1. Physics 218 Lecture 21
Dr. David Toback
Physics 218, Lecture XXI

2. Schedule for the Rest of the Semester
Today: Chapter 12
Tuesday Nov 21st: Exam 3
Thursday Nov 23rd: No class, Thanksgiving
Tuesday November 28th: Chapter 13, Part 1
Thursday November 30th: Chapter 13, Part 2
Tuesday December 5th: Final Exam Review
No lecture on December 7th (Reading day)
Final exam is Monday, December 11th, 1-3PM
We will skip Chapter 15
Physics 218, Lecture XXI

3. Newton’s gravitational law Dynamics and Gravity
Chapter 12: Overview
Newton’s gravitational law
Dynamics and Gravity
Gravity and Uniform Circular Motion
Escape Velocity
Physics 218, Lecture XXI

4. Physics 218, Lecture XXI

5. “Every particle in the universe attracts every other particle”
Gravitation
Newton’s law of Universal Gravitation
“Every particle in the universe attracts every other particle”
Physics 218, Lecture XXI

6. Large number of scales Kinda amazing!
Gravity covers the attraction between
An apple near the earth
The earth and the moon
The earth and the sun
The sun and our galaxy
Our galaxy and the universe
Every particle in the universe and an apple
The Earth and you
Bevo and Reveille
Physics 218, Lecture XXI

7. Newton’s Law
“Every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining the particles”
Gravity has a magnitude and direction
 Gravity is a force
Physics 218, Lecture XXI

8. Distance between the masses
The Force of Gravity
Direction of the force
Distance between the masses
Physics 218, Lecture XXI

9. The motion of the moon and the planets
One of the great achievements of physics is that it explains the motion of planets
It took awhile, but they eventually figured out that the motion of the planets made much more sense if one assumed that the Sun was the center of motion rather than the Earth
Newton was able to use his gravitational law and Uniform Circular Motion to “Predict” these observations
Physics 218, Lecture XXI

10. Gravity and Circular Motion
Use the force of gravity along with other forces in force diagrams
Circular motion is motion with the acceleration pointed towards the center of the circle
The Earth is a good “center” acceleration for Satellites and Moons
Physics 218, Lecture XXI

11. Binary System
Two equal mass stars maintain a constant distance RS apart and rotate about a point, midway between them, at a revolution rate of once per time T
What must be the mass of each star? RS
Physics 218, Lecture XXI

12. Gravitational Potential Energy
How much potential energy does a ball of mass m have in outer space?
Assuming you know ME and G, calculate how much work you would have to do move a ball from the surface of the earth to some distance R R
Physics 218, Lecture XXI

13. Escape Velocity
What happens if I throw a ball up in the air? Will it fall down?
What if I throw it up really fast?
What if I throw it up REALLY fast?
Can I throw it up so fast that it will never come down? How fast would that be?
Physics 218, Lecture XXI

14. Example: Geosynchronous Satellite
A satellite is in orbit around the Earth and its speed is such that it always stays above the same point on the earth throughout the day.
Assuming a spherical Earth with mass ME, determine the height of the satellite (from the center of the Earth) in terms of the period, ME and G
Physics 218, Lecture XXI

15. Next Week Monday: Chapter 11 HW due Tuesday: Exam 3
Covers Chapter 8 through 11
Only sections listed on the syllabus
Mini-practice exam is open and available for people who are caught up. Usual 5 points
Thursday: Thanksgiving, no class
Following Tuesday: Lecture on Chapter 13
Periodic Motion
Physics 218, Lecture XXI

16. End of Lecture Notes
Physics 218, Lecture XXI

17. Do not sell your textbook! You will need the same book for 208
Note:
Do not sell your textbook! You will need the same book for 208
Physics 218, Lecture XXI

18. Three Bodies
Three bodies of identical and known mass M form the vertices of an equilateral triangle of side L. They rotate in circular orbits around the center of the triangle and are held in place by their mutual gravitation.
What is the speed of each? L
Physics 218, Lecture XXI

