Transcript MOTION AND GRAVITY

MOTION AND GRAVITY
• The key LAWS of the key branch of physics
known as MECHANICS were formulated by
Isaac Newton
• Three LAWS of MOTION
• The LAW of GRAVITY
• The LAWS of CONSERVATION OF ENERGY
and MOMENTUM give a more general way to
understand motion
• While a physics course would spend the
whole first semester on these laws we’ll just
get a taste of them!
Speed, Velocity and Acceleration
• Speed = distance
traveled per time (car at
110 km/hr or 70 mph)
• Velocity is a speed + a
direction (70 mph NE)
• Acceleration is a change
in velocity per time:
speed and/or direction
(10 km/s2)
MATHEMATICS AND UNDERSTANDING
• Isaac NEWTON (1642-1727) of Woolsthorpe,
England is the most important scientist in history.
• His work completely changed the way educated
people looked at the world. Effectively, Newton
was the founder of PHYSICS as well as
THEORETICAL ASTRONOMY
• HE CO-INVENTED THE CALCULUS (w/ Leibnitz)
• HE DID PIONEERING WORK IN
• OPTICS: PRISM, REFLECTING TELESCOPE
• MECHANICS AND GRAVITY: his Philosophiae
Naturalis Principia Mathematica, (pub. 1684)
reflected work he'd mostly done in 1665-1666.
• Newton was knighted, and became first president of
the Royal Society, later Director of the Mint.
Isaac Newton
NEWTON'S LAWS OF MOTION
•
•
•
•
1. An object at rest remains at rest and
an object moving at a certain velocity
retains that velocity unless a FORCE
acts on it.
Aristotle's view: forces were needed
merely to keep something moving at a
constant speed
Newton realized friction or air resistance
were forces that slowed things down
Galileo had already understood this.
Forces Change Velocity
Gravity is a
FORCE that
causes
downward
vertical
acceleration
Which of the following is true?
•
•
•
•
•
A. You can have acceleration not equal zero, but
velocity equal to zero
B. You can have acceleration equal to zero, but velocity
not equal to zero
C. You can accelerate without changing your speed
D. A and B.
E. A, B and C.
Which of the following is true?
•
•
•
•
•
A. You can have acceleration not equal zero, but
velocity equal to zero
B You can have acceleration equal to zero, but velocity
not equal to zero
C. You can accelerate without changing your speed
D. A and B.
E. A, B, and C.
Newton’s SECOND LAW
• The core of Newtonian mechanics, it allows
trajectories of cannon balls, rockets, planets,
comets, stars and galaxies to be computed.
• F=ma
• is the most important relation in physics; one
can equivalently write
• a = F/m
• This clearly says less massive objects obtain
larger accelerations from the same force.
• Think of stepping on the gas and going from
0 mph to 60 mph in 10 seconds: your
acceleration is 6 mph/s (forwards)
• 2nd Law Applet
Momentum and Force
• Momentum = mass X velocity (p = mv)
• It takes a force to change a body’s
momentum
• Slightly more general version of
Newton’s 2nd Law: F = p/t
• Think of a 1000 kg car colliding with a 6000
kg truck head on -- if they have the same
speeds the truck has 6 times the momentum
and will push the car down the road
More on the 2nd Law
• Breaking takes you from 60 mph back to 0 in 4 sec
• or a negative acceleration of 15 mph/s.
• These are VECTOR equations -- with magnitude and
direction
• Velocity = distance covered / time
• V = d/t
• Acceleration = change in velocity/time change
• a= V/t
• - both the Speed and Direction are needed
• I.e. 50 mph to the East is the same speed, but
different velocity, from 50 mph to the North
• Going around a curve at a constant speed DOES
involve an acceleration (you feel pushed to one side
of the car, right?)
Newton’s Third Law
• 3. EVERY ACTION (FORCE) HAS AN
EQUAL AND OPPOSITE REACTION.
•
•
•
•
•
Forces don't act in isolation:
the Earth pulls the Moon and the Moon pulls back on
the Earth;
we push down and back on the ground with our
