Reading:

*Chapter 1-3, P1-2, P1-3, P1-4* (ZG4)

Key Question: | What keeps a planet or satellite in orbit? | |
---|---|---|

Key Principle: | Newton's Laws of Motion | |

Key Problem: | What is the acceleration of the Moon in its orbit? | |

Key Quote: | "Nature and nature's laws lay hid in night. God said, Let Newton be!, and all was light." - A. Pope |

- Galileo
- What is the Law of
*Inertia*? What is*acceleration*? - What was Galileo's answer to the common objection to the concept of a moving Earth that earthbound objects would be swept off and left behind?
- What were the most important of Galileo's telescopic observations to resolving the debate between the geocentric and heliocentric models of the solar system?
- What did Galileo find when he looked at the Sun and Moon?
- Was the publication of
*Dialogue on the Two Great World Systems*a politic thing to do in the climate of 17th century Italy?

- What is the Law of
- Newton's Mechanics
- What are Newton's Three
*Laws of Motion*? - What are the basic units of time and length?
- What are the differences between
*speed*,*velocity*, and*acceleration*? - What is
*momentum*? - What is the relation between
*force*and*acceleration*? - How are the
*centripetal acceleration*and gravitational force related?

- What are Newton's Three

Isaac Newton was born in Lincolnshire, England in the year 1643, a year after Galileo's death. He was appointed as the Lucasian Professor of Mathematics at Cambridge University 8 years after entering Cambridge as an undergraduate and 2 years after beginning graduate studies. Stephen Hawking, the theoretical astrophysicist and cosmologist, currently holds the Lucasian professorship that Newton once held.

As a young man Newton was interested in "Natural Philosophy", as Science
was called then, and this continued to hold his interest along with
mathematics, optics and a number of other topics. Newton was a slow
publisher, and usually needed some instigation to get him to complete his
work. For example, in 1666 and 1666, just after completing his bachelor of
arts degree and while Cambridge was closed for the plague, Newton worked out
most of his ideas on mechanics and gravitation, but did not write it up for
publication or presentation. In fact, Newton concentrated for almost two
decades on optics, on which he wrote some very important treatises, and
mathematics. He designed a type of reflecting telescope, now called the
*Newtonian telescope*. In the area of mathematics, he invented
differential and integral Calculus, which he needed to do his gravitational
calculations.

In 1684, Edmund Halley, the discoverer of Halley's comet, consulted Newton
on the problem of orbital mechanics. He was astounded to find that Newton had
worked out the solution 20 years earlier! There were several scientists, like
Halley and the physicist Robert Hooke, who were working on the problem at this
time. At Halley's urging, in the following year Newton presented his work to
the Royal Society. In 1686, he published the landmark *The Mathematical
Principles of Natural Philosophy*, usually known at the *Principia*.
It was in the *Principia* that Newton presented his *three Laws of
Motion*, placing the mechanics of gravtitation on a physical basis.

In the *Principia*, Newton stated his three Laws of Motion:

- Every body continues in a state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.
- Any change of motion is proportional to the force that acts, and it is made in the direction of the straight line in which that force is acting.
- To every action there is always an equal and opposite reaction; or, the mutual actions of two bodies upon each other are always equal and act in opposite directions.

Newton's First Law is just a restatement of Galileo's Law of Inertia.
Application of a *force* causes the *acceleration* needed to change
the *velocity* of a body. The *momentum* of a body is a measure
of the inertia. Newton's Second Law states that the *acceleration*
induced by application of a *force* is proportional to the force,
with the constant of proportionality given by the *mass* (F=ma).
Note that this means that the momentum is given by the mass x velocity.
Finally, the Third Law of action and reaction is something entirely new.
He noticed that forces between bodies alway come in pairs, and that
when you push on something it pushes back with equal force.

Newton's work was the first real mathematical formulation of mechanics that allowed true calculation of observable quantities based upon a few relatively simple principles.

There are also different types of quantities. A *scalar* quantity is
one that behaves just like a pure number (think of a "scale factor"). The
mass of a body is a scalar. On the other hand, some quantities have not only
a scale or magnitude, but a direction. These are called *vectors*, and
can be thought of as arrows with a length given by thier magnitude, and
pointing in some direction. In this section, we will denote vectors as
**boldface** variables. An example of a vectory is the 3-dimensional
position of a body in space **x**.

The *velocity* (**v**) of a body is a vector quantity expressing
the motion of a body by its speed (v) and direction in space. The units of
velocity are km/s (plus a direction). The velocity is the time derivative
of the position

The Second Law gives acceleration proportional to *force*, with the
*mass* as the constant of proportionality. Thus

The units are thus kg m/s^2 which is defined a 1 Newton of force.

From definition of acceleration and **F** = m**a**, we see that the
momentum is just

But equal forces do not mean equal accelerations! The two accelerations are related by

where G is Newton's gravitational constant (6.67 x 10^-11 N m^2 / kg). Note that the strength of the force is the same on the interchange of m_1 and m_2 (the vector direction will reverse though). Thus, the gravitational accelerations are given by

and

Thus, the centripetal force on a mass m is

The centripetal force is a useful concept, and we will use it frequently.

From the centripetal acceleration and gravitational force law, we can see why P^2 = A^3! We ignore the mass m of the Earth, and consider only the mass M of the sun. Then, if we equate

then we find that

For a circular orbit of period P, the velocity is given by

so substitution gives

or

which is Kepler's third law! Note that the relation now involves the central mass M that the body is orbiting, and thus the proportionality of Kepler's law can be used to measure the mass of the body that the satellite or planet is orbiting. The above formula was derived neglecting the mass of the satellite. We will derive the general result in the next lecture.

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*smyers@nrao.edu*
*Steven T. Myers*