Download Force, Acceleration, Momentum

Force, Acceleration, Momentum
Newton’s Law: F = m a =
Change in Momentum
Change in Time
F = ma = G M m
d2
In this picture, what mass is m ?
What mass is M?
Which of the two rocks has greater
acceleration?
Gravity and Tides
✤
Caused by the difference in strength of the moon’s
gravity on the near and far sides of the earth.
Fwater drop = mdrop a = G Mmoon mdrop
d2moon
Fdrop = G Mmoon mdrop
d21
Fdrop = G Mmoon mdrop
d22
d2
d1
As the earth rotates, we experience two high tides and two low tides each day
Gravity and Tides
✤
The sun also causes a
differential force across
the earth
✤
Spring tide: effect from
sun and moon reinforce
✤
Neap tide: sun and moon
effects fight each other,
smaller tides
✤
Total change = few feet
Spring Tides
Neap Tides
Gravity and Tides
✤
Friction drags the water along with the earth’s rotation, leads
the moon
✤
Moon’s gravity pulls the earth toward it, slows earth’s rotation
✤
Earth’s gravity tries to speed up the moon in its orbit, move it
to smaller distances from the earth.
-
Remember for a circular orbit around mass M: v =
so if v increases, d decreases
earth speeds up moon ->
<- moon slows earth’s rotation
drag force from Earth’s rotation
GM
√
d
Mass, Weight, Force
“Felix Baumgartner, Daredevil”
Red Bull “Stratos”
A balloon to 24 miles above the Earth, then jump
Mass, Weight, Force
What happens to Felix’s velocity as he drops?
Does the force he feels change as he falls?
Mass, Weight, Force
✤Radius
of the earth: ~4000 miles
✤Distance from the surface of the earth when Felix jumped: 24 miles
➡Distance from the center of the earth when Felix jumped: 4024 miles
✤40002 = 16 million
✤40242 = 16.2 million
➡1% in the denominator of the force equation. Tiny change in Force
F = GMm
d2
Mass, Weight, Force
If there were a scale under Felix’s feet as he dropped, what
would it read?
If he sat on a scale in his capsule, what would it read?
Balloon vs. Rocket
Moves away from earth due to force
(thrust) generated by the engine
Balloon vs. Rocket
Kinds of Energy
Kinetic energy = energy of motion:
m v2
2
Potential energy = stored energy
Gravitational potential energy =
Energy stored by working
against a gravitational force
- GM1M2
d
Notice resemblance
to the equation for
GM1M2
gravitational force: F = d2
Kinds of Energy
m v2
Kinetic energy = energy of motion:
2
- GM1M2
Gravitational potential energy =
d
Energy units: kg m2/s2 (Joules)
Other kinds of energy:
Chemical potential: energy stored in
chemical bonds
Thermal: kinetic energy due to random
motions of particles
Radiative: light
Energy
In an isolated system: Energy is conserved
sum of all the energy is a constant value
Energy can change form:
gravitational potential → kinetic
radiative → thermal
or be exchanged between objects
Energy
In an isolated system: Energy is conserved
sum of all the energy is a constant value
Examples of closed, isolated systems (includes all interacting
objects):
earth and pendulum
a planet or planets orbiting a star
All interacting objects may be the entire universe!
Conservation of Energy: Examples
Total Energy = Ekinetic + Egrav potential = Constant
A swinging pendulum: each swing reaches the same height
At top of swing, v=0 as the ball turns around:
Ekinetic
m v2
= 2 =0
Egrav potential = Etotal, largest possible value
At bottom of swing, v = largest value
Ekinetic = m v2 largest value
2
Egrav potential = minimum, ball is at
its smallest distance from earth
Conservation of Energy: Examples
Total Energy = Ekinetic + Egrav potential = Constant
A ball falling off a table
On table: v=0
Ekinetic
m v2
= 2 =0
Egrav potential = Etotal, largest possible value
Right before ball hits floor: v = largest value
Ekinetic = m v2 is largest
2
Egrav potential = smallest,
ball is at its smallest
distance from earth
After the ball hits the floor, energy transferred to the floor: making noise,
vibrations, thermal, ...
