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Lecture Outlines Chapter 12
Physics, 3
rdEdition
James S. Walker
Chapter 12
Gravity
Units of Chapter 12
• Newton’s Law of Universal Gravitation
• Gravitational Attraction of Spherical Bodies
• Kepler’s Laws of Orbital Motion
• Gravitational Potential Energy
• Energy Conservation
• Tides*
12-1 Newton’s Law of Universal Gravitation
Newton’s insight:
The force accelerating an apple downward is the same force that keeps the Moon in its orbit.
Hence, Universal Gravitation.
Figure 12-2
Dependence of the gravitational force on separation distance, r
12-1 Newton’s Law of Universal Gravitation
The gravitational force is always attractive, and points along the line connecting the two
masses:
The two forces shown are an action-reaction
Exercise 12-1
G är ett mycket litet tal (G = 6,67 10
-11Nm
2/kg
2).
Beräkna gravitationskraften mellan en husse (m
1=105 kg) och hans hund (m
2=11,2 kg) när de är på avståndet a) 1,0 m b) 10,0 m från varandra.
F(r = 1,00 m) = Gm
1m
2/r
2= 7,84•10
-8N
F(r = 10,0 m) = Gm
1m
2/r
2= 7,84•10
-12N
12-1 Newton’s Law of Universal Gravitation
G is a very small number; this means that the force of gravity is negligible unless there is a very large mass involved (such as the Earth).
If an object is being acted upon by several
different gravitational forces, the net force on it is the vector sum of the individual forces.
This is called the principle of superposition.
Example 12-1
How Much Force Is With You?
Exemple 12-1 How much force is with you?
Massan av rymdskeppet, Millenium Eagle, är 25,0 Mkg och vardera asteroidmassan är 350 Gkg.
Beräkna gravitationskraften på rymdskeppet i läge A och i läge B. Betrakta rymdskepp och asteroider
som punktobjekt.
r
A= [(3,00 km)
2+ (1,50 km)
2]
1/2= 3,35 km θ
1= θ
2= arctan(0,5) = 26,7°
F(r
A) = Gm
1m
2/r
A2= 52,0 N
Totala kraften (i x-led) = 2•52,0 N•cos(26,7°) = 93,0 N
r = 1,50 km
12-2 Gravitational Attraction of Spherical Bodies
Gravitational force between a point mass and a
sphere: the force is the same as if all the mass
of the sphere were concentrated at its center.
12-2 Gravitational Attraction of Spherical Bodies
What about the gravitational force on objects at
the surface of the Earth? The center of the Earth is one Earth radius away, so this is the distance we use:
Therefore,
Photo 12-2 Global model of the Earth’s gravitational strength Gravitationen är starkast vid brun färg, svagast för blå.
Example 12-2
The Dependence of Gravity on Altitude
Exemple 12-2 The Dependence of Gravity on Altitude
Vad är accelerationen på grund av gravitationen på toppen av Mount Everest (h = 8850 m)?
Vid h = 0 gäller g = 9,81 m/s
2dvs F = GMm/r
E2= gm
Vid h = 8850 m gäller
g
ME= GM/(r
E+h)
2= g/(1+h/r
E)
2= g/(1+0,00139)
2=
= g • 99,7% = 9,78 m/s
212-2 Gravitational Attraction of Spherical Bodies
The acceleration of gravity decreases “slowly”
with altitude:
12-2 Gravitational Attraction of Spherical Bodies
Once the altitude becomes comparable to the radius of the Earth, the decrease in the
acceleration of gravity is much larger:
Exercise 12-2 The Dependence of Gravity on Altitude
Vad är accelerationen på månen? (M
m= 7,35 • 10
22kg och radie R
m= 1,74 • 10
6m)
F = mGM
m/R
m2= g
mm g
m= GM
m/R
m2=
= (6,67 • 10
-11) • (7,35 • 10
22)/(1,74 • 10
6)
2= 1,62 m/s
2Månlandaren har massan 225 kg. Vad vägde den a) på jorden? 225 kg • 9,81 m/s
2= 2210 N
b) på månen? 225 kg • 1,62 m/s
2= 364 N
12-2 Gravitational Attraction of Spherical Bodies
The Cavendish experiment allows us to measure
the universal gravitation constant:
12-2 Gravitational Attraction of Spherical Bodies
Even though the gravitational force is very small, the mirror allows measurement of tiny deflections.
