Download Law of Universal Gravitation

Universal Gravitation
Chapter 12
The Falling Apple
• The idea that gravity extends through the
universe is attributed to Sir Issac Newton
• He knew that if an object undergoes a
change in direction or speed, a force is
responsible
• Newton had the insight to figure out that
the moon orbits Earth because it is falling
towards Earth, avoiding its straight-line
path
The Falling Apple
The Falling Moon
• The moon falls in the sense that it falls beneath
the straight line it would follow if no force acted
on it
• Newton imagined firing a cannonball from a
mountain top. If he fired the cannonball with a
small horizontal speed, it would follow a
parabolic path and hit Earth soon. If he fired
the same cannonball with greater horizontal
speed, it would go farther and have a less
curved path. If he fired the cannonball with
great enough horizontal speed, he reasoned
that the cannonball’s path would become a
circle and it would circle indefinitely (orbit)
The Orbital Cannon
Cannonball in Orbit
Launch Speed less than 8000 m/s
Projectile falls to Earth
Launch Speed less than 8000 m/s
Projectile falls to Earth
Launch Speed equal to 8000 m/s
Launch Speed greater than 8000 m/s
Projectile orbits Earth - Circular Path Projectile orbits Earth - Elliptical Path
Newton’s Law of Universal
Gravitation
• Law of Universal Gravitation – every
object attracts every other object with a
force that for any two objects is directly
proportional to the mass of each object,
the force also decreases as the square of
the distance between the centers of
mass of the objects increases
• The farther away the objects are from
each other, the less the force of attraction
between them
The Universal Gravitational
Constant, G
• When the proportionality constant, G, is
introduced, we can write the Law of
Universal Gravitation in the form of an
equation:
2
F = G(m1)(m2)/d
• G = 6.67 x 10-11 N∙m²/kg²
• We only sense gravitation for bodies the
size of a planet, smaller objects we can’t
sense, but the force is there!
Law of Universal
Gravitation
The Inverse-Square Law
• When a quantity varies as the inverse
square of its distance from its source, it
follows an inverse-square law
• The greater the distance from Earth’s
center, the less an object will weigh
• You may weigh 300N at sea level, but
only 299N at the top of Mount Everest
Inverse-Square Law
Universal Gravitation
• Earth is round due to gravitation, it has
“attracted” itself together to create a
spherical shape
• When planets deviate from their normal
orbits, they have been pulled due to the
attraction to other planets, this deviation
is called perturbation
• This is how we found Neptune, Uranus
was acting strangely, in a way that could
only be accounted for by another planet
perturbing its orbit
Perturbation
Gravitational
Interactions
Chapter 13
Gravitational Fields
• A force field exerts a force on any objects in its
vicinity
• A magnetic field shows the strength of magnetism
around an object, an electric field show the
strength of electricity, etc.
• Gravitational Field – a force field that exists in the
space around every mass or group of masses
• We represent fields with imaginary field lines; the
closer the lines are together, the stronger the field
• The strength of Earth’s gravitational field is the
force per unit mass exerted by Earth on any object
g = F/m = 9.8 N/kg = 9.8 m/s²
• We can verify this number by using the law of
universal gravitation and known values for Earth (it
is solved out in your textbook!)
Gravitational Fields
Gravitational Field Inside a
Planet
• The gravitational field of Earth exists inside, as
well as outside
• If you dug a tunnel from the North Pole to the
South Pole, you would gain speed all the way
to the center, and then continue going to the
South Pole while losing speed
• At the beginning of your trip, your acceleration
would be g; but by the middle of your trip, you
would have no acceleration (it has decreased
going towards the center to 0)
• Why?
• As you are being pulled down by the center of
Earth, you are also being pulled UP by the
surface (in the center, you are being pulled in
all directions equally!)
Gravitational Field Inside a
Planet
Weight and Weightlessness
• The force of gravity, like any force, causes an
acceleration
• The gravitational force between you and Earth pulls
you against the floor; by Newton’s 3rd Law, the floor
pushes up on you
• When you stand on a scale, it is reading these two
combined forces
• If I repeated this in a moving elevator, my weight
would change during any accelerated motion
(upwards, weight goes up!)
• Now, take away the support force of the floor pushing
up, and I would constantly be in free fall, weightless!