Conservation of Energy and Orbits
Total Energy = Ekinetic + Egrav potential = Constant
Kepler’s 2nd Law: Planets sweep out equal area in equal time
At perihelion: smallest distance
Egrav potential =
Ekinetic
- GM1M2
d
m v2
= 2 is largest
At aphelion: largest distance
- GM1M2
=
d
Egrav potential
is largest (least negative)
Ekinetic
m v2
= 2 is smallest
is smallest (a big negative number)
Total energy:
Kinetic Energy + Grav. Potential Energy
How doesConservation
energy conservation
help
us Orbits
understand orb
of Energy
and
Total
Energy
Ekinetic +by
Egrav
Constant
Orbits
are =defined
thepotential
total =
energy
of the orbiting
Total goes
energy:
Kinetic
Energy
+
Energy
= consta
Orbits
are
defined
total
energy
of Grav.
theon
object
orbit.energy
fasterby
or the
slower,
depending
r,Potential
butintotal
never
chan
Bound Orbit: closed ellipse, object stays with the thing it is orbiting
–  Bound:
PE
Orbits
are
definedKE
by +the
Kinetic
Energy
+ Potential
Energy
<total
0< 0 energy of the orbiting
object.
! m v2 $
goes faster or slower, depending
r, but
! onGM
m $total energy never changes (con
Kinetic Energy # 2 & + #" – R &% < 0
KE + PE > 0
"
%
–  Bound:
KE + PE
<
0
! m v 2 $ ! GM m $
&
#
&<#
total en
2
!m v $
2 m% $ " R %
! "GM
Potential #Energy
<0
&
& + #–
"
R v 2 % GM
" 2 %
<
! m v 2 $ ! GM m
2$ R
& 2GM
#
&<#
" 2 % " Rv 2 <%
KE + PE < 0
R
v 2 GM
<
2GM
2
R v<
total en
R
Total energy:
Kinetic Energy + Grav. Potential Energy = constant
Conservation of Energy and Orbits
Energy
Ekinetic
Egrav potential
= Constant
Orbits areTotal
defined
by =the
total+ energy
of the
orbiting object.
Orbitsor
areslower,
defineddepending
by the totalon
energy
the object
orbit. changes (consta
es faster
r, butoftotal
energyin never
– 
Bound Orbit: closed ellipse, object stays with the thing it is orbiting
Kinetic Energy
Potential
Bound:
KE+ +
PE < 0Energy < 0
Potential Energy
total
energ
2$
!
!
$
GM m
Kinetic m v
& <0
#
& + #–
KE + PE > 0
Energy " 2 % "
R %
! m v 2 $ ! GM m $
&
#
&<#
" 2 % " R %
v 2 GM
<
2
R
2GM
2
v <
R
2GM
v<
R
total energ
KE + PE < 0
Total energy:
Kinetic Energy + Grav. Potential Energy = constant
Conservation of Energy and Orbits
Energy
Ekinetic
Egrav potential
= Constant
Orbits areTotal
defined
by =the
total+ energy
of the
orbiting object.
Orbitsor
areslower,
defineddepending
by the totalon
energy
the object
orbit. changes (consta
es faster
r, butoftotal
energyin never
– 
Bound Orbit: closed ellipse, object stays with the thing it is orbiting
Kinetic Energy
Bound:
KE+ +Potential
PE < 0Energy < 0
Potential Energy
total
energ
2$
!
!
$
GM m
Kinetic m v
& <0
#
& + #–
KE + PE > 0
Energy " 2 % "
R %
! m v 2 $ ! GM m $
&
#
&<#
" 2 % " R %
v 2 GM
<
2
R
2GM
2
v <
R
2GM
v<
R
total energ
KE + PE < 0
otal energy:
2$
!
Kinetic
Energy
+
Grav.
Potential
Energy
=
constant
total energ
!
$
mv
GM m
&= 0
#
& + #–
"
2 %
R %
"
bits
areare
defined
bybythe
total
energy
of
the
orbiting
object.
Orbits
defined
the
total
energy
of
the
orbiting
object.
Escape
Speed
2
! m v $ ! GM m $
ter
or or
slow,
depending
on
r, but
total
energy
never
changes
(constant
#
& on
#slower,
& =depending
faster
r,
but
total
energy
never
changes
(constant!)
" 2 % " R %
Kinetic Energy + Potential Energy = 0 for an orbit that is just barely unbound:
2
v
Bound:
KE =+ GM
PE
t UN-bound:
KE<+0PE = 0
KE
PE
2
R
2
2 $v $
!GM
$
GM
m
! m !vm
!
$
m
2 + 2GM
#
&
# " &2 v+% =# – #" – R & =&% 0< 0
" RR %
" 2 %
2$
!m
!
