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Newton’s second Law
• The net external force on a body is equal to the
mass of that body times its acceleration
F = ma.
• Or: the mass of that body times its acceleration is
equal to the net force exerted on it
ma = F
• Or:
a=F/m
• Or:
m=F/a
Newton II: calculate Force from
motion
• The typical situation is the one where a
pattern of Nature, say the motion of a
planet is observed:
– x(t), or v(t), or a(t) of object are known, likely
only x(t)
• From this we deduce the force that has to
act on the object to reproduce the motion
observed
Calculate Force from motion: example
• We observe a ball of mass m=1/4kg falls to the
ground, and the position changes proportional to
time squared.
• Careful measurement yields:
xball(t)=[4.9m/s2] t2
• We can calculate v=dx/dt=2[4.9m/s2]t
a=dv/dt=2[4.9m/s2]=9.8m/s2
• Hence the force exerted on the ball must be
• F = 9.8/4 kg m/s2 = 2.45 N
– Note that the force does not change, since the
acceleration does not change: a constant force acts on
the ball and accelerates it steadily.
Newton II: calculate motion from
force
• If we know which force is acting on an object of
known mass we can calculate (predict) its motion
• Qualitatively:
– objects subject to a constant force will speed up (slow
down) in that direction
– Objects subject to a force perpendicular to their motion
(velocity!) will not speed up, but change the direction
of their motion [circular motion]
• Quantitatively: do the algebra
Newton’s 3rd law
•
For every action, there is an equal and
opposite reaction
•
Does not sound like much, but that’s
where all forces come from!
Newton’s Laws of Motion (Axioms)
1. Every body continues in a state of rest or in a
state of uniform motion in a straight line unless it
is compelled to change that state by forces acting
on it (law of inertia)
2. The change of motion is proportional to the
motive force impressed (i.e. if the mass is
constant, F = ma)
3. For every action, there is an equal and opposite
reaction (That’s where forces come from!)
Newton’s
Laws
Always the same constant pull
a) No force: particle at rest
b) Force: particle starts moving
c) Two forces: particle changes
movement
Gravity pulls baseball back to earth
by continuously changing its velocity
(and thereby its position)
Law of Universal Gravitation
Mman
MEarth
R
Force = G Mearth Mman / R2
From Newton to Einstein
• If we use Newton II and the law of universal
gravity, we can calculate how a celestial object
moves, i.e. figure out its acceleration, which leads
to its velocity, which leads to its position as a
function of time:
ma= F = GMm/r2
so its acceleration a= GM/r2 is independent of its mass!
• This prompted Einstein to formulate his
gravitational theory as pure geometry.
Orbital Motion
Cannon “Thought Experiment”
• http://www.phys.virginia.edu/classes/109N/more_stuff/Appl
ets/newt/newtmtn.html
Applications
• From the distance r between two bodies and the
gravitational acceleration a of one of the bodies,
we can compute the mass M of the other
F = ma = G Mm/r2 (m cancels out)
– From the weight of objects (i.e., the force of gravity)
near the surface of the Earth, and known radius of Earth
RE = 6.4103 km, we find ME = 61024 kg
– Your weight on another planet is F = m  GM/r2
• E.g., on the Moon your weight would be 1/6 of what it is on
Earth
Applications (cont’d)
• The mass of the Sun can be deduced from the
orbital velocity of the planets: MS = rOrbitvOrbit2/G
= 21030 kg
– actually, Sun and planets orbit their common center of
mass
• Orbital mechanics. A body in an elliptical orbit
cannot escape the mass it's orbiting unless
something increases its velocity to a certain value
called the escape velocity
– Escape velocity from Earth's surface is about 25,000
mph (7 mi/sec)
Activity: Newton’s Gravity Law
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Get out your worksheet books
Form a group of 3-4 people
Work on the questions on the sheet
Fill out the sheet and put your name on top
Hold on to the sheet until we’ve talked about
the correct answers
• Hand in a sheet with the group member’s
names at the end of the lecture
• I’ll come around to help out !
Intro to the Solar System
The Solar System
Contents of the Solar System
• Sun
• Planets – 9 known (now: 8)
– Mercury, Venus, Earth, Mars (“Terrestrials”)
– Jupiter, Saturn, Uranus, Neptune (“Jovians”)
– Pluto (a Kuiper Belt object?)
• Natural satellites (moons) – over a hundred
• Asteroids and Meteoroids
– 6 known that are larger than 300 km across
– Largest, Ceres, is about 940 km in diameter
• Comets
• Rings
• Dust
Size matters: radii of the Planets
The Astronomical Unit
• A convenient unit of length for discussing
the solar system is the Astronomical Unit
(A.U.)
• One A.U. is the average distance between
the Earth and Sun
– About 1.5  108 km or 8 light-minutes
• Entire solar system is about 80 A.U. across
Homework: Distance to Venus
• Use: distance = velocity x travel time,
where the velocity is the speed of light
• Remember that the radar signal travels to
Venus and back!
The Terrestrial Planets
• Small, dense and rocky
Mercury
Mars
Venus
Earth
The Jovian Planets
• Large, made out of gas, and low density
Saturn
Jupiter
Uranus
Neptune
Asteroids, Comets and
Meteors
Debris in the Solar System
Asteroids
Asteroid Discovery
• First (and largest) Asteroid Ceres
discovered New Year’s 1801 by G. Piazzi,
fitting exactly into Bode’s law: a=2.8 A.U.
• Today more than 100,000 asteroids known
• Largest diameter 960 km, smallest: few km
• Most of them are named
• about 20 of them are visible with binoculars
How bright does a planet, moon,
asteroid or comet appear?
• Apparent brightness of objects that reflect
sunlight do depends on three things:
– Size of the object (the bigger the brighter)
– Distance to the object (the closer the brighter)
– “Surface” properties of the object (the whiter the
brighter, the darker the dimmer)
• Technical term: Albedo (Albedo =1.00 means 100% of
incoming radiation is reflected)