• Weightlessness – the condition of free fall toward or
around Earth, in which an object experiences no
support force (and exerts no force on a scale)
Weight and
Weightlessness
Ocean Tides and the Moon
• Ocean tides are caused by differences in the
gravitational pull of the moon on opposite sides
of Earth
• The moon’s attraction is greater on Earth’s
oceans closer to the moon
• Because Earth and the moon orbit each other
around their shared center of mass (a point
inside Earth about ¾ from the center), we get
two sets of tides a day (high and low)
• Because of the mutual attraction between
Earth and the moon, both have a pair of “tidal
bulges” on them
Ocean Tides and the Sun
• The sun also contributes to ocean tides, but since it is further
away, it influences tides to a lesser extent
• Spring Tides – A high or low tide that occurs when the sun,
Earth and the moon are all lined up so that the tides due to the
sun and the moon coincide, making the high tides higher than
average and the low tides lower than average
• Lunar Eclipse – The cutoff of light from the full moon when
Earth is directly between the sun and the moon, so that Earth’s
shadow is cast on the moon
• Solar Eclipse – The cutoff of light from the sun to an observer
on Earth when the moon is directly between the sun and Earth
• Neap Tides – A tide that occurs when the moon is halfway
between a full moon and a new moon, in either direction. The
tides due to the sun and the moon partly cancel, so that the
high tides are lower than average and the low tides are not as
low as average
Ocean Tides
Black Holes
• A star is so large that it must constantly produce an outward
force to stop gravity from collapsing the star, it gets this energy
from nuclear fusion (the fusing together of hydrogen atoms to
create helium)
• When a star runs out of hydrogen, gravitation will take over and
the star will begin to collapse in on itself
• For very massive stars (much larger than our sun) this process
may result in the star crushing itself out of existence and
forming a black hole
• Black Hole – a mass that has collapsed to so great a density
that its enormous local gravitational field prevents light from
escaping
• Although black holes cannot be seen, we can see their effects;
mass emitting x-rays as it accelerates to nothingness
• Most astrophysicists believe that near the centers of most
galaxies is a black hole, causing the entire galaxy to rotate
Black Holes
Satellite Motion
Chapter 14
Earth Satellites
• An Earth satellite is a projectile that falls around Earth
rather than into it
• The surface of Earth drops a vertical distance of ~5
meters for every 8000 meters tangent to its surface;
meaning, if you were in the ocean you would only see
the top of a 5-m-tall mast on a ship that is 8 kilometers
away
• If you threw a ball fast enough to go a horizontal
distance of 8 kilometers during the time (1 second) it
takes to fall 5 meters, it would follow the curvature of
Earth!
• This would be about 29,000 km/hr (or 18,000 mi/hr)
• At this speed atmospheric friction would burn any
object to a crisp, that’s why satellites must be at 150
kilometers or more above Earth’s surface
Earth Satellites
Circular Orbits
• In a circular orbit, the speed of a circling
satellite is not changed by gravity
• A satellite in a circular orbit around Earth is
always moving perpendicular to gravity and
parallel to Earth’s surface at a constant speed
• For a satellite close to Earth, the time for a
complete orbit around Earth, its period, is
about 90 minutes
• The higher the altitude, the lower the orbital
speed, and the longer the period
• Communication satellites orbit at about 6.5
Earth radii from Earth’s center, so that their
period is 24 hours (they will always stay above
the same location)
Circular Orbits
Elliptical Orbits
• A projectile above the atmosphere moving at a
horizontal speed greater than 8km/s will overshoot
a circular path and trace an oval-shaped path – an
ellipse
• Ellipse – an oval-shaped curve that is the path of
a point that moves such that the sum of its
distances from two fixed points (foci) is constant
• Satellite speed, which is a constant for a circular
orbit, will vary in an elliptical orbit
• When the satellite moves away from Earth, it loses
speed (it’s working against gravity)
• As it comes back towards Earth, it gains speed
(it’s working with gravity)
• The satellite then rejoins its path with the same
speed it started with!
Elliptical Orbits
Energy Conservation &
Satellite Motion
• Remember, moving objects have kinetic energy (KE)
and an object above Earth’s surface has potential
energy (PE)
• A satellite has both KE and PE, such that the sum of
the KE and PE everywhere is constant
• In a circular orbit, the distance between a planet’s
center and the satellite never changes (PE), therefore,
by conservation of energy, its kinetic energy is also
constant
• In an elliptical orbit, both speed and distance change
• Apogee – the point in a satellite’s elliptical orbit
farthest from the center of Earth (PE is the greatest)
• Perigee – The point in a satellite’s elliptical orbit
where it is nearest the center of Earth (KE is the
greatest)
Energy Conservation &
Satellite Motion
Escape Speed
• In order to put an object into orbit around Earth speed
and direction are important
• Remember, “what goes up must come down”; if I shot
a rocket straight up at 8 km/s, it would back down at 8
km/s – instead I must shoot it horizontally
• Neglecting air resistance, fire anything at any speed
greater than 11.2 km/s, and it will leave Earth, going
more and more slowly, but never stopping (this is the
value of the escape speed)
• Escape Speed – the minimum speed necessary for
an object to escape permanently from a gravitational
field that holds it
• The larger the object, the greater the escape speed
(the Sun’s is 42.2 km/s)
• In order for the probe Pioneer 10 to escape the Sun’s
gravitational field NASA scientists used Jupiter’s
gravitational field to help it gain enough speed
Escape Speed
Assignment
• Read Chapters 12-14 (pg. 168-211)
• Ch. 12: Do #21-35 (pg. 181), App. F #1-6
(pg. 676)
• Ch. 13: Do #25-38 (pg. 197-198), App. F
#1-2 (pg. 677)
• Ch. 14: Do #15-28 (pg. 211), App. F #1-2
(pg. 677)