Gravitation Physics Lesson 13 by Owen Borville 11.30.2025
Isaac Newton (1643-1727) and his Law of Universal Gravitation states that all masses (including objects and particles) attract one another with a gravitational force proportional to their masses and inversely proportional to the square of the distance between them:
Force (F) = (G*m1*m2)/r^2
where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects or particles, and r is the distance between their centers. The mass of spherically symmetrical objects is calculated from the center of the object. Non-symmetrical objects can also calculate their mass from their center of mass, if the distance between the objects is large compared to the size of the two objects.
The weight of an object is the gravitational attraction between the Earth and the object. The gravitational field is represented as lines that indicate the direction of the gravitational force. The line spacing indicates the strength of the field. Apparent weight differs from actual weight due to the acceleration of the object. The acceleration due to gravity at the surface of the earth is:
g = G*Me/r^2,
where g is the acceleration of gravity at the surface, G is the gravitational constant, and Me is the mass of the Earth, r is the distance between the center of mass of the object with Earth.
Acceleration due to gravity changes as an object moves away from Earth and a different equation must be used. The total energy of a system is the sum of the kinetic and gravitational potential energy, and this total energy is conserved in orbital motion.
The escape velocity is the minimum velocity an object must have to leave a planet or mass and not return. Objects with total energy less than zero are gravitationally bounded to a planet or mass. Objects with total energy greater than zero are unbounded from a planet or mass. The gravitational potential energy beyond Earth is
U = -(G*Me*m)/r^2,
where U is gravitational potential energy, G is the gravitational constant, Me is the mass of the Earth, m is the mass of the object, and r is the distance between the object and Earth's center.
Escape velocity is v(esc) = √(2G*M/R)
The conservation of energy in terms of potential energy is E = K1 + U1 = K2 + U2, where K is kinetic energy and U is potential energy. However, the new equation for potential energy must be used:
(1/2m*v1^2)-(G*M*m/r1) = (1/2m*v2^2)-(G*M*m/r2), where M represents the mass of Earth or another planet and m is the mass of the object.
Orbital velocities of satellites and orbiting objects are determined by the mass of the body being orbited and the distance from the center of that body. The period of the orbit is independent of the orbiting object's mass.
Orbital speed = v(orbit) = √(G*Me/r)
Orbital period = T = 2π√(r^3/G*Me)
Kepler's Laws of Planetary Motion (Johannes Kepler, 1571-1630) (1) All orbital motion follows the path of a conic section. Bound or closed orbits are either circular or elliptical in pattern. Unbounded or open orbits follow a parabolic or hyperbolic orbit. (2) The areal velocity of any orbit is constant because of the conservation of angular momentum. The line connecting a planet to the sun sweeps out equal areas in equal times, so that a planet moves faster when closer to the sun and slower when farther away. (3) The square of the period of an elliptical orbit is proportional to the cube of the semi-major axis of that orbit.
Kepler's Third Law: T^2 = 4π^2*a^3/GM
Earth's tides are caused by the difference in gravitational forces from the Moon and the sun on the different sides of the Earth. Spring or neap (high) tides occur when Earth, the moon, and the sun are aligned, and neap or low tides occur when they form a right triangle. Tidal forces can create internal heating, changes in orbital motion, and even destruction of orbiting bodies.
Energy in a circular orbit = E = K (kinetic energy) + U (potential energy) = (GmMe/2r)-(GmMe/r) = -GmMe/2r
Conic sections = α /r = 1 + ecosθ, where α and e are constants determined by the total energy and angular momentum of the satellite at a given point. These constants also determine which of the four conic sections represents the path of the satellite.
According to the theory of general relativity, gravity is the result of distortions in space time created by mass and energy. The principle of equivalence states that both the mass and acceleration distort space-time and are indistinguishable in comparable circumstances.
Black holes, the result of gravitational collapse, are singularities with an event horizon that is proportional to their mass. Evidence for the existence of black holes is still circumstantial, but the amount of that evidence is overwhelming.
The Schwarzchild radius equation is Rs = 2GM/c^2, where Rs is the Schwarzhchild radius, G is the gravitational constant, M is the mass of the object, and c is the speed of light. The Schwarzchild radius is the radius around a massive object at which it becomes a black hole if its mass is compressed within this limit.