muscles, it pushes us up and forward;
a rower or gondolier pushes water (or canal bottom) in
one direction and the scull or gondola goes the other
way;
a rocket expels gases rearward and it flies forward.
Newton’s Laws of Motion, Illustrated
Changing an object’s momentum
requires
•
•
•
•
•
A. Gravity
B. Applying a force
C. Applying a torque
D. Friction
E. None of the above
Changing an object’s momentum
requires
•
•
•
•
•
A. Gravity
B. Applying a force
C. Applying a torque
D. Friction
E. None of the above
Newton’s second law, F = m·a,
(force = mass x acceleration),
means that with no force,
•
•
•
•
A. Objects remain at rest
B. An object’s speed doesn’t change
C. An object’s velocity doesn’t change
D. B and C.
Newton’s second law, F = m·a,
(force = mass x acceleration),
means that with no force,
•
•
•
•
A.
B.
C.
D.
Objects remain at rest
An object’s speed doesn’t change
An object’s velocity doesn’t change
B and C
Conservation Laws in Astronomy
• Momentum
• Angular
Momentum
• Energy
Conservation of (linear)
Momentum is implied by
Newton’s Laws of Motion.
One ball hits another, exerts
a force, which accelerates
Second ball (2nd law);
3rd Law says opposite force
decelerates the first ball
Angular Momentum Conservation
• AM = m x v x r (mass x velocity x distance)
• Orbital AM conservation says no push needed to
keep Earth orbiting and also faster motion at
perihelion than aphelion: v x r = constant
• Rotational AM conservation says Earth keeps
spinning on its axis and also faster spin when
contracted: ballerina, gas cloud making planets
Conservation of Energy
• Energy comes in many forms but three
classes can contain them all:
• Kinetic (energy of motion)
• Radiative (energy of light or
electromagnetic radiation)
• Potential (stored energy -- gravitational,
chemical, atomic, mass-energy)
Energy is Converted but not Lost
Thermal or Heat Energy
• Random kinetic energy of atoms and molecules
• Heat or thermal energy is the sum total of all of them
• Temperature is related to the average energy
Gravitational Potential and Kinetic Energy
• No KE, maximum
gravitational
potential energy
at top of throw
• Maximum KE,
minimum
gravitational PE
when thrown and
when caught
• KE = (1/2)mv2
• Energy of Thrown
Ball
Temperature is a measure of:
•
•
•
•
A. How much heat an object contains
B. How fast atoms are moving
C. How hot you feel when you touch something
D. Energy
Temperature is a measure of:
•
•
•
•
A How much heat an object contains
B How fast atoms are moving
C How hot you feel when you touch something
D Energy
A cake is baking at 400 degrees. If you
briefly touch the cake you will not be
burned. Touch the metal pan for the
same length of time and you will be
burned. Why?
•
•
•
•
•
A. The metal is hotter than the cake
B. The metal is denser than the cake–there
are more atoms per unit volume
C. The metal is a better conductor
D. B. and C.
E. All of the above
A cake is baking at 400 degrees. If you
briefly touch the cake you will not be
burned. Touch the metal pan for the
same length of time and you will be
burned. Why?
•
•
•
•
•
A. The metal is hotter than the cake
B. The metal is denser than the cake–there
are more atoms per unit volume
C. The metal is a better conductor
D. B and C.
E. All of the above
NEWTON'S LAW OF GRAVITY
•
The ATTRACTIVE FORCE OF GRAVITY IS
DIRECTLY PROPORTIONAL TO THE
PRODUCT OF THE MASSES
• AND INVERSELY PROPORTIONAL TO THE
SQUARE OF THE DISTANCE, r, BETWEEN
THEM.
Gm1m2
F
r
2
• where Newton’s gravitational constant
• G = 6.673 x 10-11 m3 kg-1 s-2
Inverse
square law
of gravity
Gravitational Acceleration: 1
• Combine 2nd Law of Motion w/ Law of Gravity
• ACCELERATION DUE TO GRAVITY, g, OF AN
OBJECT IS PROPORTIONAL TO ITS MASS AND
INVERSELY PROPORTIONAL TO THE SQUARE OF
THE DISTANCE FROM ITS CENTER.
F
a
m
Fgrav Gm1m 2
g