19. Density of a Planet
A satellite orbits very near a planet’s surface (I.e., R=RPlanet, but we don’t know R) with period T.
What is the density of the planet in terms of the measured period T?
Physics 218, Lecture XXI

20. Kepler’s 1st Law
Kepler: The path of each planet about the Sun is an ellipse with the Sun at one focus.
Newton: Gravity gives a center seeking force! Centripetal forces can lead to both elliptical motion or circular motion (circular motion is just a special case which we happen to have studied already!)
Physics 218, Lecture XXI

21. Kepler’s 2nd Law
Kepler: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out area in equal periods of time
Newton: Can show this, but it’ll be easier when we get to angular momentum. We’ll come back to this later
Physics 218, Lecture XXI

22. For a Circle S = Radius Kepler’s 3rd Law
Kepler: The ratio of the squares of the periods of any two planets revolving about the Sun is equal to the ratio of the cubes of their semi-major axes.
For a Circle S = Radius
Physics 218, Lecture XXI

23. Derivation of 3rd Law for a Circle
Let’s say there are two small planets revolving around the sun in circular motion with periods T1 and T2 and radii R1 and R2 respectively.
Use Newton’s laws to show the following:
Assume the masses of the planets are so small or that they are so far apart that we may neglect the gravitational attraction between the two.
Physics 218, Lecture XXI

24. Understanding the motion of planets
One of the great achievements of physics is that it explains the motion of planets
It took awhile, but they eventually figured out that the motion of the planets made much more sense if one assumed that the Sun was the center of motion rather than the Earth
Then Kepler made some important observations WAY before Newton
Newton was able to “Predict” these observations (and now so are we…)
Physics 218, Lecture XXI

25. A way simplify problems
It takes some fancy integration, but one can show that we can “model” the Earth as if all the mass were concentrated at its center
One can model any sphere as a point
This is why we like to model things as spheres in the first place.
Physics 218, Lecture XXI

26. Calculate the Magnitude of g
For a person on the earth what is g. Use
MEarth = 5.97*1024 kg
G = 6.67*10-11 N*m2/kg2
R = REarth = 6.38*106 m
Physics 218, Lecture XXI

27. Ok… So what? A single formula can show that:
the acceleration of an apple towards the Earth has magnitude g and is pointing down
as you may have noticed, Bevo and Reveille aren’t attracted to each other very much
Physics 218, Lecture XXI

28. Bevo and Reveille
Bevo and Reveille have masses mB and mR and are standing D meters apart.
Despite what you might like to believe, what is the attraction between them? More specifically, find Reveille’s acceleration.
Hint: Assume a spherical cow
Perhaps this explains why we’ve never observed any attraction…
Physics 218, Lecture XXI

29. How to solve these types of problems
Some thoughts
What keeps a satellite up? Its speed
|Accel| = v2/r
|Force| = ma = mv2/r
The trick is going to be to ask the question
What are the forces?
Is it in uniform circular motion? If so, we can use Newton’s law:
FGravity = FUniform Circular Motion
Physics 218, Lecture XXI

30. Space craft in Orbit
A space craft, with mass m, is circling the earth at radius R = 2rEarth.
What is the force on the space craft in terms of g and m?
Model the space craft as a single point near the Earth which we model as a point at its center
Physics 218, Lecture XXI

31. Satellites/Orbiting Problems
A satellite problem is a good example of the substantive problems we need to be able to solve. Predict the outcome of the experiment
Physics 218, Lecture XXI

32. Weightlessness What is the weight of the person in the figure?
What is the difference between being in “free fall” and being “out of the reach of large gravitational forces?”
Physics 218, Lecture XXI

33. Throwing a Baseball
A person throws a baseball at 100 km/hr, but it is attracted back towards him because of gravity.
Estimate the force 1 Meter away?
Assuming constant acceleration (Bad assumption), how long does it take to turn around?
Physics 218, Lecture XXI

34. Vector Form
Physics 218, Lecture XXI

35. More on Vector Form
Physics 218, Lecture XXI