Gravity

  • Newtonian gravity
  • Kepler's laws
  • Orbital dynamics
  • Energy approach to gravitation

Newtonian gravity

The gravitational force causes massive particles to accelerate toward each other, it is proportional to the product of the masses of the particles. Unlike the electromagnetic force, which has two kinds of charge and can be attractive or repulsive, the gravitational force is only attractive, since there is only one kind of mass.

 

The gravitational forces felt by two particles forms a Newton's third law reaction pair. The force from mass 1 on mass 2 is equal in magnitude and opposite in direction of the force from mass 2 on mass 1. The distance between the point particles is designated as r.

 

 

Please see this PH 211 course page for a discussion of mass and weight.

The gravitational force is an inverse square function. As you double the distance between the massive particles, the force between them decreases by a factor of one fourth.

The gravitational force is very weak compared to the electromagnetic force.

Recall that in general, force is the negative gradient of the scalar potential. Therefore, taking the integral of gravitational force, we can define the gravitational potential as an inverse function of the distance between two massive particles. The r-hat vector is a unit vector in the r-direction. Gravitational potential is defined as going to zero at infinity.

Video source

Newtonian mechanics works very well for low-gravity regions, like when you are not near a black hole or neutron star. We understand the gravitational force very well - well enough to trust our calculations regarding trajectories of massive objects in gravity fields. (No, stunt on the video did not really place as it is shown...)

Newton's cannon simulator

Maintaining a stable circular orbit is really just a matter of being in freefall. If an object is moving at a certain velocity and at the right height, the surface of the Earth "falls away" to match the falling of the object.

Kepler's Laws

 

Tycho Brahe (1546 - 1601) spent many years taking accurate measurements of the positions of the planets. Johannes Kepler (1571 - 1630) was Tycho's student, who compiled the data after Tycho's death. Kepler's analysis of the data gave rise to Kepler's laws, which Newton later used to formulate his law of gravity.

 

Kepler's laws can be stated as follows:

 

  1. Planet orbits are ellipses with the sun at one focus of the ellipse.
  2. A line joining a planet to the sun sweeps out equal areas in the ellipse over equal times.
  3. The square of the orbital period of a planet is proportional to the cube of its semimajor axis.

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An ellipse typically has two focal points. The farther apart the focal points, the greater the ellipticity of the ellipse. The "diameter" across the long side is called the major axis. The semi-major axis (a) is one-half the major axis. This diagram is greatly exaggerated, most planet orbits in our solar system are nearly circular. A circle is a special case of an ellipse, with only one focal point.

Kepler's second law interactive

A planet moves faster when it is closer to the sun and slower when it is farther away. In the interactive simulation above from Weber State University above, the same amount of time elapsed while tracing out the black areas as the white areas.

Kepler's third law gives the relationship between a planet's period and its distance from the star. If a planet's orbit is nearly spherical, the distance between the star and planet can be assumed to be a constant radius r.

Johannes Kepler: Determined paths of planets using detailed observation Used Earth’s distance from the sun to scale relative distances to other planets Found orbital speeds by comparing planetary motions over time Summarized planets’ motions into three laws: Planet orbits are ellipses with the sun at one focus A line joining a planet to the sun sweeps out equal areas in equal times T2 a r3

Image source

Practice problems 1. Kepler's third law states T2 a r3. Use a force approach to write this as an equation (find the constant of proportionality). 2. Find gh for a satellite in a stable orbit at a height h above the surface of the earth. Consider the case where h = rE/2. What is gh/g? 3. Suppose there was a planet orbiting the Sun that took eight Earth years to complete one orbit. How far away from the Sun would the planet be? The Earth is 150 billion meters from the Sun. 4. Consider three massive objects at the corners of an equilateral triangle, a distance L apart, in a stable circular orbit about their common center of mass. A. What is the gravitational force felt by each? B. What is the velocity of each? C. What is the orbital period? D. What is the total gravitational potential energy of the system?

5. Consider a system of two planets, each of mass m in orbit around a star of mass M. Assume the planets remain on opposite sides of the star, at radius r.

 

   A. What is the orbital period of the planets?

 

   B. What is the total gravitational potential energy of the system?

Johannes Kepler: Determined paths of planets using detailed observation Used Earth’s distance from the sun to scale relative distances to other planets Found orbital speeds by comparing planetary motions over time Summarized planets’ motions into three laws: Planet orbits are ellipses with the sun at one focus A line joining a planet to the sun sweeps out equal areas in equal times T2 a r3
Practice problems 1. Kepler's third law states T2 a r3. Use a force approach to write this as an equation (find the constant of proportionality). 2. Find gh for a satellite in a stable orbit at a height h above the surface of the earth. Consider the case where h = rE/2. What is gh/g? 3. Suppose there was a planet orbiting the Sun that took eight Earth years to complete one orbit. How far away from the Sun would the planet be? The Earth is 150 billion meters from the Sun. 4. Consider three massive objects at the corners of an equilateral triangle, a distance L apart, in a stable circular orbit about their common center of mass. A. What is the gravitational force felt by each? B. What is the velocity of each? C. What is the orbital period? D. What is the total gravitational potential energy of the system?
Johannes Kepler: Determined paths of planets using detailed observation Used Earth’s distance from the sun to scale relative distances to other planets Found orbital speeds by comparing planetary motions over time Summarized planets’ motions into three laws: Planet orbits are ellipses with the sun at one focus A line joining a planet to the sun sweeps out equal areas in equal times T2 a r3
Practice problems 1. Kepler's third law states T2 a r3. Use a force approach to write this as an equation (find the constant of proportionality). 2. Find gh for a satellite in a stable orbit at a height h above the surface of the earth. Consider the case where h = rE/2. What is gh/g? 3. Suppose there was a planet orbiting the Sun that took eight Earth years to complete one orbit. How far away from the Sun would the planet be? The Earth is 150 billion meters from the Sun. 4. Consider three massive objects at the corners of an equilateral triangle, a distance L apart, in a stable circular orbit about their common center of mass. A. What is the gravitational force felt by each? B. What is the velocity of each? C. What is the orbital period? D. What is the total gravitational potential energy of the system?
Johannes Kepler: Determined paths of planets using detailed observation Used Earth’s distance from the sun to scale relative distances to other planets Found orbital speeds by comparing planetary motions over time Summarized planets’ motions into three laws: Planet orbits are ellipses with the sun at one focus A line joining a planet to the sun sweeps out equal areas in equal times T2 a r3
Practice problems 1. Kepler's third law states T2 a r3. Use a force approach to write this as an equation (find the constant of proportionality). 2. Find gh for a satellite in a stable orbit at a height h above the surface of the earth. Consider the case where h = rE/2. What is gh/g? 3. Suppose there was a planet orbiting the Sun that took eight Earth years to complete one orbit. How far away from the Sun would the planet be? The Earth is 150 billion meters from the Sun. 4. Consider three massive objects at the corners of an equilateral triangle, a distance L apart, in a stable circular orbit about their common center of mass. A. What is the gravitational force felt by each? B. What is the velocity of each? C. What is the orbital period? D. What is the total gravitational potential energy of the system?