Transcript Slide 1
Chapter 13 Gravitation
Newton’s law of gravitation
•
Any two (or more) massive bodies attract each other
•
Gravitational force (Newton's law of gravitation)
F
G m
1
m
2
r
2
r
ˆ •
Gravitational constant 6.67*10 –11
G
= 6.67*10 –11 N*m 2 /kg 2 m 3 /(kg*s 2 ) – universal constant =
Gravitation and the superposition principle
•
For a group of interacting particles, the net gravitational force on one of the particles is
F
1 ,
net
i n
2
F
1
i
•
For a particle interacting with a continuous arrangement of masses (a massive finite object) the sum is replaced with an integral
F
1 ,
body
body
d F
Chapter 13 Problem 5 Three uniform spheres of mass 2.00 kg, 4.00 kg and 6.00 kg are placed at the corners of a right triangle. Calculate the resultant gravitational force on the 4.00-kg object, assuming the spheres are isolated from the rest of the Universe.
Shell theorem
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For a particle interacting with a uniform spherical shell of matter
F
1 ,
shell
shell d
F
•
Result of integration: a uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell's mass were concentrated at its center
Gravity force near the surface of Earth
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Earth can be though of as a nest of shells, one within another and each attracting a particle outside the Earth’s surface
•
Thus Earth behaves like a particle located at the center of Earth with a mass equal to that of Earth
F
1 ,
Earth
G m Earth m
1 2
R Earth
ˆ
j
Gm Earth
2
R Earth
m
1
j
ˆ
m
1
j
ˆ
g
= 9.8 m/s 2
•
This formula is derived for stationary spherical shape and uniform density Earth of ideal
Gravity force near the surface of Earth In reality
g
is not a constant because: Earth is rotating , Earth is approximately an ellipsoid with a non-uniform density
Gravitational field
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A gravitational field exists at every point in space
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When a particle is placed at a point where there is gravitational field, the particle experiences a force
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The field exerts a force on the particle
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The gravitational field is defined as:
g
F g m
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The gravitational field is the gravitational force experienced by a test particle placed at that point divided by the mass of the test particle
Gravitational field
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The presence of the test particle is not necessary for the field to exist
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The source particle creates the field
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The gravitational field vectors point in the direction of the acceleration a particle would experience if placed in that field
Gm Earth
2
R Earth
g
•
The magnitude is that of the freefall acceleration at that location
Gravitational potential energy
•
Gravitation is a conservative is path-independent ) force (work done by it
U
•
For conservative forces (Ch. 8):
r
f
r i
F
d r
r r i
f
Gm
1
m Earth r
2
dr
Gm
1
m Earth
1
r i
1
r f
Gravitational potential energy
U
U f
U i
Gm
1
m Earth r
1
i
1
r f
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To remove a particle from initial position to infinity
U
U i
Gm
1
m Earth
1
r i
1
Gm
1
m Earth r i
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Assuming
U ∞ =
0
U i
(
r i
)
Gm
1
m Earth r i U
(
r
)
Gm
1
m
2
r
Gravitational potential energy
U
(
r
)
Gm
1
m
2
r
Escape speed
•
Accounting for the shape of Earth, projectile motion (Ch. 4) has to be modified:
a c
v
2
R
g
v
gR
Escape speed
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Escape speed : speed required for a particle to escape from the planet into infinity (and stop there)
K i
U i
K f
U f m
1
v
2 2
Gm
1
m planet R planet
0 0
v escape
2
Gm planet R planet
Escape speed
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If for some astronomical object
v escape
2
Gm object R object
3 10 8
m
/
s
c
•
Nothing (even light) can escape from the surface of this object – a black hole
Chapter 13 Problem 30 (a) What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system, if it starts at the Earth’s orbit? (b) Voyager 1 achieved a maximum speed of 125 000 km/h on its way to photograph Jupiter. Beyond what distance from the Sun is this speed sufficient to escape the solar system?
Kepler’s laws
Tycho Brahe/ Tyge Ottesen Brahe de Knudstrup Johannes Kepler • (1546-1601)
Three Kepler’s laws 1.
The law of orbits : All planets move in
(1571-1630)
elliptical
•
orbits, with the Sun at one focus 2.
The law of areas : A line that connects the planet
•
to the Sun sweeps out equal areas planet’s orbit in equal time intervals in the plane of the 3.
The law of periods : The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit
First Kepler’s law
•
Elliptical orbits of planets are described by a semimajor axis
a
and an eccentricity
e
•
For most planets, the eccentricities are very small (Earth's
e
is 0.00167)
Second Kepler’s law
L
•
For a star-planet system, the total angular
•
momentum is constant
rp
(
r
)(
mv
) (
(no external torques)
r
)(
m
r
)
mr
For the elementary area swept by vector
2
r dA
1 2 (
r
)(
rd
)
dA dt
r
2 2
d
dt
const r
2 2
dA
dt L
2
m
Third Kepler’s law
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For a circular orbit and the Newton’s Second law
F
ma GMm
(
m
)(
r
2 )
r
2 2
GM r
3
T
•
From the definition of a period
2
T
2 4 2 2
T
2 4
GM
2
r
3 •
For elliptic orbits
T
2 4
GM
2
a
3
Satellites
•
F
For a circular
ma
GMm r
orbit and the Newton’s Second law
2 (
m
)
v r
2 •
Kinetic energy of a satellite
K
mv
2 2
GMm
2
r
U
2 •
Total mechanical energy of a satellite
E
K
U
GMm
2
r
GMm r
GMm
2
r
K
Satellites
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For an elliptic orbit it can be shown
E
GMm
2
a
•
Orbits with different
e
total mechanical energy but the same
a
have the same
Chapter 13 Problem 26 At the Earth’s surface a projectile is launched straight up at a speed of 10.0 km/s. To what height will it rise? Ignore air resistance and the rotation of the Earth.
Questions?
Answers to the even-numbered problems Chapter 13 Problem 2 2.67 × 10 −7 m/s 2
Answers to the even-numbered problems Chapter 13 Problem 4 3.00 kg and 2.00 kg
Answers to the even-numbered problems Chapter 13 Problem 10 (a) 7.61 cm/s 2 (b) 363 s (c) 3.08 km (d) 28.9 m/s at 72.9
° below the horizontal
Answers to the even-numbered problems Chapter 13 Problem 24 (a) −4.77 × 10 9 J (b) 569 N down (c) 569 N up