2
m2
r
Gm1

2
m2
r
Gravitational Acceleration: 2
• Example: if mE = ME and r = RE
11
1 2
(6.673 10 m kg s )(5.974 10 kg)
g
6
2
(6.378 10 m)
g = 9.80 m s-2
3
24
(or 32 ft/s2)
YOU SHOULD VERIFY THIS CALCULATION!
Gravitational Acceleration
Newton’s law of gravity is F = G m1 m2 / d2
Can this be used to find the force between the Sun
and a planet? If so, what is d?
•
•
•
•
A. No
B. Yes, d is the diameter of the Sun
C. Yes, d is the diameter of the planet
D. Yes, d is the distance from the Sun to the
planet
Newton’s law of gravity is F = G m1 m2 / d2
Can this be used to find the force between the Sun
and a planet? If so, what is d?
•
•
•
•
A. No
B. Yes, d is the diameter of the Sun
C. Yes, d is the diameter of the planet
D. Yes, d is the distance from the Sun to the
planet
When I drive my car at 30 miles per hour, it
has more kinetic energy than it does at 10
miles per hour.
A.
B.
C.
D.
E.
Yes, it has three times as much kinetic energy.
Yes, it has nine times as much kinetic energy.
No, it has the same kinetic energy.
No, it has three times less kinetic energy.
No, it has nine times less kinetic energy.
When I drive my car at 30 miles per hour, it
has more kinetic energy than it does at 10
miles per hour.
A.
B.
C.
D.
E.
Yes, it has three times as much kinetic energy.
Yes, it has nine times as much kinetic energy.
No, it has the same kinetic energy.
No, it has three times less kinetic energy.
No, it has nine times less kinetic energy.
Gravitational Acceleration
Weight v. Mass
• Weight (Newtons, dynes) is the force due to
gravity acting on a mass (amount of matter,
kilograms, grams) so
• W=mg
(special case of F = m a).
Since gravity gets weaker a greater distances, you
actually weigh less at the top of a building than
you do at its base, even though your
mass hasn't changed.
• Since Atlanta is about 300 m above sea level, you
weigh a little less here than in Savannah
• -- at sea level, and closer to the center of the earth.
• You weigh more in an elevator as it just accelerates
to go up and less in one that accelerates to go down;
• you are weightless in one that is falling w/o support!
Weight and Weightlessness
• Take a scale in an elevator with you. No cable  free fall
• Fast leap from a tower  constant free-fall (weightlessness)
Gravitational Acceleration
Figuring Out the Law of Gravity: 1
• Newton compared the acceleration the Moon feels
compared to that felt at the surface of the Earth.
• Knew the Earth-Moon distance was about 60 x RE
• Found the inertial ("centripetal") acceleration, a, due
to rotation at speed v and at distance r (experiment:
rock swung on string)
2
v
a
r
Figuring Out the Law of Gravity: 2
Start from vM = circumference of orbit divided by period
2 rEM
vM 
P
2 (3.84 10 8 m)
vM 
27.32 days  24  60  60
v M  1.02 km/s
• This gives, aM = 2.7 x 10-3 m s-2
• Newton realized: aM = g/(3625) = g/(rEM /RE )2

• concluded the INVERSE SQUARE RELATION OF
GRAVITY ON DISTANCE was LIKELY to be true
EVERYWHERE.
Gravity Keeps the Moon from Flying off
on a Tangent: “Constantly Falling”
Circular Velocity and Escape Velocity
• Newton also showed that the general shape of a BOUND
ORBIT was an ELLIPSE (with a circle as a special situation)
• and that the general shape of an ESCAPE ORBIT was a
HYPERBOLA (with a parabola as a special case).
• The simplest case: a CIRCULAR orbit, just skimming the earth
v c2
RE
v
2
c
vc
 a  g

GM E
RE2
GM E

RE

GM E
RE
 7.90 km/s
Orbital Types
Orbit Shape Depends on Speed
2GM
v esc 
 2 vc
R
•
•
•
•
v = vc : circular orbit
vc < v < vesc : elliptical orbit w/ center of E at near focus;
Both BOUND (NEGATIVE ENERGY ORBITS)

v = vesc: parabolic escape orbit: reaches infinity with no
energy left (ZERO ENERGY ORBIT)
• v > vesc
• hyperbolic escape orbit: reaches infinity still moving
away (POSITIVE ENERGY ORBIT)
• For earth, vesc = 11.2 km/s, or about 25,000 mph!
Newton DERIVED Kepler’s Laws
• FUNDAMENTAL LAWS explain EMPIRICAL ONES
• Consider a general circular orbit of a low mass
object around a much more massive one:
Gm
v
r
P

P  r3/2
2 r
r3
 2
v
Gm
; P 2  a3 ; a  P 2 / 3
• This is Kepler's Third Law!
• Newton also derived Kepler’s first and second laws,
but these are actually harder (you do this in a

sophomore, not freshman, physics course).
Orbital acceleration
Two Bodies Orbit Each Other
Weighing Astronomical Bodies
2 3
4

r
2
P 
Gm
2
3
4 r
m(total) 
2
G P
• For example, to get the mass of the Sun +
Earth (basically just Sun)
 m 
4
(1.496 10 m)
11 3
1 2
7 2
6.67 10 m kg s (3.15 10 s)
2
m = 2.0 x 1030 kg
11
3