$
v
GM
m
2
! m #v $ &!2GM
$ &
<GM
m
#
# " v 2=
& = %# " R & %
" 2 % " RR %
2
v
GM
2
< ellipse, object
v closed
GM
Bound Orbit:
R
=2
stays with the
it is orbiting
2 thing
R
2GM
2
<Potential Energy < 0
Kinetic Energyv +2GM
R
v2 =
R2GM
v<
2GMR
v=
R
total
energy
>
total
energ
total
energ
total energy <
total energ
From last time: Orbits and Circular Motion
✤
Combine:
Acceleration required to keep an object in circular motion:
G
M
with acceleration from Gravity
=
d2
v2
GM
=
d
d2
v=
GM
√
d
v = speed for an object in
stable circular motion around
mass M at distance d
v2
d
Energy and Orbits
v=
GM
√
d
speed for an object in stable circular
motion around mass M at distance d
escape speed for an object at
2GM
distance d from mass M
√
d
For an orbit, larger d = smaller v
vescape =
For an object in orbit at d, increase v
by √2 and it will leave its orbit and
escape.
d
Energy and Orbits
v=
GM
√
d
speed for an object in stable circular
motion around mass M at distance d
escape speed for an object at
2GM
distance d from mass M
√
d
For an orbit, larger d = smaller v
vescape =
For an object in orbit at d, increase v
by √2 and it will leave its orbit and
escape.
What happens if v increases by less
than √2 ?
A Speeds up
B Slows down
d
Energy and Orbits
v=
GM
√
d
speed for an object in stable circular
motion around mass M at distance d
escape speed for an object at
2GM
distance d from mass M
√
d
For an orbit, larger d = smaller v
vescape =
For an object in orbit at d, increase v
by √2 and it will leave its orbit and
escape.
What happens if v increases by less
than √2 ?
A Speeds up
B Slows down
d
Energy and Orbits
v=
GM
√
d
vescape =
speed for an object in stable circular
motion around mass M at distance d
2GM
√
d
escape speed for an object at
distance d from mass M
iClicker question:
You are building a rocket to send
the next rover to Mars.
If the cost of the rocket depends on
its escape speed from the Earth,
where should you build the rocket?
A
B
Earth
the International Space Station
Energy and Orbits
v=
GM
√
d
vescape =
speed for an object in stable circular
motion around mass M at distance d
2GM
√
d
escape speed for an object at
distance d from mass M
iClicker question:
You are building a rocket to send
the next rover to Mars.
If the cost of the rocket depends on
its escape speed from the Earth,
where should you build the rocket?
A
B
Earth
the International Space Station
Angular Momentum
Momentum:
mass x velocity
change in momentum requires force
Angular momentum:
mass x velocity x radius
The angular momentum of an isolated system is conserved.
Just like linear momentum.
r
Angular Momentum
Momentum:
mass x velocity
change in momentum requires force
Angular momentum:
mass x velocity x radius
The angular momentum of an isolated system is conserved.
Just like linear momentum.
iClicker question:
If this is a closed system, what happens
to the speed v as I make the string
length r shorter?
A Stays constant, velocity is conserved
B Slower
C Faster
r
Angular Momentum
Momentum:
mass x velocity
change in momentum requires force
Angular momentum:
mass x velocity x radius
The angular momentum of an isolated system is conserved.
Just like linear momentum.
iClicker question:
If this is a closed system, what happens
to the speed v as I make the string
length r shorter?
A Stays constant, velocity is conserved
B Slower
C Faster
r
Angular Momentum and Orbits
Kepler’s 2nd Law: Planets sweep out equal area in equal time
At perihelion: smallest distance
v must be largest
At aphelion: largest distance
v must be smallest
Angular Momentum and Orbits
Kepler’s 2nd Law: Planets sweep out equal area in equal time
At perihelion: smallest distance
v must be largest
At aphelion: largest distance
v must be smallest
We came to the same conclusion
thinking about energy:
Egrav potential =
Ekinetic
- GM1M2
d
m v2
= 2
At perihelion, d is smallest so potential energy is smallest, kinetic energy
must be largest -> velocity is largest
Angular Momentum
Galaxies and the solar system form when clouds of stuff,
mostly hydrogen, collapse due to their own gravitational force
The clouds are very large. As they
collapse, they become smaller.
Conserve angular momentum → Spin up,
become flat, disk-like.
Angular Momentum and Galaxies