Measuring G also allowed the mass of the Earth to
be calculated, as the local acceleration of gravity
and the radius of the Earth were known.
Exercise 12-3 Hur stor är jordens massa?
Vid havets nivå gäller g = 9,81 m/s
2dvs F = GMm/r
E2= gm
M = g • r
E2/G = 9,81•(6,37•10
6m)
2/(6,67•10
-11) =
= 5,97•10
24kg
Example 12-3 Mars Attracts!
Exemple 12-3 Mars attracts!
Hur stor är Mars massa?
Dess radie är 3,39•10
6m och vid Marsytan gäller
g
M= 3,73 m/s
2dvs
M = g
M• R
M2/G = 3,73•(3,39•10
6m)
2/(6,67•10
-11) =
= 6,43 • 10
23kg
12-3 Kepler’s Laws of Orbital Motion
Johannes Kepler made detailed studies of the
apparent motions of the planets over many years, and was able to formulate three empirical laws:
1. Planets follow elliptical orbits, with the Sun at
one focus (brännpunkt) of the ellipse.
12-3 Kepler’s Laws of Orbital Motion
2. As a planet moves in its orbit, it sweeps out an
equal amount of area in an equal amount of time.
Conceptual Checkpoint 12-1 Compare speeds
Jordens bana runt solen är lätt elliptisk, så jorden är något närmare solen under vintern (!). Är då jordens hastighet < = > än hastigheten under
sommaren?
12-3 Kepler’s Laws of Orbital Motion
3. The period, T, of a planet increases as its
mean distance from the Sun, r, raised to the 3/2 power.
This can be shown to be a consequence of the
inverse square form of the gravitational force.
Figure 12-10
Kepler’s third law and some near misses
Härledning av Keplers tredje lag a
cp= v
2/r
F = ma
cp= m v
2/r = m (2πr/T)
2/r = mr 4π
2/T
2men kraften är ju också
F = GM
sm/r
2sätter man dessa uttryck lika får man
T
2= r
3(4π
2/GM
s)
12-3 Kepler’s Laws of Orbital Motion
A geosynchronous satellite is one whose orbital period is equal to one day. If such a satellite is orbiting above the equator, it will be in a fixed position with respect to the ground.
These satellites are used for communications and
and weather forecasting.
Example 12-4
The Sun and Mercury
Example 12-4 The Sun and Mercury
Jorden kretsar kring solen på ett medelavstånd till solen av 1,50•10
11m. Beräkna solens massa.
T
2= r
3(4π
2/GM
s) M
s= r
34π
2/GT
2=
= (1,5•10
11)
3•4π
2/{(6,67•10
-11)•(π•10
7)
2}
= 2,02•10
30kg
Beräkna periodtiden för Merkurius vars
medelavstånd till solen är 5,79•10
10m.
Photo 12-5
Geosynchronous orbit
Active Example 12-1 Find the altitude of a Geosynchronous Satellite
Satelliten kretsar kring jorden med en periodtid av 24h. (M
E= 5,97 • 10
24kg, R
E= 6370 km)
T
2= r
3(4π
2/GM
E)
r
3= M
EGT
2/4π
2= (5,97•10
24)(6,67•10
-11)(86400)
2/4π
2r = (7,52953•10
22)
1/3= 42226909 m
h = r - R
E= 42,23 Mm – 6,37 Mm ≈ 35,8 Mm
12-3 Kepler’s Laws of Orbital Motion
GPS satellites are not in geosynchronous orbits;
their orbit period is 12 hours. Triangulation of
signals from several satellites allows precise
location of objects on Earth.
