WEBSITE NOTES: Gravitation.

1. Much of what we know about gravitation is due to the work of the

following astronomers and mathematicians.

(i) Nicolaus Copernicus (1473-1543), Poland, suggested that the

Earth and all other planets revolve in circular orbits around the Sun, a

heliocentric system, not the geocentric model that persisted for 1400

years.

(ii) Tycho Brahe (1564-1601), Denmark, charted the positions of the

planets and 777 stars for 20 years.

(iii) Johannes Kepler (1571-1630), Germany, Brahe's assistant who

studied the data from the charts for 16 years and finally formulated 3

laws of planetary motion.

(iv) Galileo Galilei (1564-1642), Italy, who perfected the telescope and

later was placed under house arrest and force to recant for supporting

the heliocentric theory.

(v) Isaac Newton (1642-1727), England, developed the Law of Universal

Gravitation which states that all masses attract each other with a mutual

force that varies with the inverse-square of the distance.

(vi) Henry Cavendish (1731-1810), England, was the first experimenter to

measure the force of gravity between masses in the laboratory, and the first

to yield accurate values for Newton’s gravitational constant and the mass of

the Earth. The Cavendish experiment, done in 1797-98, consisted of a

torsion apparatus with metal spheres attached to rods.

2. Kepler's 3 laws of planetary motion state the following:

Law (1): All planets revolve in elliptical, nearly circular, orbits around the

Sun.

Law (2): A straight line from a planet to the sun sweeps out equal areas in

equal time intervals.

Law (3): The cube of the orbital radius of any planet divided by the square

of its period is constant.  r³/T² = k

3. Newton's law of Universal Gravitation: "The force of attraction between

two bodies is directly proportional to the product of their masses but varies

inversely with the square of the distance between them."  F = G·m₁·m₂/r² .

4. The value of the Universal Gravitational constant, G, was also predicted

by Newton.

5. In 1798 the value of G was carefully measured with a torsion apparatus

by Henry Cavendish (1731-1810), England, confirming Newton's prediction.

G = 6.67x10^-11 Nm²/kg²  (known as "Big G")

6. The mass of the Earth can be found by using Newton's Gravitation Law.

It is M_E = 5.98x10²⁴ kg. The mass of the Sun can be found from the

period and radius of a planet's orbit. The Sun's mass is computed to be

M_S = 2.0x10³⁰ kg.

7. The mass of a planet can be found only if it has a satellite orbiting it.

8. A satellite in a circular orbit, radius R,  accelerates centripetally toward

Earth at a rate equal to the acceleration of gravity at its orbital radius.

The following properties of satellite motion can all be proven:

(ii) acceleration due to gravity at the orbital radius, R, is g = (G·M_E)/R²

9. All bodies have gravitational fields around them, which can be

represented by a collection of vectors representing the force per unit

mass at all locations.

10. The mass of an object can be determined in two ways, gravitationally

and inertially. Both result in equivalent determinations of mass.

11. Albert Einstein (1879-1955), proposed that gravity is not a force,

but a property of space itself. Mass curves space causing objects to

be accelerated toward these massive bodies.

12. Einstein's theory, called the General Theory of Relativity, makes

predictions slightly different from Newton's laws, but when tested, gives

correct results.

13. Light has also shown to be deflected by massive celestial objects,

and if a mass is large enough, light leaving it will be totally bent back

to the object. This predicts that black holes in space exist.

14. And still, we need these steps to solve any problem in Physics:

(i) read the problem and identify the given variables

(ii) determine what you are asked to solve for

(iii) find the correct motion formula to use

(iv) use algebra to isolate the unknown

(v) substitute-in the given information and simplify.

Kepler's Law Lab: Mars diagram.