1. Gravitational Potential Energy & Orbital Energy
Textbook: 6.3
Homework: pg. 294 # 1 – 9
Ex. How much work is done to move an object from a position r1 to a position r2 in a gravitational field?
Discuss - To derive this properly we need integral calculus. Refer to pg. 285 in textbook.

2. 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.
Eg = mgh
Eg = m(GM/r^2)r
Eg = GMm/r
W = delta(Eg) = Eg2 – Eg1
W = Area under curve = GMm/r1 – GMm/r2

3. 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:
Ex. Pg. 287 #1 - 3

4. 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”
Ex. Explore the graph of Eg and discuss what a potential well means
Ex. Pg #4, 5
Ex. What is the

5. 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
Ex. Kepler’s 2nd Law in terms of Energy Conservation
Pg 293 #11
Pg. 293 #8

6. Binding Energy
The kinetic energy required to become unbound is called the binding energy
Ebinding = r
Discuss Binding Energy
Ex. Derive escape speed
Ex. What is the escape speed of the planet Earth
Ex. Pg #12

7. 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

8. 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.

9. 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 attain the geosynchronous orbit? (Assume all fuel is burned in a short period. Neglect air resistance.) [1.1 x 104 m/s] (c) How much additional energy would have to be supplied to the satellite once it was in orbit, to allow it to escape from Earth’s gravitational field? [2.36 x 109 J]

10. 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

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

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