Download Study Notes Lesson 13 Gravitational Interactions

Physics Study Notes
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Lesson 13 Gravitational Interactions
Gravitational Interactions
We will investigate the role of gravity at, below, and above Earth’s surface.
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Gravitational Field
a.
Force Field— A force field exerts a force on objects in its vicinity. A magnetic field is a force
field that surrounds a magnet and exerts a magnetic force on magnetic substances. The pattern of
the iron filings shows the strength and direction of the magnetic field at different points in the
space around the magnet. Where the filings are closest together, the field is strongest. An electric
field is a force field surrounding electric charges. A gravitational field is a force field that
surrounds massive objects.
b.
Earth’s gravitational field— is represented by imaginary field lines. Where
the field lines are closer together, the gravitational field is stronger. The
direction of the field at any point is along the line the point lies on. Arrows
show the field direction. Any mass in the vicinity of Earth will be
accelerated in the direction of the field line at that location.
c.
Strength of the gravitational field— is the force per unit mass exerted by
Earth on any object.
Gravitational Field:
g=
F
M
=G 2
m
R
F:
weight of the object
G:
universal gravitational constant (6.67 x 10-11 N·m2/kg2)
m:
€mass of the object
M:
mass of Earth (5.98 x 1024 kg)
R:
Earth’s radius (6.37 x 106 m)
Near the surface of Earth, g = 9.8 m/s2. The g is a vector quantity, for it has both magnitude
(strength) and direction. Gravitational field vector g and acceleration due to gravity g have the
same magnitude and their units are equivalent: m/s2. The numerical value of g at Earth’s surface
depends only on the mass of Earth and its radius. So, all free falling objects have the same
acceleration. If Earth had a different mass or radius, g at its surface would have a different value.
As R increases, g decreases. By knowing the mass and radius of any planet, we can calculate the
acceleration due to gravity at the surface of that planet.
d.
Calculate the gravitational field strength:
Relative Earth’s Mass
M
2M
M
2M
M
2M
3
Relative Earth’s Radius
R
R
R/2
R/2
2R
2R
Gravitational Field Strength
g
2g
4g
8g
g/4
g/2
Gravitational Field Inside a Planet
a.
Gravitational Field Inside Earth: gravitational field exists both inside and outside Earth.
b.
Imaginary tunnel from the North Pole to the South Pole: If you started falling from the North
Pole end, you’d fall and gain speed all the way down to the center, and then overshoot and lose
speed all the way to the South Pole. You’d gain speed moving toward the center, and lose speed
moving away from the center. Without air drag, the trip would take nearly 45 minutes. The
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Physics Study Notes
Lesson 13 Gravitational Interactions
a=g
gravitational field strength is steadily decreasing as you
continue toward the center. At the center of Earth, you are
pulled in every direction equally, so that the net force is zero
and the gravitational field is zero. There is no acceleration as
you whiz with maximum speed past the center of Earth. If you
did not grab the edges at either end, you would oscillate back
and force, approximating simple harmonic motion. Each round
trip would take about 90 minutes, similar to an Earth satellite in
close orbit about Earth.
c.
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a=0
a = g/2
a=g
Cancellation of gravitational force: If Earth were of uniform density, half way to the center
gravitational force would be exactly half that at the surface. Cancellation is of the entire
surrounding shell of inner radius equal to your radial distance from the center. Half way down, for
example, all mass in the shell that surrounds you contributes zero gravitational force on you. You
are pulled only by the mass within this shell – below you. It is this Earth mass you use in
Newton’s equation for gravitation. At Earth’s center, the whole Earth is the shell and complete
cancellation occurs.
Weight and Weightlessness
a.
Weight: The force of gravity, like any force, cause acceleration. We are almost always in contact
with Earth. For this reason, we think of gravity primarily as something that presses us against
Earth rather than as something that accelerates us. The pressing against Earth is so called weight.
b.
Weight and motion:
c.
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a = g/2
Steady Motion
Accelerate Upward
Accelerate Downward
Normal Weight
Greater Weight
Less Weight
Broken Rope
No Weight
Weightless: Weightless is not because of the missing of gravity. Rather than define your weight as
the force of gravity that acts on you, it is more practical to define weight as the force you exert
against a supporting floor. The condition of weightless is not the absence of gravity; rather, it is
the absence of a support force. Astronauts in orbit are without a support force and are in sustained
state of weightlessness.
Ocean Tides
a.
Ocean tides: Newton showed that the ocean tides are caused by the
differences in the gravitational pull of the moon on opposite sides
of Earth. It is because the gravitational force is weaker with
increased distance.
b.
Two high tide per day:
i)
Both Earth and the moon are circling about their combined
center of mass— center of mass of the Earth-moon system. A
point about 3/4 of the way from Earth’s center to its surface.
ii)
The ocean nearest the moon is pulled upward toward to moon., while the main body of Earth
is pulled toward the moon also— away from the ocean on the far side. It is because Earth is
closer to the moon than the far-side ocean is.
iii) Earth’s water gets slightly elongated at both ends.
