1. Universal Gravitation & Satellites
Textbook: 3.3 & 3.4
Homework: pg. 141 # 3, 5
pg. 143 # 9, 11, 12
pg. 147 #2

2. Universal Gravitation
Gravity is the mutual force of attraction between any two objects that contain matter, regardless of their size.
The strength of the gravitational force between two objects depends on two variables: mass and distance.

3. The Law of Universal Gravitation
Universal Gravitational Constant,
Mass of object 1, m1
Mass of object 2, m2
Distance between the centres of the object, r
Mass of the Earth = 5.92*10^24 kg
Radius of the Earth = 6371 km

4. Universal Gravitation
Any two objects of mass m and M, separated by a distance r, will attract each other with a force FG
G = 6.67 x Nm2/kg2

5. Gravitational Fields
The gravitational field strength of an object M a distance r away is:
[g] = N/kg = m/s2

6. Ex. 1.

7. Satellites and Space Stations
Any object that is orbiting Earth is undergoing constant free fall toward Earth
Therefore,
Minimum orbiting speed:

8. Ex. 1: The ISS travels at an altitude of 450 km above the surface of the Earth.
Ex. 2: How fast must you run to be able to hover above the surface of Earth? 7.9x103m/s
(b) How many times will the astronaut circle the Earth in 1 hour?
(a) Determine the gravitational force on a 64 kg astronaut at that altitude.

9. Apparent Weight and Artificial Gravity
Apparent weight: is the contact force exerted on an accelerating object in its non-inertial frame of reference.
Artificial gravity: where the apparent weight of an object is similar to its weight on Earth

10. Universal Gravitation & Kepler’s Laws
Textbook: 6.1 – 6.2
Homework: pg. 277 # 1 – 8
pg. 284 # 2 – 9

11. “geocentric model” of the universe: Earth is the frame of reference
In 1543, Polish astronomer Nicolas Copernicus
(1473–1543) published the “heliocentric model” of the solar system: planets revolve around the Sun
Danish astronomer Tycho Brahe (1546–1601): naked-eye observations using unusually large instruments – quadrant – orbital shapes???
Before his death, Tycho hired a brilliant young mathematician to assist in the analysis. That mathematician was Johannes Kepler (1571–1630).

12. Kepler’s Laws of Planetary Motion
K1: All planet’s move around the Sun in an elliptical orbit with the Sun at one focus
K2: A straight line connecting a planet and the Sun sweeps out equal areas in equal times.
K3: The period T of a planet’s orbit is related to the average radius of the planet’s orbit, r

13. Circular Motion
Kepler’s Constant for objects travel around a planet

14. Between March 21 and September 21, there are three days more than between September 21 and March 21. These two dates are the spring and fall equinoxes when the days and nights are of equal length. Between the equinoxes, Earth moves 180º around its orbit with respect to the Sun. Using Kepler’s laws, explain how you can determine the part of the year during which the Earth is closer to the Sun. Draw an Earth’s revolution around the Sun.

15. (a) Use the data of the Moon’s motion (refer to Appendix C) to determine Kepler’s third-law constant CE to three significant digits for objects orbiting Earth. [1.023 x 1013 m3/s2 ]
(b) If a satellite is to have a circular orbit about Earth (mE = x 1024 kg) with a period of 4.0 h, how far, in kilometres, above the surface of Earth must it be? What must be its speed? [6.5 x 104 km; 5.63 x 103 m/s]

17. Gravitational Potential Energy & Orbital Energy
Textbook: 6.3
Homework: pg. 294 # 1 – 9

18. Newton’s New Mathematics
Newton saw the need to accurately
calculate areas. To do so,
he developed a whole new branch
of mathematics called calculus.
At approximately the same time,
independently of Newton, Gottfried
Wilhelm Leibniz (1646–1716), a
German natural philosopher, also
developed calculus.

19. Gravitational Potential Energy
The change in gravitational potential energy of an object m, due to the field of M, when m moves from r1 to r2 is:
We choose Eg to be zero at infinity so:

20. Orbits
Gravity creates a “potential well” that prevents objects from escaping
An orbit in a gravitational field is defined by the total mechanical energy
ET < 0  Objects is “bound”
ET  0  Object is “unbound”

21. Orbital Energy The total energy of an orbit is found by:
If ET < 0 then the object is bound
If ET > 0 then the object is unbound

22. Binding Energy
The kinetic energy required to become unbound is called the binding energy
Ebinding = r

23. Escape Speed
Escape speed is the minimum speed needed to project a mass m at radius R from the centre of mass M to escape the gravity of the mass M.
Ek + Eg = 0

24. True/False
As a space probe travels away from Earth, its change in gravitational potential energy is positive, even though its gravitational potential energy is negative.

25. Pg 293
11. Consider a geosynchronous satellite with an orbital period of 24 h. (a) What is the satellite’s speed in orbit? [3.1 x 103 m/s]
(b) What speed must the satellite reach during launch to escape Earth’s gravitational field? (Assume all fuel is burned in a short period. Neglect air resistance.) [1.1 x 104 m/s]

26. Black hole: is a small, very dense body with a gravitational field so strong that nothing can escape from it, even light.
Event horizon: is The surface of a black hole
A singularity: is an unbelievably dense centre, inside the event horizon, at the very core of the black hole.
(Karl) Schwartzschild radius: is the distance from the centre of the singularity to the event horizon

27. True/False
X rays and gamma rays can escape from a black hole, even though visible light cannot.

28. Pg 293
12. Determine the Schwarztschild radius, in kilometres, of a black hole of mass 4.00 times the Sun’s mass.