Download The gravitational force between two objects is and more Exercises Physics in PDF only on Docsity!
Recap: Newton’s Gravitational Law
The gravitational force between two objects is
proportional to their masses and inversely
proportional to the square of the distance
between their centers.
G m
1
m
2
r
2
F =
(Newtons)
- F is an attractive force vector acting along line
joining the two centers of masses.
- G = Universal Gravitational
Constant
G = 6.67 x 10
- 11
N.m
2
/kg
2
Note: G was not measured until > 100 years
after Newton! - by Henry Cavendish (
th
cen.)
F
1
F
2
r
m
2
(F 1
= -F 2
)
m
1
(very
small)
m
How is Weight Related to Gravitation?
Gravitational force of attraction:
F (^) F
m
e
r
e
m
e
= mass of Earth = 5.98 x 10
24
kg
r
e
= radius of Earth = 6370 km
m = mass of an object
G m
e
m
r
e
2
if m = 150 kg, F = 1472 N (or ~ 330 lbs wt)
By Newton’s 2
nd
law (F=ma) we can also calculate weight:
W = m g = 9.81 x 150 = 1472 N
By equating these expressions for gravitational force:
m g = (^) or at surface: g =
G m
e
r
e
2
G m
1
m
2
r
2
F =
(N)
Result: ‘g’ is independent of mass of object !!
But this force creates the object’s weight:
F
F
m
2
m
1
r
2
rd
law: Each
body feels same force
acting on it (but
in opposite directions)
Gm
1
m
2
r
2
F=
- Thus each body experiences an acceleration!
Example: Boy 40 kg jumps off a box:
Force on boy: F = m g = 40 x 9.81 = 392 N
Force on Earth: F = m
e
a = 392 N
5.98 x 10
24
or a = = 6.56 x 10
m/s
2
ie. almost zero!
Example: 3 billion people jumping off boxes all at same time
(mass 100 kg each)
Conclusion: The Earth is so massive, we have essentially
no effect on its motion!
3 x 10
9
x 100 x 9.
5.98 x 10
24
= 5 x 10
m/s
2
a =
Planetary Motions & Orbits (Chapter 5)
- Heavenly bodies: sun, planets, stars… How planets move?
Greeks:
- Stars remain in the same relative position to one another as
they move across the sky.
- Several bright “stars” exhibit motion relative to other stars.
- Bright “ wanderers ” called planets.
- Planets roam in regular but curious manner.
Hypothesis:
- Geocentric “Earth-centered” universe!
- Sun moves around the Earth - like on a long rope with
Earth at its center.
- Stars – lying on a giant sphere with Earth at center.
- Moon too – exhibits phases as it orbits Earth.
Heliocentric Model: Copernicus (
th
century)
- Sun centered view that was later proven by Galileo – using
telescope observations of Jupiter and its satellite moons.
- Bad news : demoted Earth to status of just another planet!
- Revolutionary concept – required the Earth to spin (to
explain Sun’s motion).
- If Earth spinning why are we not thrown off? (at 1000 mph).
- Good news : no more need for complex epicycles to explain
retrograde motion!
Result: Earth moves faster in orbit and Mars appears
to move backwards at certain times.
Earth orbit
Mars orbit
Same
direction
Fixed
stars
field
- Copernicus heliocentric model assumed circular orbits – but
careful observations by Tycho Brahe (the last great “naked
eye” astronomer) showed not true…
th
century, Brahe’s student) developed three laws
based on emperical analysis of Brahe’s extensive data…
1. Orbits of planets around the sun are ellipses with
Sun at one focus.
two foci
Sun
planet
Note: A circle is a special
case of an ellipse with 2
foci coincident.
In reality, the planets’ orbits are very close to
circular but nevertheless are slightly elliptical.
- This means that the “outer” planets (i.e. further from the
Sun than Earth) all have much larger orbital periods than
Earth (and vice versa).
i.e. Τ
2
∝ r
3
- So if we know Τ (by observations) we can find “r” for each
planet!
Conclusion:
- These careful observations and new formula set the scene for
Newton’s theory of gravitation…
G m
1
m
2
r
2
F =
rd
law , Newton calculated:
where: m = mass of Sun for the planetary motions, but
m = mass of Earth for the Moon’s motion.
- Hence Kepler’s different result for the constant
for moon compared with other planets!
constant (fora given 'm' )
G m
4 D
r
Ô
2
3
2
2
r
3
= a constant number
equator
Example: Geosynchronous Orbit
- Orbital period = 24 hours
- Permits satellites to remain “stationary” over a given
equatorial longitude.
For Τ = 24 hrs
=> r = 42,000 km (to center Earth)
i.e. altitude ≈7 R
e
(compared with 60 R
e
for Moon.)
- Geostationary orbit is very important for Earth observing and
communications satellites – a very busy orbit!
- In general there are many, many possible orbits, e.g.:
- Circular and elliptical
- Low Earth orbits (LEO)
- Geostationary orbit
- Polar orbit
- e.g. GPS system uses many
orbiting satellites.
circular
elliptical
polar
- We can equate centripetal force to gravitational
attraction force to determine orbital speed (v
or
For circular motion:
Centripetal force = gravitational force (F
C
= F
G
2
2
or
r
G M m
r
m v
=
Orbital Velocity
M = planet’s mass
m = satellite’s mass
M » m
r
G M
v or
=
Results:
- Any satellite regardless of its mass (provided M » m)
will move in a circular orbit or radius r and velocity v
or
- The larger the orbital altitude, the lower the required
tangential velocity!
Qu: How to achieve orbit?
- Launch vehicle rises initially vertical (minimum air drag).
- Gradually rolls over and on separation of payload is moving
tangentially at speed v
or
produces circular orbit.
or
, craft will
descend to Earth in an
(decaying) elliptical orbits.
or
it will
ascend into a large elliptical
orbit.
it will escape earths gravity on
parabolic orbit!
Earth
circular (v
or
)
elliptical
parabolic
or
2v
- Not a simple ellipse… due to gravitational force of
Earth and Sun.
- Gravitational attraction between Earth and Moon provides
centripetal acceleration for orbit.
- Sun’s gravitation distorts lunar orbital ellipse. (Orbit
oscillates about true elliptical path.)
Lunar Orbit
Sun
Moon
Earth 1.5 x 10
8
km
(≈ 400 R
moon
)
F ∝
r
2
- Moon on same side of Earth as the Sun.
- Essentially invisible (crescent moons seen
either side of New moon).
- Rises at sunrise and sets at sunset (i.e. up all
day).
- Solar eclipse condition.
- At other times during moon’s orbit of Earth, we see
only a part of illuminated disk.
New moon:
- When moon is in-between full and new phase, it can often
be seen during daylight too.
- Example: half moon rises at noon and sets at midnight (and
vice versa).
- Under good observing conditions (at sunset or sunrise) you
can see dark parts of moon illuminated by Earthshine!
