Evans SPH 4U1

Physics Grade 12

Unit 4: Energy & Momentum

Note 6: Gravitational Fields and Kepler's Laws

 

Reference: Chapter 6.1 & 6.2

 

Gravitational Field Strengths: can be determined by ( it is easy to derive) in N/kg

                                                                                   

where G is universal gravitation constant, M is the mass of the central body, r is the distance between the centres of the 2 bodies.

Example 1:

Calculate the magnitude of Saturn's surface gravitational field strength. The mass of Saturn is 5.67 x 1023 kg and the radius is 6.03 x 107 m.

Answer: 10.4 N/kg

 

Do Text Practice Page 276 #'s 3-5

 

Orbital Speed:

In Chapter 3 we discussed how satellites must maintain a certain speed for a particular orbit and we calculated it was: (remember r is the distance from the centre of the earth to the satellite)..

                                                                               

Just remember any object with mass can have a body in orbit around it.

 

Planetary Mechanics:

Plato (427-347 B.C.) - This ancient Greek assumed that the earth is fixed and the rest of the universe revolved around the earth in spherical orbits. This geocentric view lasted for about 2000 years.

Copernicus (1473-1543 A.D.) - Polish astronomer had a heliocentric view and thought the sun was the centre of the universe and the rest of the universe revolved around the sun in circular orbits.

Tycho Brahe (1546-1601) - Danish astronomer who charted the heavens and proposed the sun went around the earth and that all other planets revolved around the sum.

Johann Kepler (1571-1630) - German mathematician who worked with Brahe.  After Brahe's death Kepler found that the orbits of the planets were elliptical with the sun at one focus.

Galileo (1564-1642) - Galileo defended the Copernican system (heliocentred) and published a book "Dialogue on the Two Chief World Systems". He wrote in Italian (the people's language) not Latin.  The Catholic Church and Pope were offended and Galileo was imprisoned.  His works were denied to Roman Catholics for over 200 years and only in 1979 did the Church start to clear Galileo's name.

 

Movie

Kepler's Laws:

1. The planets move about the sun in elliptical orbits, with the sun at one focus of the ellipses.

2. The straight line joining the sun and a planet sweeps out equal areas in equal time intervals. The significance of this is tat the linear speed of the satellite is not constant.  It varies - fast when the planet is close to the sun and slower when it is at a greater distance away.

3. The cube of the average radius, r, of a planet's orbit is directly proportional to the square of the period, T, of the planet's orbit.    In the following, Cs is the constant of proportionality in m3/s2

Example 1:

The planet, Hoth , has two moons, Seti1 and Seti2.  If Seti1 orbits at a mean orbital radius of 2.0 glucks and has an orbital period of 2.0 years, calculate Seti2's orbital period if it orbits with a mean orbital radius of 8.0 glucks. (Answer: 16)

 

Example 2:

The mean radius of orbit of the earth around the sun is 1.49 x 1011m and its period around the sun is 365 days. The mean radius of orbit of the moon around the earth is 3.84 x 108 m and its period around the earth is 27.3 days. Find

a) Kepler's constant for the solar system in m3/s2  (Answer: 3.33 x 1018 m3/s2 - although actual value of Ks is 3.35 x 1018 m3/s2)

b) Kepler's constant for the earth's system of satellites in m3/s2  (Answer: 1.02 x 1013 m3/s2)

c) The period of revolution in seconds for an earth satellite if its mean radius of orbit is 8.65 x 106 m. (Answer: 7.97 x 103s)

 

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