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

Empirical Science • Scientists do not always set out to discover a particular phenomenon • Instead they first note and then accurately Our Place Our Place in in the the Cosmos Cosmos describe patterns in nature • They then look for the simplest physical model Lecture 6 which explains these observations, a process Orbits and the Laws of Motion known as empirical science • The laws of gravity were discovered by such a process Heliocentric Model Occam’s Razor • A principle attributed to the 14th- • Nicolaus Copernicus (1473-1543) and century English logician William of before him Aristarchus (310-230 BC) did Ockham that the explanation of any not understand why the planets orbit phenomenon should make as few the Sun, but they did realise that a assumptions as possible Sun-centred (heliocentric) system • When given two equally valid explanations for a phenomenon, one should choose the provided a much simpler description of less complicated formulation the observations (retrograde orbits) • The fewer assumptions an explanation of than an Earth-centred model a phenomenon depends on, the better it is Brahe and Kepler Kepler I • Tycho Brahe (1546-1601) was a firm believer in • Planets move on the geocentric model but using primitive elliptical orbits with equipment he made careful and accurate the Sun at one focus 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

Eccentricity e of an ellipse is the separation of the two foci divided by the length Earth 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 Pluto Earth has e = 0.017 meaning Sun-Earth distance varies by 1.7% from average Kepler II Kepler III • The square of the orbital period in years • A planet sweeps equals the cube of the semi-major axis of the out equal areas in orbit in astronomical units (AU): p 2 = A 3 equal times • The period of an orbit is just the time it • If time intervals t 2 - t 1 , takes the planet to go once round on its orbit t 4 - t 3 , t 6 - t 5 are the • The semi-major axis of the orbit is just half same, then areas A, B, of the long length of the orbit C are equal • To square a number multiply it by itself: • Planets thus move more p 2 = p x p ; to cube a number multiply it by rapidly when they are itself three times: A 3 = A x A x A closer to the Sun 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 (P years ) 2 = ( A AU ) 3

Kepler’s Laws Summary Isaac Newton (1642-1723) Kepler I describes the shape of planetary • Kepler’s laws were derived empirically from • orbits (elliptical) observations of planetary motion Kepler II describes how the speed of a • • Isaac Newton proposed three hypothetical planet varies about its orbit (“equal areas in laws of motion which are more general then equal times”) Kepler’s laws Kepler III relates the period of an orbit to • • They govern the motion of falling apples, its size, p 2 = A 3 (“harmony of the worlds”) cannonballs as well as planets They are empirically determined: they enable • • Success of Newton’s laws has led them to be us to predict how a planet will move, but do not tell us why - they are descriptive but accepted as physical laws not explanatory Newton I Galileo • Galileo challenged Aristotle’s authority by • Objects at rest stay at rest, objects in arguing that a force must act upon moving motion stay in motion objects to slow them down • “Newton’s First Law” is actually due to Galileo • “An object in motion will continue moving Galilei (1564 -1642) along a straight line with a constant velocity • The Greek philosopher Aristotle, around 2000 until an unbalanced force acts on it to change years earlier, believed that the natural state its state of motion” of objects was to be at rest - an object in • Galileo also referred to the resistance of an motion would tend toward this natural state - object to changes in its state of motion as a reasonable empirical rule due to friction inertia • Galileo’s (and Newton’s first) law is sometimes referred to as the law of inertia Inertial Frame of Reference 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 The coffee in your cup will be level whether frame of reference you are stationary or moving at a constant • Any inertial frame is as good as any other velocity

Newton II Acceleration • Motion is changed by unbalanced forces • Acceleration is the rate of change of the velocity of an object, that is the change in • Acceleration = Force/Mass velocity divided by the time over which the change takes place • Sybolically: a = F / m , or F = ma • 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 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 Accelerations are detectable from within a car 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 Discussion Topics • A 100 kg astronaut doing some repair work is • In what ways are Kepler’s laws empirical? adrift in space 10 metres from the space • Are Newton’s Laws empirical? shuttle • A planet is on a perfectly circular orbit about • How can they get back to the shuttle without a star and so is moving with constant speed - a tether line to pull on? is the planet accelerating? • Click for answer • How can we reconcile the Coriolis effect with • The spanner has mass 1 kg and it is Newton’s 1st law of motion? accelerated to a velocity of 10 m/s • In there were a planet with an orbit 1/4 the • How long will it take the astronaut to reach size of Mercury’s, what would be its orbital the shuttle? period relative to that of Mercury?