1. Review Chap. 12 Gravitation
PHYS 218 sec
Review
Chap. 12
Gravitation

2. What you have to know Newton’s law of gravitation
Gravitational potential energy
Motion of satellites
Kepler’s three laws
We skip from section 12.6 through section 12.8

3. N. Copernicus
G. Galilei
I. Newton
J. Kepler
T. Brahe

4. Newton’s law of gravity
Always attractive
In vector form, the gravitational force exerted by m2 on m1 is
The negative sign means that it is an attractive force.

5. Gravity for spherically symmetric bodies
For an object which has spherically symmetric mass distribution: concentrate all the mass of the object at its center.
Earth of mass mE

6. Acceleration due to gravity
Ex 12.2
Acceleration due to gravity

7. Superposition of gravitational forces
Ex 12.3
Superposition of gravitational forces
Gravitational force is a vector.
The gravitation force exerted on m
= vector sum of two forces

8. At the surface of the Earth, we can neglect other stellar objects.
Weight
Weight of a body: the total gravitational force exerted on the body by ALL other bodies in the universe
At the surface of the Earth, we can neglect other stellar objects.
Mass of the Earth
Radius of the Earth

9. Use this information to know the mass of the Mars lander
Ex 12.4
Gravity on Mars
Use this information to know the mass of the Mars lander
At d = m above the surface of Mars

10. Gravitational potential energy
When the gravitational acceleration is constant
In general, the gravitational acceleration depends on r
Gravitational force
displacement

11. Gravitational potential energy II
Gravitational force is conservative
At the surface of the Earth
= constant, so can be dropped

12. From the earth to the moon
Ex 12.5
From the earth to the moon
Muzzle speed needed to shoot the shell from RE to 2RE
To obtain the speed, we use energy conservation.

13. From the earth to the infinity
Ex 12.5b
From the earth to the infinity
Muzzle speed needed to shoot the shell from RE to infinity
Independent of the mass of the object
This is called the escape speed

14. Motion of satellites
Closed orbits
Open orbits

15. Satellites: circular orbits
The radius of the circular orbit of the satellite is determined by its speed.
Independent of the satellite mass

16. Satellites: circular orbits
For a given radius, satellite speed is determined, so is its energy

17. From the earth to the infinity
Ex 12.6
From the earth to the infinity
Speed, period, acceleration

18. The work needed to place this satellite in orbit
Ex 12.6
Cont’d
The work needed to place this satellite in orbit
The additional work to make this satellite escape the earth

19. Kepler’s laws
Kepler’s First Law: each planet moves in an elliptical orbit, with the sun at one focus of the ellipse
This can be shown by solving the equation of motion based on Newton’s theory on gravity and Newton’s second law of motion. (but needs higher level of math)
e: eccentricity in most cases, e is very small and the orbit is close to a circle
Aphelion: distance between P and S is maximum.
Perihelion: distance between P and S is minimum.

20. A result of angular momentum conservation
Kepler’s Second Law: A line from the sun to a given planet sweeps out equal area in equal times
A result of angular momentum conservation
The line SP sweeps out equal areas in equal times
See the textbook for the proof.
Kepler’s Third Law: The period of the planets are proportional to the 3/2 powers of the major axis lengths of their orbits
We have seen this for the case of circular orbit. But this is true even for elliptic orbits.