Transcript Our Place in the Cosmos Elective Course

Our Place in the
Cosmos
Lecture 6
Orbits and the Laws of Motion
Empirical Science
• Scientists do not always set out to discover a
particular phenomenon
• Instead they first note and then accurately
describe patterns in nature
• They then look for the simplest physical model
which explains these observations, a process
known as empirical science
• The laws of gravity were discovered by such a
process
Heliocentric Model
• Nicolaus Copernicus (1473-1543) and
before him Aristarchus (310-230 BC) did
not understand why the planets orbit the
Sun, but they did realise that a Suncentred (heliocentric) system provided a
much simpler description of the
observations (retrograde orbits) than an
Earth-centred model
Occam’s Razor
• A principle attributed to the 14th-century
English logician William of Ockham that the
explanation of any phenomenon should
make as few assumptions as possible
• When given two equally valid explanations
for a phenomenon, one should choose the
less complicated formulation
• The fewer assumptions an explanation of a
phenomenon depends on, the better it is
Brahe and Kepler
• Tycho Brahe (1546-1601) was a firm believer in
the geocentric model but using primitive
equipment he made careful and accurate
observations of planetary positions over several
decades
• His assistant Johannes Kepler (1571-1630)
studied Tycho’s observations and deduced three
empirical rules of planetary motion now known
as Kepler’s laws
Kepler I
• Planets move on
elliptical orbits with the
Sun at one focus
Eccentricity e of an ellipse is
the separation of the two foci
divided by the length of the
long (major) axis of the
ellipse
A circle is a special case of
an ellipse where the two foci
coincide and e = 0
The larger the eccentricity
the more elongated is the
ellipse
Earth has e = 0.017 meaning
Sun-Earth distance varies by
1.7% from average
Earth
Pluto
Kepler II
• A planet sweeps
out equal areas in
equal times
• If time intervals t2 -t1, t4 t3, t6 -t5 are the same,
then areas A, B, C are
equal
• Planets thus move more
rapidly when they are
closer to the Sun
Kepler III
• The square of the orbital period in years equals
the cube of the semi-major axis of the orbit in
astronomical units (AU): p2 = A3
• The period of an orbit is just the time it takes the
planet to go once round on its orbit
• The semi-major axis of the orbit is just half of the
long length of the orbit
• To square a number multiply it by itself:
p2 = p x p; to cube a number multiply it by itself
three times: A3 = A x A x A
Kepler III
• Note that the length of a planet’s orbit is
proportional to its semi-major axis, L  A
• So the period of an outer planet is not only
longer than that of an inner planet because it
has further to travel (in that case we would have
P  A), it is also moving more slowly
• As we go out from the Sun, a planet has both
further to travel round its orbit and also moves
more slowly
• Together, these give (Pyears)2 = (AAU)3
Kepler’s Laws Summary
• Kepler I describes the shape of planetary orbits
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(elliptical)
Kepler II describes how the speed of a planet
varies about its orbit (“equal areas in equal
times”)
Kepler III relates the period of an orbit to its
size, p2 = A3 (“harmony of the worlds”)
They are empirically determined: they enable
us to predict how a planet will move, but do not
tell us why - they are descriptive but not
explanatory
Isaac Newton (1642-1723)
• Kepler’s laws were derived empirically from
observations of planetary motion
• Isaac Newton proposed three hypothetical laws
of motion which are more general then Kepler’s
laws
• They govern the motion of falling apples,
cannonballs as well as planets
• Success of Newton’s laws has led them to be
accepted as physical laws
Newton I
• Objects at rest stay at rest, objects in motion
stay in motion
• “Newton’s First Law” is actually due to Galileo
Galilei (1564 -1642)
• The Greek philosopher Aristotle, around 2000
years earlier, believed that the natural state of
objects was to be at rest - an object in motion
would tend toward this natural state - a
reasonable empirical rule due to friction
Galileo
• Galileo challenged Aristotle’s authority by
arguing that a force must act upon moving
objects to slow them down
• “An object in motion will continue moving along
a straight line with a constant velocity until an
unbalanced force acts on it to change its state of
motion”
• Galileo also referred to the resistance of an
object to changes in its state of motion as inertia
• Galileo’s (and Newton’s first) law is sometimes
referred to as the law of inertia
Inertial Frame of Reference
• An object at rest beside you in your car is
moving at 60 mph according to a bystander on
the side of the road, and is moving at 120 mph
according to a car in oncoming traffic
• All three perspectives are equally valid: the laws
of physics do not depend on the relative motion
of the observer
• A reference frame moving in a straight line at
constant speed is referred to as an inertial frame
of reference
• Any inertial frame is as good as any other
Inertial Frame of Reference
The coffee in your cup will be level whether you
are stationary or moving at a constant velocity
Newton II
• Motion is changed by unbalanced forces
• Acceleration = Force/Mass
• Sybolically: a = F/m, or F = ma
Acceleration
• Acceleration is the rate of change of the velocity
of an object, that is the change in velocity
divided by the time over which the change takes
place
• Mathematically a = v/t
• Note that velocity has a direction - an
acceleration may result in a change in speed
and/or a change in direction
• Any object not moving in a straight line at
constant speed is undergoing an acceleration
Accelerations are detectable from within a car
Acceleration
• The larger the force applied to an object, the
larger its acceleration
• The acceleration is proportional to the force
applied
• An object resists acceleration according to its
mass - it is twice as hard to accelerate a trolley
weighing 200 kg than one weighing 100 kg
• Mass is defined as the degree to which an
object resists changes in its motion
Newton III
• “Whatever is pushed, pushes back”, or
• “Every action has an equal and opposite
reaction”
• This is why you can push yourself along on a
skateboard: as you push on the ground with
your foot, the ground pushes back at you
• Your weight pushes you down on the floor and
the floor pushes back with an equal force (hence
unbalanced forces are required to accelerate an
object)
Summary of Newton’s Laws
1. An object in motion will remain in motion
unless an unbalanced force acts upon it;
an object at rest will remain at rest until a
force acts upon it
2. Acceleration = Force/Mass
3. Every action has an equal and opposite
reaction
Application of Newton’s Laws
• A 100 kg astronaut doing some repair work is
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adrift in space 10 metres from the space shuttle
How can they get back to the shuttle without a
tether line to pull on?
Click for answer
The spanner has mass 1 kg and it is accelerated
to a velocity of 10 m/s
How long will it take the astronaut to reach the
shuttle?
Discussion Topics
• In what ways are Kepler’s laws empirical?
• Are Newton’s Laws empirical?
• A planet is on a perfectly circular orbit about a
star and so is moving with constant speed - is
the planet accelerating?
• How can we reconcile the Coriolis effect with
Newton’s 1st law of motion?
• In there were a planet with an orbit 1/4 the size
of Mercury’s, what would be its orbital period
relative to that of Mercury?