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Physics Study Notes
c.
Lesson 13 Gravitational Interactions
Moon contributes more than sun:
i)
Not because the pull from the moon is stronger, but because the difference in pulls on either
sides of Earth is greater for the closer moon.
ii) The sun’s contribution: The sun’s pull on Earth is about 180 times stronger than the moon’s
pull on Earth. Because the sun’s great distance from Earth, there is not much difference. That
is, The small difference in the solar pull on opposite sides of Earth only slightly elongates
Earth’s shape and produce tidal bulges less than half those produced by the moon.
iii) The sphere of Earth: is torn into an elliptical shape by difference in gravitational forces
exerted by moon.
iv) Spring Tides and neap tides: The tides are extra high when the moon and sun are lined up
(called spring tides) because the pulls add and an the two tides due to the moon and sun
overlap. When the moon and sun are at right angles to each other, the high tides of the sun
overlaps the low tides of the moon (called neap tides).
d.
Earth caused tides on the moon:
i)
The solid moon is slightly football shaped. The center of gravity is slightly displaced from its
center of mass. Both lie along the moon’s long axis.
ii) Whenever the long axis of the moon is not lined up toward Earth, Earth exerted a small torque
on the moon.
iii) This will twist the moon toward aligning with Earth’s gravitational field.
iv) This is why the moon always shows us its same face.
v) This tidal lock is also working on Earth.
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e.
High and low tides: Average high tide is about 1 meter above the average surface level of the
ocean. Average low tide is about 1 meter below the average surface level of the ocean. They are
six hours apart.
f.
Tides do not occur at the same time: While Earth spinning, the moon revolves about Earth and
appear at the same position in our sky every 24 hours and 50 minutes. That’s the period of two
high tides. That’s why tides do not occur at the same time everyday.
g.
Inverse cube relation: Newton deduced that the difference in pulls decreases as cube of the
distance between the centers of Earth and the moon.
h.
Size of the body: The size of the tides also depends on the size of the body. So, there is no
significant difference in the moon’s pull on lake, human fluid, etc.
i.
Tilt of Earth’s axis: Thought the opposite tidal bulges are equal, because of the tilt of Earth’s axis,
the daily high tides are never the same.
j.
Ignored factors: interfering land masses, tidal inertia, and friction with the ocean bottom.
k.
Comet Shoemaker-Levy 9(SL9): SL9 was in pieces ranging in size up to 2 kilometers in diameter,
and is believed to have been pulled apart by Jupiter's tidal forces during a close encounter in July
1992. These fragments collided with Jupiter's southern hemisphere over a period of time between
July 16 and July 22, 1994, at a speed of approximately 60 kilometers per second (37 miles per
second).
Tides in Earth and the Atmosphere
a.
Earth tides: Earth is not rigid but molten liquid covered by a thin solid and pliable crust. The
moon-sun tidal forces produce Earth tides as well as ocean tides. Twice each day the solid surface
of Earth rises and falls by as much as 25 cm.
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Physics Study Notes
b.
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Lesson 13 Gravitational Interactions
Atmospheric tides: Tidal effects in the ionosphere produce electric currents that alter the magnetic
field that surrounds Earth. These are magnetic tides which regulate the degree to which cosmic ray
penetrate into the lower atmosphere. The penetration affects the composition of our atmosphere,
which in turn changes the behavior of living things.
Black Holes
a.
Two processes in the sun:
i)
Gravitation: will crunch all solar materials.
ii) Thermonuclear fusion: Hydrogen bomb-like reaction, will blow solar materials outwards.
b.
Generation of black hole:
i)
A star which is 2-3 times more massive than sun.
ii) Run out of fusion fuel (hydrogen).
iii) Gravitational force dominates and starts to collapse.
iv) The collapse ignites fusion of the nuclear ashes (helium) and fuses them into carbon.
v) The star (such as the sun) will expand and extend beyond Earth’s orbit and swallows Earth.
(Red Giant)
vi) All the helium are burned out and start to collapse and die out. No longer give out heat and
light.
vii) Gravitational collapse starts, caves on itself and this process never stops. No empty space and
infinite density. Nothing can get back out, and even light cannot get out, and can not be
observed. (Black hole)
viii) Thought near the black hole the gravity is huge, gravity has no change beyond its original
radius.
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Problems about Gravitational Interactions
a.
An object is falling into a tunnel which connects the North Pole and the South Pole. a) Describe the gravity
change of the object along the path from A to D. b) Identify where can we find the maximum speed? c) What
will happen if nobody will catch the object on the other side of Earth. Why? (Ignore the friction, air drag and
technical feasibility.)
b.
There are gravitational attraction among the sun, the moon and Earth. Calculate the net force pulling on Earth
during a) a new moon, b) a full moon and c) a first quarter moon. (MMoon = 7.35 x 1022 kg, MEarth = 5.98 x
1024 kg, MSun = 1.99 x 1030 kg, dE-M = 3.84 x 108 m, dE-S = 1.5 x 1011 m)
1st Quarter Moon
New Moon
Full Moon
EARTH
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