For Wednesday, Feb. 18
Reading: Section 4.1 (including Math Tools 4.1)
Assignments: Homework #2 (due today)
Homework #3 (due Wed. Feb 23)
The “Asteroid Tugboat”
spacecraft lands and fires rocket to
push asteroid
• what direction will the asteroid go?
• what if the asteroid is spinning?
• how far in advance of a collision
do we need to start?
Thought Question:
Which of the following is an example of
constant velocity?
A. a car speeding up on a straight road
B. a car slowing down on a straight road
C. a car maintaining a steady speed on a straight road
D. a car speeding up on a curved road
E. a car slowing down on a curved road
F. a car maintaining a steady speed on a curved road
Newton’s Laws of Motion
• speed: how fast an object’s position changes
measured by speedometers, radar guns
• velocity: speed and direction of travel
measured by weather vanes
• First Law: An object will maintain a constant
velocity if there is no net force acting on it.
Newton’s Laws of Motion
• acceleration: how fast velocity changes
3 ways to accelerate a car:
 GAS PEDAL (change speed)
 BRAKE PEDAL
 STEERING WHEEL (change direction)
change in velocity
acceleration=
time for change
Newton’s Laws of Motion
• force: strength and direction of a push or pull
any effort that can cause acceleration
• Second Law: For an unbalanced force,
F
m
a=
a: acceleration (units: 2 )
s
m
m: mass (units: kg)
kg × m
F: force (units: Newton = 2 )
MASS REDUCES
s
ACCELERATION
Explosion…Bad!
trying to blow up a BIG asteroid
is not a great idea…
… even a big explosion wouldn’t
accelerate it much
 Newton’s Second Law
… fragments would keep moving
in mostly the same direction
 Newton’s First Law
Thought Question:
A ball is attached to a string and swung in a circular path
above my head. At the point shown below, I suddenly
release the string. If this is viewed from directly above,
which of the paths below would the ball most closely
follow when released?
D
E
B
A
VIEW FROM ABOVE:
C
Orbits are Curved Paths
Newton’s First Law says: If object travels on a curved
path, there MUST BE an unbalanced force.
TOP VIEW:
VELOCITY
FORCE
(friction
between tires
and road)
VELOCITY
FORCE
(gravity)
PATH
PATH
Newton’s Second Law says: object accelerates (turns)
in direction of unbalanced force
 force is NOT pushing planet forward
 force IS pulling toward inside of orbit (toward Sun)
Thought Question:
The picture below shows the velocity of a planet at different
times in its orbit (larger arrow means larger speed).
2
Draw the direction of the
force on the planet at the
different positions shown.
For position #3, which combination of
these describes the direction the
force is acting:
A. To the right of the velocity arrow
B. To the left of the velocity arrow
C. Forward (with the velocity arrow)
D. Back (against the velocity arrow)
Enter all that apply.
1
3
Ellipse Orbit:
PLANET’S
VELOCITY
(ALWAYS TANGENT TO
ORBIT)
SUN’S FORCE
TURNS AND
SPEEDS PLANET
SUN’S FORCE JUST
TURNS PLANET
SUN’S FORCE
TURNS AND
SLOWS PLANET
THE FORCE POINTS
TOWARD THE SUN!
PLANET’S
VELOCITY
Newton’s Laws of Motion
• Third Law: When one object exerts a force on a
second object, second object exerts an equal
and opposite force back on the first.
EXAMPLES: GAS FORCE ON ROCKET
ROCKET’S FORCE ON GAS
USA
SKATER FORCES ON EACH OTHER
ICE
Measuring New Planets
Newton’s Third Law: star moves slightly
as planet orbits
“wobble” of star depends
on star and planet
masses, and size of
planet orbit
PLANET’S
FORCE ON
STAR
STAR’S
FORCE ON
PLANET
Thought Questions:
A compact car and a large truck collide head-on and
stick together.
1) Which one feels the largest force during the
collision?
2) Which one receives the largest acceleration?
(Enter both letters of your answers, then hit “Enter”.)
A. The car.
B. The truck.
C. Both experience the same amount.
D. You can’t tell without knowing how fast they were
moving before the collision.
Newton’s Third Law
Forces have equal strength, but accelerations may differ:
MAN’S FORCE ON BOAT
FORCE
BOAT’S FORCE ON MAN
FORCE
MORE MASS,
LESS
ACCELERATION
For Friday, Feb. 20
Reading: Section 4.2 (including Math Tools 4.2),
Appendix 7
Assignments: Homework #3 (due Mon. Feb. 23)
Asteroid Ida (and Dactyl)
54 km
The “Gravity Tractor”
satellite uses rocket to hover
near asteroid
• What direction will the satellite
pull the asteroid?
• How long would it take to
deflect the asteroid enough?
Thought Question
A hypothetical planet system has planets in equally-spaced
circular orbits. The planet masses are given in terms of the
mass of the innermost planet. Which of the planets exerts the
greatest gravitational force on the star?
A.
B.
C.
D.
16 M
10 M
3M
1M
4 AU
3 AU
2 AU
1 AU
Universal Gravitation
m1
m2
d
Gm1m2
Fg =
2
d
Fg: force
m1, m2: masses
d: distance between centers of objects
G: universal gravitational constant
3
m
-11
G = 6.67 ´10
kg × s2
attractive force: always pulls masses together
equal strength forces pull on both masses
Thought Question:
At which positions does a rocket feel a greater
gravitational force from Earth than from the
Moon? Earth is about 80 times more massive
than the Moon. (There may be more than one
answer.)