12-3 Kepler’s Laws of Orbital Motion
Kepler’s laws also give us an insight into possible orbital maneuvers.
[Conceptualcheckpoint 12-2 Which rockets to use?]
12-4 Gravitational Potential Energy[+ Exercise 12-4]
Gravitational potential energy of an object of
mass m a distance r from the Earth’s center:
12-4 Gravitational Potential Energy
Very close to the Earth’s surface, the
gravitational potential increases linearly with altitude:
Gravitational potential energy, just like all other forms of energy, is a scalar. It
therefore has no components; just a sign.
Potentiell Energi p.376
U = - mGM
E/(R
E+h)
bilda skillnaden och serieutveckla
ΔU = U
h– U
0= {R
E>>h} ≈ - mGM
E/R
E•(1 – h/R
E) - 1} =
= m(GM
E/R
E2)h = mgh
Example 12-5 Simple Addition
Example 12-5 Simple Addition p.377
Beräkna systemets potentiella energi för de tre massorna i exempel 12-5 med m
1= 2,5 kg, m
2= 0,75 kg och m
3= 0,75 kg.
U
ab= - Gm
am
b/r
abU
12= - G • 2,5 kg • 0,75 kg/1,25 m = - 1,0 • 10
-10J
U
13= - G • 2,5 kg • 0,75 kg/1,25√2 m = - 0,71 • 10
-10J U
23= - G • 0,75 kg • 0,75 kg/1,25 m = - 0,30 • 10
-10J
U
total= - 2,0 • 10
-10J
12-5 Energy Conservation
Total mechanical energy of an object of mass m a distance r from the center of the Earth:
This confirms what we already know – as an
object approaches the Earth, it moves faster
and faster.
12-5 Energy Conservation
12-5 Energy Conservation
Another way of visualizing the gravitational
potential well:
Example 12-6a
Armageddon Rendezvous
Example 12-6a Armageddon Rendez-vous
Anta att asteroiden startat i vila på oändligt avstånd från jorden. Beräkna dess hastighet då den är på
”månavstånd” från jorden.
E
i= E
f(konservativt kraftfält!) 0 = mv
f2/2 - GmM
E/60R
Ev
f= (2GM
E/60R
E)
1/2=
= [2•(6,67•10
-11)•(5,97•10
24)/60•6,37•10
6]
1/2=
= 1440 m/s
Example 12-6b
Armageddon Rendezvous
12-5 Energy Conservation: Escape speed (p.381) Anta att man vill skjuta iväg en raket med massan m med en begynnelsehastighet så att den kan
undfly jordens dragningskraft.
E
i= mv
i2/2 - GmM
E/R
ENär raketen är på oändligt avstånd från jorden så är avtar dess kinetiska och potentiella energi till noll.
Då kan den (lägsta) flykthastigheten beräknas v
flykt= (2GM
E/R
E)
1/2=
= [2•(6,67•10
-11)•(5,97•10
24)/6,37•10
6]
1/2=
Exercise 12-5 Calculate the escape speed from the Moon (p.382) + Conceptual checkpoint 12-3 (m>< ?)
Anta att man vill skjuta iväg en raket med massan m med en begynnelsehastighet så att den kan
undfly månens dragningskraft.
E
i= mv
i2/2 - GmM
M/R
MNär raketen är på oändligt avstånd från månen så är avtar dess kinetiska och potentiella energi till noll.
Då kan den (lägsta) flykthastigheten beräknas v
flykt,månen= (2GM
M/R
M)
1/2=
= [2•(6,67•10
-11)•(7,35•10
22)/1,74•10
6]
1/2=
≈ 2370 m/s
Example 12-7 Half Escape