PHYS 2211 Module 12

12: Gravitation

Our visible Universe contains billions of galaxies, whose very existence is due to the force of gravity. Gravity is ultimately responsible for the energy output of all stars—initiating thermonuclear reactions in stars, allowing the Sun to heat Earth, and making galaxies visible from unfathomable distances. Most of the dots you see in this image are not stars, but galaxies. (credit: modification of work by NASA/ESA)

In this module, we study the nature of the gravitational force for objects as small as ourselves and for systems as massive as entire galaxies. We show how the gravitational force affects objects on Earth and the motion of the Universe itself. Gravity is the first force to be postulated as an action-at-a-distance force, that is, objects exert a gravitational force on one another without physical contact and that force falls to zero only at an infinite distance. Earth exerts a gravitational force on you, but so do our Sun, the Milky Way galaxy, and the billions of galaxies, like those shown above, which are so distant that we cannot see them with the naked eye.

Gravity is remarkable. An apple falls out of a tree- what made it accelerate? Nothing touches it. Saying the motion is due to “gravity” doesn’t explain it. Einstein, in his general theory of relativity (1917) went very far in explaining why there is gravity. But let’s follow Newton’s path, and merely try to describe it…

Gravity acts at a distance. Since the acceleration due to gravity is constant, gravity apparently acts at the top of trees as well as the bottom. How high does it reach? In Australia, gravity still acts towards the center of the earth. It appears that it is the earth itself, which is doing the attracting.

Let’s think about projectiles again.

Suppose v0 is very big. So big, that in the few seconds it takes to fall “h”, it has traveled a very long way horizontally. Suppose it travels so far, that you finally notice the earth is not flat:

This projectile goes further than we previously thought. It hits the ground later. What if v0 is bigger still? It hits further and farther away. Could it ever be going so fast that it never hits, that it keeps missing the ground? The answer is yes – the moon does this!

This was Newton’s epiphany: the moon’s motion is of the exact same nature – due to gravity – as falling apples. Gravity reaches up to the top of a tree, or mountains. Why not up to the moon? Why not further? Newton realized that there is one universal law of gravity, affecting all masses. One law that could explain and unify falling objects, projectiles, orbiting objects, and indeed all astronomy.

By 1650, the distance to the moon was well known: about 60RE (60 earth radii). The moon’s period T was also well known: about 27 days. Newton could thus easily compute  (using ).

Put in the numbers, you’ll find .

Conclusions: the moon is about 60 times further from the Earth’s center than us, and (whereas a (for any body near the earth) = g ).

12.1 Newton’s Law of Universal Gravitation

  • List the significant milestones in the history of gravitation
  • Calculate the gravitational force between two point masses
  • Estimate the gravitational force between collections of mass

12.2 Gravitation Near Earth’s Surface

  • Explain the connection between the constants G and g
  • Determine the mass of an astronomical body from free-fall acceleration at its surface
  • Describe how the value of g varies due to location and Earth’s rotation

12.3 Gravitational Potential Energy and Total Energy

  • Determine changes in gravitational potential energy over great distances
  • Apply conservation of energy to determine escape velocity
  • Determine whether astronomical bodies are gravitationally bound

12.4 Satellite Orbits and Energy

  • Describe the mechanism for circular orbits
  • Find the orbital periods and speeds of satellites
  • Determine whether objects are gravitationally bound

Basics of Space Flight

12.5 Kepler’s Laws of Planetary Motion

  • Describe the conic sections and how they relate to orbital motion
  • Describe how orbital velocity is related to conservation of angular momentum
  • Determine the period of an elliptical orbit from its major axis

12.6 Tidal Forces

  • Explain the origins of Earth’s ocean tides
  • Describe how neap and leap tides differ
  • Describe how tidal forces affect binary systems

12.7 Einstein’s Theory of Gravity

  • Describe how the theory of general relativity approaches gravitation
  • Explain the principle of equivalence
  • Calculate the Schwarzschild radius of an object
  • Summarize the evidence for black holes

Module 12 Self Assessment Practice Problems