Transcript Physics 131: Lecture 14 Notes

Physics 151: Lecture 28
Today’s Agenda

Today’s Topics
Gravity and planetary motion (Chapter 13)
Physics 151: Lecture 28, Pg 1
See text: 13.1
Gravitation
according to Sir Isaac Newton
Newton found that amoon / g = .000278
 and noticed that RE2 / R2 = .000273

amoon
g
R

RE
This inspired him to propose the
Universal Law of Gravitation:
|FMm |= GMm / R2
G = 6.67 x 10 -11 m3 kg-1 s-2
Physics 151: Lecture 28, Pg 2
See text: 13.1
Gravity...


The magnitude of the gravitational force F12
exerted on an object having mass m1 by another
object having mass m2 a distance R12 away is:
m1m2
F12  G 2
R12
The direction of F12 is attractive, and lies along
the line connecting the centers of the masses.
m1
F12
F21
m2
R12
Physics 151: Lecture 28, Pg 3
Example

What is the magnitude of the free-fall acceleration at a
point that is a distance 2R above the surface of the Earth,
where R is the radius of the Earth ?

a.
b.
c.
d.
e.




4.8 m/s2
1.1 m/s2
3.3 m/s2
2.5 m/s2
6.5 m/s2
Physics 151: Lecture 28, Pg 4
See text: 13.4
Kepler’s Laws
1st
All planets move in elliptical orbits with the sun
at one focal point.
2nd
The radius vector drawn from the sun to a
planet sweeps out equal areas in equal times.
3rd
The square of the orbital period of any planet is
proportional to the cube of the semimajor axis
of the elliptical orbit.

It was later shown that all three of these laws are a
result of Newton’s laws of gravity and motion.
Physics 151: Lecture 28, Pg 5
Example

Which of the following quantities is conserved for a planet
orbiting a star in a circular orbit? Only the planet itself is to
be taken as the system; the star is not included.

a.
b.
c.
d.
e.




Momentum and energy.
Energy and angular momentum.
Momentum and angular momentum.
Momentum, angular momentum and energy.
None of the above.
Physics 151: Lecture 28, Pg 6
Example

The figure below shows a planet traveling in a clockwise
direction on an elliptical path around a star located at one
focus of the ellipse. When the planet is at point A,

a.
b.
c.
d.
e.




its speed is constant.
its speed is increasing.
its speed is decreasing.
its speed is a maximum.
its speed is a maximum.
Animation
Physics 151: Lecture 28, Pg 7
Example

A satellite is in a circular orbit about the Earth at an altitude at
which air resistance is negligible. Which of the following
statements is true?


a.
b.

c.

d.
e.

There is only one force acting on the satellite.
There are two forces acting on the satellite, and their
resultant is zero.
There are two forces acting on the satellite, and their
resultant is not zero.
There are three forces acting on the satellite.
None of the preceding statements are correct.
Physics 151: Lecture 28, Pg 8
Example

A satellite is placed in a geosynchronous orbit. In this
equatorial orbit with a period of 24 hours, the satellite
hovers over one point on the equator. Which statement
is true for a satellite in such an orbit ?
a. There is no gravitational force on the satellite.
b. There is no acceleration toward the center of the Earth.
c. The satellite is in a state of free fall toward the Earth.
d. There is a tangential force that helps the satellite keep
up with the rotation of the Earth.
e. The force toward the center of the Earth is balanced by
a force away from the center of the Earth.
Physics 151: Lecture 28, Pg 9
Example

Normally, if I throw a ball up in the air it will eventually
come back down and hit the ground.

What if I throw it REALLY hard so that I put it into an
orbit !

