5. Universal Laws of Motion

Transcription

1 5. Universal Laws of Motion If I have seen farther than others, it is because I have stood on the shoulders of giants. Sir Isaac Newton ( ) Physicist Image courtesy of NASA/JPL

2 Sir Isaac Newton ( ) Invented the reflecting telescope Invented calculus Connected gravity and planetary forces Philosophiae naturalis principia mathematica

3 The Acceleration of Gravity As objects fall, they accelerate. The acceleration due to Earth s gravity is 10 m/s each second, or g = 10 m/s. The higher you drop the ball, the greater its velocity will be at impact. Gravity of the moon is 1/6 as much..on Mars its 1/3 Why?

4 The Acceleration of Gravity (g) Galileo demonstrated that g is the same for all objects, regardless of their mass! This was confirmed by the Apollo astronauts on the Moon, where there is no air resistance.

5 Is Mass the Same Thing as Weight? mass the amount of matter in an object weight a measurement of the force which acts upon an object What is your mass on the Moon? What is your weight on Mars? Do you weight the same on a tall mountain as at Sea level?

6 Weight and Mass Image courtesy of NASA/JPL In Free fall, you are weightless

7 Newton s Laws of Motion 1 A body at rest or in motion at a constant speed along a straight line remains in that state of rest or motion unless acted upon by an outside force. If you are moving at constant speed in a circle, are you accelerating?

8 Centripetal Force Don t forget centripetal acceleration: In a circle, a c = v r V = πr/t so So F = 4π rm/t v F c = m r

9 Newton s Laws of Motion The change in a body s velocity due to an applied force is in the same direction as the force and proportional to it, but is inversely proportional to the body s mass. F/m = a is the ratio of F/m always g? Why doesn t m matter?

10 New force law? Connect the dots 1- F/m independent of m, so F = something x m Acceleration of moon towards earth =.007m/s so So, that s g/3600! So, that s g/60 So, the moon is 60 Earth radii away! So.

11 More connections So among = g/(60r) So a proportional to 1/distance Since a = F/m, F proportional to 1/distance So what do we have? F = something x m/d Finally Force is an interaction between two masses! I.e. Ball pulls on Earth and Earth pulls on ball So.F proportional to M e m/d

12 Universal Law of Gravitation Between every two objects there is an attractive force, the magnitude of which is directly proportional to the mass of each object and inversely proportional to the square of the distance between the centers of the objects.

13 Newton, Kepler, Galileo Newton used Galileo s law of Inertia and Galileo s formula for calculating centripetal acceleration and His formula relating Force and Accleration to Derive Kepler s laws Formulate the law of Universal gravitation

14 Copernicus ( ) by 1400 the planetary positions were no longer predicted by the almagest Copernicus Proposed all the following fix : 1. Earth spins on its axis once every 3 hrs, 56 min. Earth and all known planets orbited the sun in circular orbits with sun at center. 3. distant stars were so far that no parallax could be seen. 4. Polar axis precessed every 6,000 years. 5. All the above just a mathematical model to make accurate predictions easier easier than updating the Ptolemaic model

15 So was Copernicus or Ptolemy s model correct? Tycho Brahe, Johanes Kepler, and Galileo were the greatest contributors to the debate. Brahe (pronounced Bray) was the last and greatest pre-telescopic astronomer. Brahe felt that better observations were needed. interested in proving that the Tyconic Universe" was correct

16 Johannes Kepler ( ) Mathematician...sought out Brahe for his famous data, and was hired by Brahe to fit data to the tyconic model universe. But. Kepler was trying to fit data to the Kepler model" Universe the orbits of the six known planets fit into the largest spheres which could be inscribed into the six regular geometric solids --crazy by today's standards, but at least the orbits were centered on the sun! Wrote: Harmony of the worlds relating music, geometry, astronomy

17 Kepler and Brahe...continued Brahe died and family wouldn't release data after Brahe died (don t t ask how he died). : Eventually Kepler "acquired" Brahe's data and found that: --the orbit of Mars just isn't a circle! Plato was wrong! The door to a true understanding of the solar system was now wide open! Brahe s tombstone, from:

18 From Brahe s s data, Kepler deduced three laws: 1. Planets orbit the sun in Ellipses with the sun at one focus. A line joining a planet and the sun sweeps out equal areas in equal times Kepler s results These three relations are now known as Kepler's three laws. Extra for experts: x and y are the positions of the Earth with 0,0 at intersection of major and minor axis.