A
B
USA
C
Weight on Planets


Rplanet
d
(radius of
planet)
 planet’s gravity is made up of
pulls from every bit of its mass…
The bigger the planet, the farther
you are from most of the mass
 strength of force is same as if
WHOLE mass is distance
d = Rplanet away
Fweight =
GM planet myou
2
Rplanet
Thought Question:
An astronaut goes on the first mission to Mars. Mars has a
mass that is only about 1/10th the mass of Earth, and it is
½ the size. How will the astronaut’s weight on Mars
compare to their weight on Earth?
A. The same as on Earth
B. 1/2 (50%) the weight on Earth
C. 2/5ths (40%) the weight on Earth
D. 1/4th (25%) the weight on Earth
E. 1/5th (20%) the weight on Earth
F. 1/10th (10%) the weight on Earth
Your weight will be:
1/10
22 4
=
=
2
(1/2)
10 10
or 40% what it is on Earth
Mass in Astronomy
 Moon: 7  1022 kg
1/80th Earth
 Earth: 6  1024 kg
 Jupiter: 2  1027 kg
300 Earth
 Sun: 2  1030 kg
1000 Jupiter
300,000 Earth
.
Center of Mass
m1
v1
r1
r2
v2
m2
center of mass
r1
r2
Center of Mass
“balance point” is closer
to more massive object:
r2 m1
=
r1 m2
For Monday, Feb. 23
Reading: Review Section 4.2 (and Math Tools 4.3)
Assignments: Homework #3 (due Mon. Feb. 23)
QUIZ #2 is NEXT FRIDAY
Dwarf Planet Ceres
Thought Question:
If two stars are orbiting each other (and star 1 is more
massive than star 2), how does the distance of the
center of mass from star 1 (r1) compare to the total
separation A?
1
2
11
2 2
A = r +r
æ m2 ö
A. r1 = ç ÷ A
è m1 ø
B.
æ m1 ö
r1 = ç ÷ A
è m2 ø
r1
C.
æ m2 ö
r1 = ç
÷A
è m1 + m2 ø
æ m1 ö
D. r1 = ç
÷A
è m1 + m2 ø
mr =m r
E.
æ m1 + m2 ö
r1 = ç
÷A
è m1 ø
F.
æ m1 + m2 ö
r1 = ç
÷A
è m2 ø
A
m2
m1
center of mass
Example: Earth and Moon
The Earth has about 80 times the mass of the Moon, and
the distance to the Moon is about 60 Earth radii. How far
is the center of mass from the center of Earth?
m1
center of
mass?
r1
m2
r2
r1 + r 2 = 60RE
m1
r1 + ( )r1 = 60RE
m2
m1
(1+ )r1 = 60RE
m2
81r1 = 60RE
60
r1 =
RE
81
Thought Question:
For the Earth and the Moon, answer one letter for each of
these:
A. More force acts on Earth than Moon
B. More force acts on Moon than Earth
C. The same amount of force acts on both
A.
B.
C.
Earth is accelerated more than Moon
Moon is accelerated more than Earth
Both are accelerated the same amount
A.
B.
C.
Earth moves faster than Moon
Moon moves faster than Earth
Both move at the same speed
mMoon
mEarth
center of mass
v1t
Center of Mass
m1
• If center of mass is to remain
between two moving objects,
less massive object must move
faster in exact opposite direction
r1
v 2 r2 m1
= =
v1 r1 m2
• From Newton’s 3rd Law:
F1 = m1a1 = m2a2 = F2
r2
m2
v 2t
a2 m1
=
a1 m2
less massive object accelerates more
Acceleration due to Gravity
Galileo’s Experiment:
Two different masses dropped at same
time hit ground at same time…
 equal accelerations!
At Earth’s surface, force is
GM E m
F = mg =
2
RE
which creates an acceleration:
GM E
g=
2
RE
that doesn’t depend on the mass of the object dropping!
Thought Question
In which of the following
situations would it be
harder to drive along the
curve without skidding out?
A.
B.
C.
D.
Slow speed into a sharp turn
Slow speed into a gentle turn
High speed into a sharp turn
High speed into a gentle turn
TOP VIEW:
Acceleration Needed for Circular Path
a
a
v
v
r
v circ
2pr
=
P
distance
traveled by
ball in one
circuit
(total change
in position
during orbit)
“distance” traveled by tip of
velocity arrow during one circuit
2p v æ v ö
v2
acirc =
= ç ÷v =
P èrø
r
(total change to velocity arrow during
orbit caused by the acceleration)
Circular Orbit Speed
If gravity is providing the acceleration to keep something on a circular path:
a
v
r
GM
vcirc =
r
mass of object pulling
(NOT the one orbiting)
distance between objects
(usually radius of orbit)
r
M
General Form of Kepler’s Third Law
4p
3
P =
A
GM total
2
2
Mtotal: total amount of mass involved (example: star plus planet)
• applies to any elliptical orbit
• applies to any pair of orbiting masses (Sun + planet;
Earth + satellite; Jupiter + moon;…)
AN ASTRONOMER’S MAIN WAY TO DETERMINE MASS!
…just need orbital period and average orbital distance
Forms of Kepler’s Third Law
æ P ö æ A ö
÷
ç
÷ =ç
è 1 yr ø è 1 AU ø
2
3
• only applies to objects
orbiting Sun
• meant to be used with units
of yrs and AU
4p
3
P =
A
GM total
2
2
• applies to any pair of orbiting
masses (Sun + planet;
Earth + satellite; Jupiter + moon;
star + star; …)
• one object might not dominate
mass Mtot (like Pluto + Charon)
• make sure units work out!
Measuring Planet Orbits
Can measure size of planet’s
orbit (A) if:
PLANET’S
FORCE ON
STAR
• measure P from star’s wobble
• determine M for star from its light
(and as long as Mplanet << Mstar)
STAR’S
FORCE ON
PLANET
4p
3
P =
A
GM total
2
2