Isaac Newton (1643-1727) and his Law of Universal Gravitation states that all masses (including objects and particles) attract one another with a gravitational force proportional to their masses and inversely proportional to the square of the distance between them:
Force (F) = (G*m1*m2)/r^2
where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects or particles, and r is the distance between their centers. The mass of spherically symmetrical objects is calculated from the center of the object. Non-symmetrical objects can also calculate their mass from their center of mass, if the distance between the objects is large compared to the size of the two objects.
The weight of an object is the gravitational attraction between the Earth and the object. The gravitational field is represented as lines that indicate the direction of the gravitational force. The line spacing indicates the strength of the field. Apparent weight differs from actual weight due to the acceleration of the object. The acceleration due to gravity at the surface of the earth is:
g = G*Me/r^2,
where g is the acceleration of gravity at the surface, G is the gravitational constant, and Me is the mass of the Earth, r is the distance between the center of mass of the object with Earth.
Acceleration due to gravity changes as an object moves away from Earth and a different equation must be used. The total energy of a system is the sum of the kinetic and gravitational potential energy, and this total energy is conserved in orbital motion.
The escape velocity is the minimum velocity an object must have to leave a planet or mass and not return. Objects with total energy less than zero are gravitationally bounded to a planet or mass. Objects with total energy greater than zero are unbounded from a planet or mass. The gravitational potential energy beyond Earth is
U = -(G*Me*m)/r^2,
where U is gravitational potential energy, G is the gravitational constant, Me is the mass of the Earth, m is the mass of the object, and r is the distance between the object and Earth's center.
Escape velocity is v(esc) = √(2G*M/R)
The conservation of energy in terms of potential energy is E = K1 + U1 = K2 + U2, where K is kinetic energy and U is potential energy. However, the new equation for potential energy must be used:
(1/2m*v1^2)-(G*M*m/r1) = (1/2m*v2^2)-(G*M*m/r2), where M represents the mass of Earth or another planet and m is the mass of the object.
Orbital velocities of satellites and orbiting objects are determined by the mass of the body being orbited and the distance from the center of that body. The period of the orbit is independent of the orbiting object's mass.
Orbital speed = v(orbit) = √(G*Me/r)
Orbital period = T = 2π√(r^3/G*Me)
Kepler's Laws of Planetary Motion (Johannes Kepler, 1571-1630) (1) All orbital motion follows the path of a conic section. Bound or closed orbits are either circular or elliptical in pattern. Unbounded or open orbits follow a parabolic or hyperbolic orbit. (2) The areal velocity of any orbit is constant because of the conservation of angular momentum. The line connecting a planet to the sun sweeps out equal areas in equal times, so that a planet moves faster when closer to the sun and slower when farther away. (3) The square of the period of an elliptical orbit is proportional to the cube of the semi-major axis of that orbit.
Kepler's Third Law: T^2 = 4π^2*a^3/GM
Earth's tides are caused by the difference in gravitational forces from the Moon and the sun on the different sides of the Earth. Spring or neap (high) tides occur when Earth, the moon, and the sun are aligned, and neap or low tides occur when they form a right triangle. Tidal forces can create internal heating, changes in orbital motion, and even destruction of orbiting bodies.
Energy in a circular orbit = E = K (kinetic energy) + U (potential energy) = (GmMe/2r)-(GmMe/r) = -GmMe/2r
Conic sections = α /r = 1 + ecosθ, where α and e are constants determined by the total energy and angular momentum of the satellite at a given point. These constants also determine which of the four conic sections represents the path of the satellite.
According to the theory of general relativity, gravity is the result of distortions in space time created by mass and energy. The principle of equivalence states that both the mass and acceleration distort space-time and are indistinguishable in comparable circumstances.
Black holes, the result of gravitational collapse, are singularities with an event horizon that is proportional to their mass. Evidence for the existence of black holes is still circumstantial, but the amount of that evidence is overwhelming.
The Schwarzchild radius equation is Rs = 2GM/c^2, where Rs is the Schwarzhchild radius, G is the gravitational constant, M is the mass of the object, and c is the speed of light. The Schwarzchild radius is the radius around a massive object at which it becomes a black hole if its mass is compressed within this limit.