19 1. The cube of a planets semi- major axis is proportional to the square of its orbital period a 3 = T. a = semi-major major axis also average distance from planet to sun (also written as d or r). Units are AU s. 3. T = orbital period (also written as P). Units are years. 4. Newton later used this discovery to develop and prove the law of universal gravitation. Kepler s third law A graph of semi-major axis cubed on vertical axis, and orbital period squared on horizontal axis. Clearly not a coincidence!

20 Newton s Version of Kepler s Third Law Using the calculus, Newton was able to derive Kepler s Third Law from his own Law of Gravity. In its most general form: P = 4π a 3 / G (m 1 + m ) If you can measure the orbital period of two objects (P) and the distance between them (a), then you can calculate the sum of the masses of both objects (m 1 + m ).

21 Determining the Mass Image courtesy of NASA/JPL Often one mass is so much smaller than the other, the small mass can be ignored 4 3 so... P = π a GM where M is the larger mass G is the constant of Universal Gravitation G = 6.67 x N m kg Constants determie the units used Newton didn' t know G...so he used ratios : P 1 a 1 = note the constants drop out! P a Let P and a be the period and semi major axis of the Earth or P = 1year, and a = 1AU so 3 3 ( P ) = ( a ) as long as the object is orbiting the sun! 1 3

22 Extending Kepler s Law #1, Newton found that ellipses were not the only orbital paths. possible orbital paths ellipse (bound) parabola (unbound) hyperbola (unbound) Orbital Paths

23 Changing Orbits orbital energy = kinetic energy + gravitational potential energy conservation of energy implies: orbits can t change spontaneously An object can t crash into a planet unless its orbit takes it there. An orbit can only change if it gains/loses energy from another object, such as a gravitational encounter: If an object gains enough energy so that its new orbit is unbound, we say that it has reached escape velocity.

24 Tides Since gravitational force decreases with (distance), the Moon s pull on Earth is strongest on the side facing the Moon, and weakest on the opposite side. The Earth gets stretched along the Earth-Moon line. Greatest force pulls water away from Earth towards moon. Both Earth orbits center of mass of Earth Moon System Weaker force allows water to slide away from Earth on side opposite moon

25 Tidal Friction This fight between Moon s pull & Earth s rotation causes friction. Earth s rotation slows down (1 sec every 50,000 yrs.) Conservation of angular momentum causes the Moon to move farther away from Earth.

26 Synchronous Rotation is when the rotation period of a moon, planet, or star equals its orbital period about another object. Tidal friction on the Moon (caused by Earth) has slowed its rotation down to a period of one month. The Moon now rotates synchronously. We always see the same side of the Moon. Tidal friction on the Moon has ceased since its tidal bulges are always aligned with Earth.

27 Escape Velocity If you want to leave the planet, you have to do work! dr W r = r F r ( r) dr F(r) r 1 R

28 So you d better have kinetic energy! W r = r 1 r r F( r) dr Note the tricky negative signs! 1 E = mv o GMm R

29 Leaving a sphere GMm GMm W= ( ) r1 r r 1 = r, r = GMm So W = ( r If you escape r = infinity GMm So W = ( ) r Recall W = -ΔE p E p = 0 at infinity. GMm ) R F(r)

30 Energy conservation E p (r)= E total = E = GMm ( ) r 1 1 mv mv + ( GMm r GMm r )

31 Lets escape finally 1 mv o At surface, E = At r =, E = This means you ve lost all your speed getting to infinity! So E initial = E final = 0 For Earth, v 0 is 11km/s For Black hole this is c! actually, newtonian mechanics don t quite apply there v v escape GMm R 1 mvescape = GMm R GM R c blackhole = = escape = 0 GM R

32 Black Holes For a Black Hole, we can use this to find R R = GM/c = Radius of event Horizon This works out to about 3km/solar mass Largest black holes have about 3 billion solar masses Smallest (so far) about 3 solar masses.