocity is constant. This implies that the linear velocity is greater when the planet is nearest the sun. This law is the conservative equivalent of L for the planet from the Sun.
3. Law of Periods: The squares of the periods of revolution are proportional to the cube of the semi-major axis of the orbit. The linear velocity of planets is not constant but depends on the orbital radius; a planet rotates faster if its radius is smaller. These laws were later demonstrated by Newton.
Law of Universal Gravitation
It was enunciated by Newton in the 17th century and enabled the explanation of all the gravitational effects known at the time. The law states that every body in the universe attracts every other body with a central force that is proportional to the mass of both and inversely proportional to the square of the distance. Formula: F = -G(m₁m₂/r²) * ur, where F is the gravitational force between two bodies of masses m₁ and m₂, r is the distance between them, and ur is the unit vector from the body that exerts the force to the body that suffers it. The sign – indicates that the force is attractive. G is the universal gravitational constant, measured experimentally, and whose value is 6.67 × 10⁻¹¹ N⋅m²/kg².
The equation of the gravitational force applies equally to the two masses. So the force of Earth on the Moon is the same, but in the opposite direction, to the force of the Moon on Earth. If there is a set of particles, the F on each one of them is the vector sum of the forces produced by the rest of the particles.
Gravitational Potential Energy
The gravitational force, being conservative, has an associated gravitational potential energy function, such that the work done by the force between two points is equal to the decrease of this potential energy. Deduction: the gravitational potential energy (Ep) of a particle of mass m at a distance r from another mass M is equal to Ep = -G(mM/r), which is taken at infinity Ep = 0. As it comes to energy, it is a scalar quantity whose SI unit is the joule (J).
For a system consisting of more than two masses, the Ep of the system is the sum of the energies of all distinct pairs of masses that can be formed. Due to the action of gravitational force, the bodies tend to fall spontaneously into regions of lower Ep.
Ep near the Earth’s surface: The gravitational force acting on a body of mass m is its weight: F = mg = -mgj
Whereas the constant value of g in the vicinity of the Earth, the work (W) done by the force when the body’s weight shifts
vertically from point A to B is Wab = mghA – mghB. So the Ep at a point at height h is: Ep = mgh, where
we have chosen the source of energy h = 0.