Download Lecture 6 Review

Lecture 6 Review
1) Galileo A) observed the moons of Jupiter wit a telescope
B) observed the phases of Venus
C) observed sun spots
D) (1642) published the Dialogue of the Two Chief World Systems
2) Uranus was discovered accidentally by Herschel in 1781.
Neptune was discovered in 1846 from perturbations to the orbit of Uranus.
Pluto was discovered in 1930 - orbit quite elliptical and at an angle to the ecliptic.
3) Planets: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto
circumference 2 π r r
=
~ .
period
T
T
r2 1
3
2
According to Kepler’s Third Law, r ~ T , then 2 ~ .
T
r
r
1
Thus, v ~ ~
.
T
r
4) Given velocity = v =
If we assume that most of the mass in the solar system is in the Sun and that the
planets are spherically symmetrical, then, if we plot v versus r we should find the
planetary velocity has the above radial dependence.
We note that the moons of Jupiter and Saturn behave similarly with respect to the
planets they revolve around.
5) Isaac Newton (1642-1727) published the Principea in 1687 at age 41.
Newton invented The Calculus (along with Leibnitz, independently). The slope is
called the derivative (the tangent at any point of the curve) of the function x(t) and
represents the velocity. The derivative of the velocity is called the acceleration. In
the cases shown the velocities and accelerations are not constant.
For
form circular motion (constant speed v, constant radius r) what is the mechanical
relationship between the vector quantities of acceleration, velocity and position?
where ∆ v = θ v
For the left hand figure
radian measure is defined
as θ =
s v ∆t
.
=
r
r
Plugging this into the
equation above you find
v2
∆v =
∆ t . Thus the centripetal acceleration is calculated to be
r
∆ v v2
centripetal acceleration = a =
inward
=
∆t
r
uni
C) Newton’s Three Laws
1) A body remains at rest or in constant velocity unless acted upon by a force F( in
Newtons).
r
r
2) F = ma
F
= m = inertial mass
a
or
3) For every action there is an equal and opposite reaction.
D) Newton’s Universal Law of Gravitation says there is an attractive force acting
between any two mass points m and M.
r
GMm This is known as the Inverse Square Law, where
F= −
$r
r2
G = 6.67 x 10-11 Nm2/kg2 is the gravitation constant and r$ is a unit vector pointing
in the direction between the masses.
6) The acceleration due to gravity on the Earth
F = ma = mg =
⇒ g=
GM E
=
R 2E
GmM E
= weight of mass m
R E2
2

−11 Nm 
24
kg)
 6 .6 7x1 0
2  ( 6 x 10
kg 

(6 .3 7 x 10 6 m)
2
= 9 .8
m
s2
On the Moon
g Moon
GM Moon
=
=
R 2Moon
2

− 11 Nm 
7.36x10 22 kg
 6.67x10
2 
kg 

(
(1.74x10 m )
6
2
)
m
= 1.6 2
s
7) The circular velocity necessary to orbit a large mass without falling = vcirc.
For this to happen implies the centripetal force = Newton’s gravitational force
2
mv circ
GmM
ma circ =
=
R
R2
GM
solving vcirc =
R
Near the surface of the Earth we find v circ =
GM E
m
= 7981 = 17 ,700 mph
RE
s
This is the velocity needed to circle the Earth at, or near, the surface of the Earth.
Thus, satellites 100 miles up need this speed to remain in orbit. Less than this and the
rocket does not reach orbit. Some type of orbit is reached for all velocities less than
the escape velocity.
8) The escape velocity from a planets surface is
v esc =
2 GM
=
R
2 v circ ( r )
From the Earth’s Surface vesc = 11,287 m/s = 25,000 mph.
9) Now, we are not in stable orbit standing on the Earth, the Earth doesn’t rotate that
fast, so we can’t set centripetal force equal to gravitational force.
But we can determine the acceleration due to gravity, g, on the Earth by watching
things fall.
g ~ 10 m/s2 for all objects at RE = 6.4 x 106 m
What about the centripetal acceleration acting on the Moon to keep it in orbit?
The Moon’s circular velocity is
2 π (3 .8 4 x10 8 m)
circumference 2 π R Moon
m
v=
=
=
= 1023
sec
sidereal period
TMoon
s
27 .3 days ⋅ 86400
day
The centripetal acceleration is
a Moon
1.023 x 10 3 )
(
v2
m
=
=
=
.
00272
R Moon
8 .84 x 10 8
s2
What is the ratio of accelerations?
What is the ratio of radii?
earth surface
10
=
= 3676
Moon
.00272
R Moon 3 .84 x 10 8
=
= 60
R Earth
6 .4 x 10 6
R Moon
= 3600
R Earth
That the ratio of accelerations and the inverse ratio of radii squared is no
coincidence. Newton used this fact to propose his universal Law of Gravitation and
the Inverse Square Law.
The arrangement of our solar system as it is now known and shown approximately to
scale is shown.
Newton realized that a vehicle
launched parallel to the surface of the
Earth from a mountaintop could travel
varying distances depending on its
launch velocity. At low speed a
projectile falls to Earth (A and B). At
one special speed the projectile
maintains a circular orbit (C). At
higher speeds the orbit becomes
elliptical (D and E). At the escape
velocity the orbit is parabolic and the
projectile never returns to the Earth
(F). At still higher velocities the orbit
is hyperbolic (G) and the projectile never returns to the Earth.
Bode’s Rule: Distances of Planets from the Sun
Mercury
Venus
Earth
Mars
Asteroids
Jupiter
Saturn
Uranus
Neptune
Pluto
4
4
4
4
4
4
4
4
4
4
0
3
6
12
24
48
96
192
anomaly
384
0.4
0.7
1.0
1.6
2.8
5.2
10.0
19.6
--
38.8
0.4
0.7
1.0
1.5
2.8
5.2
Note: All distances are expressed in astronomical units
9.5
19.2
30.0
39.4
Predicted
Actual
Dis tan ce in AU =
3(0,1,2,4,8,16,32,64, ,128) + 4
10