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Transcript
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 • • • (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 • • • • 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?