How HARD do I have to throw ?
Physics 151: Lecture 28, Pg 10
See text: 13.7
Energy of Planetary Motion
A planet, or a satellite, in orbit has some energy
associated with that motion.
Let’s consider the potential energy due to gravity in
general.
Mm
F  G
r2
s
p
R2
r2
M sm p
W   F(r)dr    G 2 dr
r
r1
r1

U
1 1
U  U f U i  W  GMsm p (  )
rf ri

Define ri as infinity

U 
GM s m p
RE
r
0
U
1
r
r
 151: Lecture 28, Pg 11
Physics
See text: 13.7
Energy of a Satellite
A planet, or a satellite, also has kinetic energy.
1 2 GM s m p
E  K  U  mv 
2
r
We can solve for v using Newton’s Laws,
GM s m p
r2
mv 2
 ma 
r
Plugging in and solving,
GM s m p GM s m p
GM s m p
E


2r
r
2r
Physics 151: Lecture 28, Pg 12
See text: 13.7
E
GM s m p
Energy of a Satellite
2r
So, an orbiting satellite always has negative total
energy.
A satellite with more energy goes higher, so r gets
larger, and E gets larger (less negative).
It’s interesting to go back to the solution for v.
1
GmM
K  mv 2 
2
2r
GM
v
r
v is smaller for higher orbits (most of the energy
goes into potential energy).
Physics 151: Lecture 28, Pg 13
Lecture 28, Act 2
Satellite Energies

A satellite is in orbit about the earth a distance of
0.5R above the earth’s surface. To change orbit it
fires its booster rockets to double its height above the
Earth’s surface. By what factor did its total energy
change ?
(a)
(b)
(c)
(d)
(e)
1/2
3/4
4/3
3/2
2
Physics 151: Lecture 28, Pg 14
Lecture 28, Act 2
Satellite Energies
GM E ms
E
Note : E2/E1 = 3/4 actually
2r
GM E ms
GM E ms means that the energy is
E1  

larger (because it is
2( RE  0.5RE )
3RE
negative)
GM E ms
GM E ms
E2  

2( RE  RE )
4 RE
(b)
GM E ms

E2
3
4 RE
Ratio 


GM
m
E1 
4
E S
3 RE
Physics 151: Lecture 28, Pg 15
Escape Velocity



Normally, if I throw a ball up in the air it will
eventually come back down and hit the ground.
What if I throw it REALLY hard ?
Two other options
1) I put it into orbit.
2) I throw it and it just moves away forever
– i.e. moves away to infinity
Physics 151: Lecture 28, Pg 16
Orbiting



How fast to make the ball orbit.
I throw the ball horizontal to the ground.
We had an expression for v above,
GM
v
r
(6.67 10 11 Nm 2 / kg 2 )(5.98 10 24 kg )
v
6.37 106 m
v  7.9km / s  16,000mi / hr
Physics 151: Lecture 28, Pg 17
Escape Velocity




What if I want to make the ball just go away from
the earth and never come back ?
(This is something like sending a space ship out
into space.)
We want to get to infinity, but don’t need any
velocity when we get there.
This means ETOT = 0 Why ??
1 2 GMm
Ei  mv 
0
2
R
v
2GM
R
(2)(6.67 10 11 Nm 2 / kg 2 )(5.98 10 24 kg )
v
 11.2km / s
6
6.37 10 m
Physics 151: Lecture 28, Pg 18
Example

A projectile is launched from the surface of a planet (mass
= M, radius = R). What minimum launch speed is required if
the projectile is to rise to a height of 2R above the surface
of the planet? Disregard any dissipative effects of the
atmosphere.
Physics 151: Lecture 28, Pg 19
Example

A satellite circles planet Roton every 2.8 h in an orbit
having a radius of 1.2x107 m. If the radius of Roton is
5.0x106 m, what is the magnitude of the free-fall
acceleration on the surface of Roton?

a.
b.
c.
d.
e.




31 m/s2
27 m/s2
34 m/s2
40 m/s2
19 m/s2
Physics 151: Lecture 28, Pg 20
Example

A spacecraft (mass = m) orbits a planet (mass =
M) in a circular orbit (radius = R). What is the
minimum energy required to send this spacecraft
to a distant point in space where the gravitational
force on the spacecraft by the planet is
negligible?
a. GmM/(4R)
b. GmM/R
c. GmM/(2R)
d. GmM/(3R)
e. 2GmM/(5R)
Physics 151: Lecture 28, Pg 21
Recap of today’s lecture

Today’s Topics
Gravity and planetary motion (Chapter 13)
Physics 151: Lecture 28, Pg 22