1. Gravitation Sections 9.1 – 9.3

2. Reminders Reading quiz for Chapter 11 (Sections 1-2) due by start of class on Tuesday. Lab this week: Lab A7-CE – Conservation of Energy – due by Friday at 4 p.m. In-class Quiz #4 on Thursday, October 23 – addresses chapters 8 and 9.

3. Newton’s Apple Perhaps in contemplating the fall of an apple, Newton realized that its acceleration was (in modern metric terms). He realized that the moon’s acceleration toward Earth was

4. Working… Earth acts as though all its matter is concentrated in a point at its center.

5. Working…

6. Newton Finds Kepler’s Third Law P 2 = r 3 (P in years; r in astronomical units) P 2 = kr 3 (arbitrary units) Newton derives this relationship

7. A Note about Dark Matter

8. Relationship Weight versus Mass Weight is a force defined by (kg. m/s 2 ). Recall that 1kg. m/s 2 is a Newton, N What weighs 1 N? – A medium apple – A cooked quarter-pounder burger (meat only) Mass is the amount of matter in a body. To find relationship between weight and mass make a graph of weight versus mass.

9. Time for an Experiment! Let’s graph gravitational force (weight) versus mass to find the relationship.

10. Gravitational Field Strength W t = km where k = g; ergo, W t = mg g = gravitational field strength in units of N/kg g = -9.81N/kg When simplified, Hence, g = -9.81m/s 2 Gravitational field strength has the units of acceleration and is directed downward.

11. Warning! Do not confuse g with G. g = -9.81N/kg = -9.81m/s 2 G = 6.67x10 -11 Nm 2 /kg 2 W t = mg = -GMm/r 2 g = -GM/r 2 = -(6.67x10 -11 Nm 2 /kg 2 ) * (5.97x10 24 kg)/(6,371,000m) 2 = -9.81m/s 2

12. Gravitational PE on Universal Scale PE = mgh = -GMm/r At cosmic distances, – E total = KE + PE (where PE = 0 at ∞) – E total = ½mv 2 – GMm/r

13. Black Holes Recall, E total = KE + PE = ½mv 2 – GMm/r If E total = 0, then v escape = √(2GM/r) If E total > 0, then v > v escape If E total < 0, then v < v escape In a black hole, M is “normal” but r can be vanishingly small. Hence, when something gets close enough to a black hole it can not escape – even light!

14. Sample Problems #1 and #2 #1 Calculate the weight of a 0.454kg loaf of bread on the surface of Earth. Give the magnitude and direction of that force. #2 Show that the local value of “g” (the gravitational field strength) on the surface of the moon is approximately 1/6 that on the surface of Earth. Note that g Earth = -9.81N/kg

15. Sample Problems #3 and #4 #3 How much gravitational force exists between two 1kg lead spheres whose centers are 1m apart from one another? #4 Show mathematically that the acceleration due to gravity in a 300km high orbit is 91.2% that on the surface of Earth. Assume a radius of 6,371km for Earth.