1. TUESDAY(11/25): Newton's Law of Universal Gravitation. Mass and

Radius of Earth. HW: Read pages 390-401 and solve probs. 3, 5, 7, 9, and

11 on pages 412-13.

Website Notes for Ch.13: Universal Gravitation.

1. Much of what we know about universal gravitation is because of the

work of the early 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 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 force

that varies with the inverse-square of the distance.

(vi) Henry Cavendish(1731-1810), England, in 1798 experimentally

confirmed the numerical value of the constant in Newton's Inverse-Square

Law with a torsion apparatus.

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. The equation we get is  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." The law can be

stated mathematically as F_G = Gm₁m₂/r² .

4. The value of the universal gravitational constant, G, was predicted by

Newton to be  G = 6.67x10^-11 Nm²/kg² .

5. The fact that gravitational force is centripetal allows the computation of

planetary periods and orbital radii. Recall that F_C = mv²/r .

6. 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 2.0x10³⁰ kg.

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

example the Earth's mass can be calculated to be 5.98x10²⁴ kg.

8. A satellite in a circular orbit 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) the acceleration due to gravity at the orbital radius, R, is g = G∙M_E / R²

(iii) the minimum or critical velocity for stable orbit is v = √(R∙g) .

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 Gravitational Field at a point in space is defined by: g = F_G /m ,

measured in N/kg.

17. 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.

18. Einstein's theory, called the general theory of relativity, makes predictions

slightly different from Newton's laws, but when tested, gives correct results.

19. 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.

20. 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 vector formula to use

(iv) use Algebra, Trigonometry, and/or Calculus to isolate the unknown

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

ARCHIVES:   CH.1   CH.2  CH.3  CH.4  CH.5&6 CH.7&8  CH.9  CH.10&11  CH.12