Our Place Our Place in in the the Cosmos Cosmos describe - - PDF document

Our Our Place Place in in the the Cosmos 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 Sun-centred (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

• bservations 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

• f 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
• ut 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

• rbit 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
• f 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
• rbits (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
• bservations 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

• f 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

• bjects 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
• bject 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

The coffee in your cup will be level whether you are stationary or moving at a constant velocity

Inertial Frame of Reference

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
• bject 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

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?