Angular Momentum, Kepler’s Laws, and Universal Gravitation

Posted on Dec 17, 2024 in Physics

Angular Momentum of a Particle

The angular momentum of a particle about a point O is the vector product of its position vector with respect to that point and its momentum:

The angular momentum is measured in SI units of kg * m2/s. It is a vector quantity, perpendicular to r and v. Its magnitude is “sen” where is the angle between r and v. Whenever r and v are parallel, the angular momentum is 0. The angular momentum characterizes the rotational motion of the particle.

Variation of Angular Momentum

This defines the moment of force M, over the same point O as the vector product of r and F. This result is fundamental to the study of rotations: its physical meaning is that the moment of the force tends to change the direction of motion.

If the net force acting on the particle is zero, the angular momentum is conserved. This is the conservation theorem. This occurs when the net force is zero, or when the force is parallel to r, as in the case of central forces.

Kepler’s Laws

Empirical laws enunciated by Kepler in the seventeenth century to describe the motion of planets around the Sun are three:

  1. 1st Law (Law of the Orbits): The planets describe elliptical orbits, in whose foci is the Sun.
  2. 2nd Law (Area Law): The vector of position with respect to the Sun of a planet sweeps out equal areas in equal times. That is, the velocity is constant. This implies that the linear velocity of the planet is greater the closer it is to the Sun. This law is equivalent to the conservation of angular momentum of the planet with respect to the Sun.
  3. 3rd Law (Law of Periods): The squares of the periods of revolution of the planets are proportional to the cube of their mean distances from the Sun.

One consequence is that the linear velocity of the planets is not constant but depends on the orbital radius: a planet spins faster the smaller the orbit described. Kepler’s laws were demonstrated theoretically later thanks to Newton’s law of gravitation.

Newton’s Law of Universal Gravitation

It was enunciated by Newton in the seventeenth century and allowed to explain all the gravitational effects known in his day (among them: the movement of the stars in the solar system, tides, or falling bodies on Earth). The law states:

Every body in the universe attracts every other body with a central force that is proportional to both the mass and inversely proportional to the square of the distance that separates them.

Mathematically formulated as follows:

where F is the gravitational force between two bodies of masses m1 and m2, r is the distance that separates them, and ur is a unit vector from the body that exerts the force to which the sufferer. The minus sign indicates that the force is attractive. G is a constant called the universal gravitational constant measured experimentally and its value. The equation of gravitational force applies equally to the two masses. For example, the force of attraction of the Earth on the Moon is equal and opposite to the force on the Moon from Earth. If we have a set of particles, the gravitational force on each is the vector sum of the forces produced by the other particles.

Gravitational Potential Energy

The gravitational force, being conservative, has an associated gravitational potential energy function. So it follows that the gravitational potential energy of a particle of mass m1 at a distance r from another mass m2 is equal to:

where we take the potential energy at infinity equal to zero. As this energy is a scalar quantity whose SI unit is the joule. For a system comprising more than two masses, the gravitational potential energy of the system is the sum of the potential energies of all distinct pairs of masses that can be formed.

Due to the action of gravitational force, the bodies tend to fall spontaneously to the regions of lower potential energy.

Types of Waves

There are several possible classifications.

According to the Medium in Which the Wave Propagates

  • Waves that do not need a material medium to propagate and can therefore spread in the empty. These are the electromagnetic and gravitational waves. Examples of electromagnetic waves are: light, radio waves, television and mobile phones, microwaves, ultraviolet rays, gamma rays, etc.
  • Waves that require a material medium to propagate. To meet such other wave phenomena we know, for example, the sound waves, vibrations of a string, etc. Such waves are the result of the orderly movement of many particles.

According to the Vibration Direction

  • Transverse waves: vibration occurs in a direction perpendicular to the direction of propagation. Examples: shock cord transversely and electromagnetic waves.
  • Longitudinal waves: vibration occurs in the direction of propagation. Example: sound waves.

Depending on the Number of Dimensions of Space to Spread

  • One-dimensional (vibrations on a string).
  • Two-dimensional (waves on the surface of a liquid or vibration in a membrane).
  • Three-dimensional (light and sound).

Huygens’ Principle

This is a simple mechanism for the construction of wave fronts, from front above. A wavefront is becoming one of the surfaces that pass through the points where a wave oscillates with the same phase. The principle states that: The points in a wavefront become sources of secondary waves, whose envelope forms a new primary wavefront. How to apply is: are plotted as small circles of radius with centers at different points on a wavefront, and then plotted the envelope of the circles, which is the new wavefront. The figure shows an example of application to a spherical wavefront and another example to explain the diffraction of a plane against an obstacle. new front secondary wavelets explanation ray diffraction barrier

One consequence of Huygens’ principle is that all the rays take the same time between two consecutive wavefronts. Rays are lines perpendicular to the wave fronts, and correspond to the line of propagation of the wave.

Although Huygens was formulated for the matter waves, which were the only known in his time, his principle is valid for all types of waves. Kirchhoff extended the method to electromagnetic waves, once they were discovered.