Transcript Unit 1

Homework 2
Unit 11 Problems 16, 19
Unit 12 Problems 10, 11, 16, 17, 18
Unit 14 Problems 17, 19
Unit 15. Problems 16, 17
Unit 16. Problems 12, 17
Unit 17, Problems 10, 19
Units to be covered:
Units 16-18
Newton’s Universal Law of Gravitation
• Every mass exerts a force of
attraction on every other mass. The
strength of the force is proportional
to the product of the masses divided
by the square of the distance
between them
– Simply put, everything pulls on
everything else
– Larger masses have a greater pull
– Objects close together pull more on
each other than objects farther apart
• This is true everywhere, and for all
objects
– The Sun and the planets exert a
gravitational force on each other
– You exert a gravitational force on
other people in the room!
Surface Gravity
• Objects on the Moon weigh less
than objects on Earth
• This is because surface gravity is
less
– The Moon has less mass than the
Earth, so the gravitational force is
less
• We let the letter g represent
surface gravity, or the acceleration
of a body due to gravity
• F = mg
• g= (GmM/R2)/m=GM/R2
• On Earth, g = 9.8 m/s2
• g on the Moon is around 1/6 as
much as on the Earth!
Centripetal Force
•
•
FC =
m ´V 2
d
•
•
If we tie a mass to a string and
swing the mass around in a circle,
some force is required to keep the
mass from flying off in a straight
line
This is a centripetal force, a force
directed towards the center of the
system
The tension in the string provides
this force.
Newton determined that this force
can be described by the following
equation:
m ´V 2
FC =
d
Masses from Orbital Speeds
• We know that for planets, the
centripetal force that keeps the
planets moving on an elliptical
path is the gravitational force.
• We can set FG and FC equal to
each other, and solve for M!
d ´V 2
M=
G
• Now, if we know the orbital speed
of a small object orbiting a much
larger one, and we know the
distance between the two objects,
we can calculate the larger
object’s mass!
Newton’s Modification of Kepler’s 3rd Law
• Newton applied his ideas to
Kepler’s 3rd Law, and
developed a version that works
for any two massive bodies, not
just the Sun and its planets!
a 3AU
MA + MB = 2
PYR
• Here, MA and MB are the two
object’s masses expressed in
units of the Sun’s mass.
• This expression is useful for
calculating the mass of binary
star systems, and other
astronomical phenomena
Orbits
• As we saw in Unit 17, we can
find the mass of a large object
by measuring the velocity of a
smaller object orbiting it, and
the distance between the two
bodies.
d ´V 2
M=
G
• We can re-arrange this
expression to get something
very useful:
Vcirc
GM
=
d
We can use this expression to determine
the orbital velocity (V) of a small mass orbiting
a distance d from the center of a much larger
mass (M)
Calculating Escape Velocity
• From Newton’s laws of
motion and gravity, we can
calculate the velocity
necessary for an object to
have in order to escape from
a planet, called the escape
velocity
Vesc
2GM
=
R
What Escape Velocity Means
• If an object, say a rocket, is
launched with a velocity less than
the escape velocity, it will
eventually return to Earth
• If the rocket achieves a speed
higher than the escape velocity, it
will leave the Earth, and will not
return!
Escape Velocity is for more
than just Rockets!
• The concept of escape velocity is useful
for more than just rockets!
• It helps determine which planets have an
atmosphere, and which don’t
– Object with a smaller mass (such as the
Moon, or Mercury) have a low escape
velocity. Gas particles near the planet can
escape easily, so these bodies don’t have
much of an atmosphere.
– Planets with a high mass, such as Jupiter,
have very high escape velocities, so gas
particles have a difficult time escaping.
Massive planets tend to have thick
atmospheres.
The Origin of Tides
• The Moon exerts a
gravitational force
on the Earth,
stretching it!
– Water responds to
this pull by
flowing towards
the source of the
force, creating
tidal bulges both
beneath the Moon
and on the
opposite side of
the Earth
High and Low Tides
As the Earth rotates beneath
the Moon, the surface of the
Earth experiences high
and low tides
The Sun creates tides, too!
•
•
The Sun is much more massive than the
Moon, so one might think it would create
far larger tides!
The Sun is much farther away, so its tidal
forces are smaller, but still noticeable!
•
•
When the Sun and the Moon line up,
higher tides, call “spring tides” are formed
When the Sun and the Moon are at right
angles to each other, their tidal forces work
against each other, and smaller “neap
tides” result.
The Conservation of Energy
• The energy in a closed system may change
form, but the total amount of energy does not
change as a result of any process
Kinetic Energy
• Kinetic Energy is simply the energy of
motion
• Both mass (m) and velocity (V) contribute
to kinetic energy
1
2
EK = m ´ V
2
• Imagine catching a thrown ball.
– If the ball is thrown gently, it hits your hand
with very little pain
– If the ball is thrown very hard, it hurts to
catch!
Thermal Energy
• Thermal energy is the energy
associated with heat
• It is the energy of the random motion
of individual atoms within an object.
• What you perceive as heat on a
stovetop is the energy of the individual
atoms in the heating element striking
your finger
Potential Energy
• You can think of potential
energy as stored energy,
energy ready to be converted
into another form
• Gravitational potential energy
is the energy stored as a result
of an object being lifted
upwards against the pull of
gravity
• Potential energy is released
when the object is put into
motion, or allowed to fall.
Conversion of Potential Energy
• Example:
– A bowling ball is lifted from the floor
onto a table
• Converts chemical energy in your
muscles into potential energy of the ball
– The ball is allowed to roll off the table
• As the ball accelerates downward
toward the floor, gravitational potential
energy is converted to kinetic energy
– When the ball hits the floor, it makes a
sound, and the floor trembles
• Kinetic energy of the ball is converted
into sound energy in the air and floor, as
well as some heat energy as the atoms
in the floor and ball get knocked around
by the impact
Definition of Angular Momentum
• Angular momentum is the rotational equivalent of
inertia
• Can be expressed mathematically as the product of the
objects mass, rotational velocity, and radius
• If no external forces are acting on an object, then its
angular momentum is conserved, or a constant:
L = m ´V ´ r = constant
Conservation of Angular Momentum
• Since angular momentum is
conserved, if either the mass,
size or speed of a spinning
object changes, the other
values must change to
maintain the same value of
momentum
– As a spinning figure skater
pulls her arms inward, she
changes her value of r in
angular momentum.
– Mass cannot increase, so her
rotational speed must increase
to maintain a constant angular
momentum
• Works for stars, planets
orbiting the Sun, and satellites
orbiting the Earth, too!