Transcript Gravity - Don't Let It Get You Down

Gravity
Don’t Let It Get You Down!
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The Truth About Gravity
 Gravity
is a phenomenon
 The
phenomenon results in a force
which can accelerate objects with mass
 “g”
represents the acceleration due to
the force caused by the phenomenon of
gravity
Back to the Beginning

Astronomy drove
our understanding
and discovery

It did so without
knowing it, however
The Importance of Being Gravity

Gravity has basic properties that set it apart from
the other forces: (1) it is long-ranged and thus
can act over cosmological distances; (2) it always
supplies an attractive force between any two
pieces of matter in the Universe.

Thus, although extremely weak, it always wins
over cosmological distances and is the most
important force for the understanding of the large
scale structure and evolution of the Universe.
So, let us deal with GRAVITY
We’ll need a bit of a history lesson:
•Brahe
•Kepler
•Newton
•Einstein
Pay close attention, gravity has
many
implications!
Tycho Brahe
1546 - 1601
A wild Dane, but he made
and recorded large
quantities of accurate
measurements of the
motions of the planets
around the Sun.
Began working with
Johannes Kepler in 1600.
Kepler’s “Laws” of Planetary Motion
1) The planets move abort the sun in elliptical orbits
with the Sun at one focus.
2) The radius vector joining a planet
to the sun sweeps over equal areas
in equal intervals of time.
3) The square of the time of one compete
revolution of a planet about its orbit is proportional
to the cube of the orbit's semi-major axis
T12/ T22 =R13/ R23
or
T2=k.R3
The empirical discovery of these laws
from Tycho Brahe's mass of data
constitutes one of the most remarkable
inductions ever made in science.
Isaac Newton





He put the physics and
mathematics to
Kepler’s Laws!


Born 1642, the year Galileo
died
Loner, tinkerer, paranoid
1665-1666 Plague was very
good for him
Suffered mental breakdown
1675
Math, Chemistry, Theology,
Parliament
Died 1727
Has his picture on the British
pound note
Newton’s Laws of Motion

First Law - A body remains in its state of motion
unless acted upon by an outside force
 Second
Law - A body acted upon by an external
force will change its momentum in the direction of the
force such that the greater the force the greater the
change in momentum (F= ma).

Third Law - Forces always occur in pairs, i.e. for
every action there is an equal and opposite reaction
Universal Law of Gravitation
 All
objects in the Universe attract each
other with a force that varies directly as
the product of their masses and
inversely as the square of their
separation from each other.
F gravity
= G m 1m
r2
2
General Relativity

Einstein’s Theory of Gravity, published 1915

Principle of Equivalence: Accelerations are
indistinguishable from gravitational fields. They are
equivalent.
So,
for example, when you are in freefall (like in an
orbiting Shuttle), your downward acceleration is just
enough to cancel the gravitational force.
Einstein’s View of Gravity
 Gravity is due to the curvature of spacetime.

Spacetime is curved by mass.
GR Made Predictions

Light would be bent by gravity
 Tested by Arthur Eddington during solar eclipse.
 Confirmed!

Emission of gravitational radiation by accelerating objects
 Tested by observations of binary pulsars.
 Confirmed to 14 decimal places!
Applications of Newton’s 2nd
Law and Einstein’s GR

Projectile Motion

Pendulums

Black Holes
Projectile Motion
Projectile - any object given an initial velocity which
subsequently follows a path determined by the
gravitational force acting on it, and by the frictional
resistance of the atmosphere
bullet shot from a gun, rocket after the fuel is exhausted,
thrown or batted baseball
Trajectory
Trajectory - the path followed by a projectile
Our Assumptions
We will consider only short length trajectories so that:
 Gravitational force is considered constant in
magnitude and direction
 Earth is an inertial system
 Air resistance is negligible
In other words, we will examine motion in a
vacuum on a flat, non-rotating Earth. In
physics, we call this creating an
First, the Forces
Only force acting on the projectile is its
weight…remember we are in the Ideal Case.

X-axis is horizontal; y-axis is vertical; origin is point
where projectile starts its free flight

So x-component of the force on the projectile is zero
and y-component of the force is the weight, mg.

Acceleration
Netwon’s 2nd law then tells us that the xcomponent of the acceleration is zero and the ycomponent is -g.
In other words, trajectory is a combination of a
horizontal motion with constant velocity and a
vertical component with constant acceleration.
Result

Under these conditions, projectiles travel in
trajectories which are parabolas.

You can derive the equation for the
parabolic motion from Newton’s Laws!
Let’s Play
Pendulums
For small angles,
sinQ = Q
Q
Simple Harmonic
Motion
L
Period = 2p L/g
Measure period of
oscillation and
length of pendulum,
determine g!
T
mg sinQ
x
Q
mg
Black Holes
A huge great enormous thing, like — like nothing. A
huge big — well, like a — I don’t know — like an
enormous big nothing …
Piglet describes the Heffalump,
in Winnie the Pooh by A.A. Milne
History
 Based
on Newton’s theory of gravity
 Proposed
independently by:
– 1783 Rev. John Mitchell
– 1796 Pierre Simon Laplace
Structure of a Blackhole
event horizon
curvature
singularity
Artist’s View
HST View
Calculation of Critical Radius
Calculation Continued
Earth: normal size
Earth: Normal Size
Earth: half size
Earth: Half Size
Earth: Quarter Size
Earth: Black Hole
Rbh = 2GM/c2
R = 9 mm
Earth as Depression in
Spacetime
Blackhole is Bottomless Abyss
in Spacetime
Bending of Light
D
Q
D
R
d
photon
d is the approximate
distance the photon
falls over the
diameter D of an
object with mass M
From Newton, We Know...
F = ma = GMm/R2
Solving for a,
a = GM/R2
If you accelerate at rate
a for time t, you move a
distance d of
d = 1/2 at2
Recall that t = D/c. Putting everything together, we see:
d = 1/2 GMD2/(c2R2)
Solving the Equations
D
Q
d
d = 1/2 GMD2/(c2R2)
D
R
but D=2R
and Q = d/D
RESULT: Q = GM/c2R
Using GR, we would get 4 GM/c2R!
Making the Prediction

When you plug in the values for G, c, and
the mass and radius of the Sun, you predict
that light should be bent by 8.5 x 10-6 radians.

This corresponds to 1.75 seconds of arc.

Eddington measured 1.75…probably.
Light Benders, Mind Benders
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Mass and Weight
Weight is a force. It is the resultant
gravitational force exerted on a body with mass
m by all the other bodies on the Universe.

Near Earth’s surface, the gravitational force
from our planet dominates all others, so
according to Newton’s Laws, we can write:

Weight = Fg = G m Me / R2 = mg
where Me is the mass of the Earth and R is the
radius of the Earth.
But also...
 Since weight is a force, it is measured in units
of force, namely Newtons.
Remember what mass is measured in units of?
Remember This!

Mass is a fundamental, universal property. You
have the same amount of mass no matter where you
are in the Universe. The only thing that can change
your mass is velocity…according to General
Relativity. As you approach the speed of light, you
become infinitely heavy.

Weight is not fundamental; its value depends
on what your circumstances are in the Universe.