Physics 218, Lecture XXI 1

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

Chapter 12: Overview
Physics 218, Lecture XXI 3

Physics 218, Lecture XXI 4

Gravitation
Physics 218, Lecture XXI 5

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 6

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 7

The Force of Gravity

F gravity

G m
1 m
2 r
2
(

Direction of the
Distance between force the masses r
ˆ
)
G

6.67

10

11
N
 m
2
/kg
2
Physics 218, Lecture XXI 8

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 9

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 10

Binary System
Two equal mass stars maintain a constant distance R
S 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?
R
S
Physics 218, Lecture XXI 11

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

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 13

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
M
E
, determine the height of the satellite
(from the center of the Earth) in terms of the period, M
E and G
Physics 218, Lecture XXI 14

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 15

Physics 218, Lecture XXI 16

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

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 L
L
Physics 218, Lecture XXI 18

Density of a Planet
A satellite orbits very near a planet’s surface (I.e., R=R
Planet
, 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 19

Kepler’s 1 st 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 20

Kepler’s 2 nd 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 21

Kepler’s 3 rd 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.



T
T
2
1


 2




S
S
2
1


 3
Physics 218, Lecture XXI 22

Derivation of 3 rd Law for a Circle
Let’s say there are two small planets revolving around the sun in circular motion with periods
T
1 and T
2 and radii R
1 and R
2 respectively.
Use Newton’s laws to show the following:
Assume the masses of



T
T
1
2


 2




R
R
1
2


 3 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 23

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 24

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 25

Calculate the Magnitude of g
For a person on the earth what is
g. Use
–
M
Earth
= 5.97*10 24 kg
–
G = 6.67*10 -11 N*m 2 /kg 2
–
R = R
Earth
= 6.38*10 6 m
Physics 218, Lecture XXI 26

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

Bevo and Reveille
Bevo and Reveille have masses m
B are standing D meters apart. and m
R and
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 28

How to solve these types of problems
•
Some thoughts
–
What keeps a satellite up? Its speed
–
|Accel| = v 2 /r
–
|Force| = ma = mv 2 /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:
F
Gravity
= F
Uniform Circular Motion
Physics 218, Lecture XXI 29

Space craft in Orbit
A space craft, with mass m, is circling the earth at radius R = 2r
Earth
.
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 30

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 31

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 32

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 33

Vector Form

21
 
1
2
21
2 r
12
Physics 218, Lecture XXI 34

More on Vector Form
21
 
12

21

 
12
Physics 218, Lecture XXI 35