Newton's Theory of Gravitation

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Contents

Scientific Theories

  • scientific theory is an axiom system
    • designed to explain certain kinds of phenomena
    • defined by its postulates
    • supported or disproved by its predictions

Big Picture

Principia Mathematica

  • Isaac Newton set forth his theory of gravitation in Philosophiae Naturalis Principia Mathematica (Principia for short), published in 1687
  • The Principia is “not only Newton’s masterpiece but also the fundamental work for the whole of modern science.”
  • Newton’s theory was the first to use differential equations, enabling its postulates to generate instant-by-instant predictions of the goings-on of a physical system.

Postulates

  • Law of Universal Gravitation
    • Informal: Things with mass attract each other
    • Precise: For any pair of physical bodies there’s a force on each, toward the other, with magnitude Gm1m2/r2
      • where G = 6.67384 10-11 (the gravitational constant), m1 and m2 are the masses of the bodies, and r is the distance between them
    • Formula: F = Gm1m2/r2
  • Equation of Motion
    • Informal: Forces makes things move
    • Precise: The acceleration A a physical body equals the net force F on the body divided by its mass M
      • Acceleration is the rate of change of velocity
      • Formula: F = MA
  • Law of Action and Reaction
    • Forces come in pairs, equal and opposite
    • For example, if a body exerts a gravitational force on another body, the latter exerts an equal and oppositely-directed force on the first.

Predictions

  • Kepler’s Laws of Planetary Motions
    • Ellipses: the orbit of each planet is an ellipse with the Sun at one focus. (1609).
    • Equal Areas: an imaginary line drawn from a planet to the Sun sweeps out equal areas in equal times (1609).
    • Harmony: the square of the orbital period of a planet is directly proportional to the cube of the average distance from the Sun (1618).
  • Galileo’s Laws of Motion
    • Falling Bodies: the distance a body falls is directly proportional to the square of the elapsed time (1609).
    • Projectile Motion: the trajectory of a projectile is a parabola (1609).
  • Law of Inertia (Newton’s First Law of Motion)
    • A body remains at rest or moving in a straight line at constant speed, unless acted on by a force. (Follows from F = MA)
  • Conservation of Momentum
    • The total momentum of a physical system remains the same.

Deriving Predictions

Quiz: What does Newton’s Theory Predict?

  1. Two stationary 16-pound bowling balls are placed 10 meters apart (center to center) in intergalactic space.  What does NTG predict? 
  2. A feather is dropped down an evacuated tube at sea level on the equator. Does NTG predict it falls faster, slower, or at the same rate as a lead ball?
  3. A lead ball is dropped down an evacuated tube on the surface of the Moon. Does NTG predict it falls faster, slower, or at the same rate as a ball dropped on Earth?
  4. A lead ball is dropped down an evacuated tube at sea level at the North Pole. Does NTG predict it falls faster, slower, or at the same rate as a ball dropped at the Equator?
  5. A lead ball is dropped down an evacuated tube inside the International Space Station, orbiting 250 miles above Earth. Does NTG predict it falls faster, slower, or at the same rate as a ball on the Equator?

Answers

  1. The bowling balls slowly accelerate toward each other, colliding in roughly 13 days. 
  2. NTG predicts the same acceleration: 32.14 f/s2 
  3. NTG predicts the ball falls slower, accelerating at 5.33 f/s2.   Gravity is weaker on the Moon because it’s less massive than Earth.
  4. NTG predicts the ball falls faster, accelerating at 32.36 f/s2.  The Earth is an oblate spheroid so the distance r to Earth’s center is less at the poles than at the Equator. After 3 seconds the object at the Pole has fallen 11 inches further
  5. International Space Station
  • Suppose a 250-mile tower is built with a platform on top.  You set up an evacuated tube on the platform and drop a ball down the tube.  Newton’s theory predicts it will fall at 28.44 feet per second per second (versus 32.14 at sea level on the Equator).  And you confirm that prediction. So gravity at 250 miles is 88% of gravity at the Earth’s surface.
  • Now the International Space Station zooms by at 17,000 mph and you see Scott Kelly through the window doing flips. 
  • This is strange.  Gravity is the same for you and Kelly, 88% of gravity at the Earth’s surface. But you’re walking on a platform and Kelly is floating around.  What’s different between you and Kelly?
  • The difference is that Kelly experiences an equal and opposite force: centrifugal force due to being in orbit.

Shortcomings of the Theory

  • Mercury’s Orbit
    • In 1859 Urbain Le Verrier pointed out that Newton’s Theory was unable to fully predict the observed precession of Mercury’s orbit around the Sun, even factoring in the gravitational forces by nearby planets. He proposed the existence of another planet, Vulcan, to explain Mercury’s orbit.
  • Newton’s Theory is incompatible with Einstein’s Special Relativity
    • Newton’s Law of Universal Gravitation says:
      • For any pair of physical objects there’s a force on each, toward the other, with magnitude Gm1m2/r2
    • According to Newton the forces on each object act simultaneously. But, per Special Relativity, simultaneity is relative to an inertial reference frame.
    • quora.com/Why-is-Newtonian-gravity-incompatible-with-special-relativity
      • If an effect is simultaneous with its cause in one frame of reference, there will be another frame of reference where the effect precedes the cause.

Fictitious Forces

  • In addition to gravity, a so-called real force, there are fictitious forces, e.g. the Centrifugal and Coriolis forces.
  • The difference can be illustrated by contrasting two scenarios:
    • The real scenario, where the International Space Station orbits Earth at an altitude between 200 and 250 miles. 
    • A pretend scenario where ISS has been moved to intergalactic space.
  • In both scenarios the crew is weightless, floating around the cabin.
  • ISS in intergalactic space
    • The crew is weightless.
    • The acceleration A of the crew = 0
    • The gravitational force F on the crew = 0
      • That is, the crew experiences zero-gravity.
  • ISS orbiting Earth
    • The crew is weightless.
    • The acceleration A of the crew = 0
    • But the gravitational force F on the crew > 0
      • At an altitude of 200 and 250 miles Earth’s gravity is about 88 percent of what it is at the surface.
    • Since F = MA, it seems reasonable that there’s a force offsetting the force of gravity, making the crew weightless
      • The centrifugal force is the outward force experienced by a body along a curve.
  • So the crew of the ISS orbiting Earth is weightless because they are subject to offsetting gravitational and centrifugal forces.
  • But the crew of the ISS in intergalactic space is weightless because they are subject to no forces at all.
  • The centrifugal force is fictitious, unlike gravity.
    • britannica.com/science/centrifugal-force
      • Centrifugal force, a fictitious force, peculiar to a particle moving on a circular path, that has the same magnitude and dimensions as the force that keeps the particle on its circular path (the centripetal force) but points in the opposite direction.
  • The problem is that the centrifugal force violates the Law of Action and Reaction, that forces come in equal and opposite pairs.  There’s only one red centrifugal-force arrow in the diagram (but two blue gravity arrows).  
  • The cause is that the ISS in orbit is accelerating.
  • Moreover, F in F = MA is the sum of all real forces on an object.
    • In his textbook Classical Mechanics, referring to F = MA, John Taylor writes “where, as usual, F denotes the sum of all the forces as identified in any inertial frame.” (page 343)
  • So in the ISS-orbiting-Earth scenario, F = MA must be extended by adding a term to the left-side representing the centrifugal force:
  • The math establishing the weightlessness of the orbiting crew is then:

Magic of Differential Equations

  • Differential equations are magic:
    • Differential equations enable the instant-by-instant calculation of the state of a physical system from an initial time to any future time.
  • A differential equation is an equation with derivatives.
  • A derivative is the rate of change of a function relative to a variable.  For example:
    • Velocity is the rate of change of location relative to time.
    • Acceleration is the rate of change of velocity relative to time.
  • How differential equations are solved:
    • Analytic Methods, by manipulating formulas
    • Numerical Analysis, by crunching numbers
  • britannica.com/science/differential-equation
    • Differential equations are very common in science and engineering, as well as in many other fields of quantitative study, because what can be directly observed and measured for systems undergoing changes are their rates of change.
    • The solution of a differential equation produces a function that can be used to predict the behaviour of the original system, at least within certain constraints.
  • How Newton’s Equation of Motion works:
  • You first determine a particle’s acceleration by dividing the net force F on it by its mass M, e.g. 
  • Then, using calculus, you calculate the particle’s future velocity and location from its initial velocity and location:
    • Velocity at future time t is 9.8t + the initial velocity
    • Location at future time t is 4.9t2 + initial location + initial velocity x t