N t ew on L s aws Monday, February 2

Transcription

1 Newton s Laws Monday, February

2 Isaac Newton ( ): English Discovered: three laws of motion, one law of universal gravitation.

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4 Newton s great book: Newton s laws are universal in scope, and mathematical in form.

5 Newton s First Law of Motion: An object remains at rest, or moves in a straight line at constant speed, unless acted on by an outside force. Precise mathematical laws require precise definitions of terms: SPEED = rate at which an object changes its position. Example: 65 miles/hour. VELOCITY = speed plus direction of travel. Example: 65 miles/hour to the north.

6 Acceleration = rate at which an object changes its velocity. Acceleration can involve: (1) increase in speed () decrease in speed OR (3) change in direction.

7 Force = a push or pull acting on an object. Examples: gravity = pull electrostatic t ti attraction ti = pull electrostatic repulsion = push

8 Newton s Second Law of Motion: The acceleration of an object is directly proportional to the fo orce acting on it, and inversely proportional to its mass. In mathematical form: F a m Or alternatively: F G Mm r

9 Example of Newton s Second Law: A package of cookies has mass m = kilograms, And experiences gravitatio ional acceleration g = 9.8 meters/second How large is the force acting on the cookies? F F F m a (0.454 kg)(9.8 m/se ec ) 4.4 kg m/sec 4.44 Newtons (4.4 Newtons 1pound)

10 Newton s Third Law of Motion: For every action, there is an equal and opposite reaction. Whenever A exerts a forc e on B, B exerts a force on A that s equal in size and opposite in direction. i All forces come in pairs.

11 The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Example of Newton s Third Law: Cookies push on hand: F = 1 pound, downward. Hand pushes on cookie es: F = 1 pound, upward. Remove hand! Earth pulls on cookies: F = 1 pound, downward. Cookies pull on earth: F = 1 pound, upward.

12 THIRD Law states: force on Earth = force on cookies SECOND Law states: acceleration = force divided by mass Mass of Earth = 10 5 x mass of cookies Therefore, acceleration of cookies = 10 5 x acceleration of Ea arth. (Cookies reach a high spe ed while the Earth hardly budges.)

13 But why do the cookies force on each other? and the Earth exert a Newton s Law of Gravity attractive force acting massive objects. states that gravity is an between ALL pairs of Gravity depends on: (1) () MASSES of the two objects, () DISTANCES between the objects.

14 Newton s Law of Gravity: The gravitational forc ce between two objects Mm F G r F = gravitational i force M = mass of one object m = mass of the second object r = distance between centers of objects G = universal constant t of gravitation ti

15 inversely G ravitational force varies directly with mass and with square of distance. Double the distance between objects: Force 1/4 as large. Triple the distance between objects: Force 1/9 as large. G Newtons m / kg (very small!)

16 Example: What is gravitational force between Earth and cookies? Mm F G r M mass of Earth r radius of Earth 6.01 m mass of cookies kg G Newtons m 0 4 / 6 kg kg m Let's plug the numbers in: F 4.4 Newtons 1pound

17 Example Encore: What is acceleration of cookies? Second Law of F a m Motion : Law of Gravity : Mm F G r Therefore... Mm 1 a G r m Independent of GM r 9.8 m/sec mass of cookies!!

18 The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Newton s question: can GRAVITY be the force keeping the Moon in its orbit? Newton s approximation: Moon is on a circular orbit. Even if its orbit were perfectly circular, the Moon would still be accelerated.

19 The Moon s orbital speed: radius of orbit: r = 3.8 x 10 8 m circumference of orbit: r =.4 x 10 9 m orbital period: P = 7.3 days =.4 x 10 6 sec orbital speed: v = (r)/p = 10 3 m/sec = 1k km/sec!

20 Acceleration required to keep Moon on a circular orbit The required acceleration is : a v r v orbital speed, r orbital radius For the Moon : v 10 3 m/sec r m a v r 3 (10 m) m m/sec

21 Acceleration provided d by gravity GM a r At the surface of the Earth ( r At the orbit of a a 9.8 m/sec the Moon ( r 9.8 m/sec (60) a m/ /sec radius of (close enough, given rounding errors) m/sec 3600 radius of Earth) Earth)

22 Bottom Line If gravity goes as one over the square of the distance, Then it provides the rig ght acceleration to keep the Moon on its orbit ( to keep it falling ). Triumph for Newton!!

23 Newwton modified and expanded Kepler s Laws of Planetary Motion Kepler s First Law: The orbits of the planets around the Sun are ellipses with the Sun at one focus. Newton s revision: The orbits of any pair of objects are conic sections with the center of mass at one focus.

24 As the Earth pulls on Moon n, Moon pulls on Earth. Both hearth and Moon orbit the center of mass of the Earth-Moon system: Center of mass = balance point: closer to more massive object.

25 Artificial satellites as envisaged by Isaac Newton: To put an object into orbit, launch it sideways with a large enough speed. How large is large enough?

26 The shape of the orbit depends on the speed of the satellite at launch: Low speed = closed orbit, a circle or ellipse. High speed = open orbit, a pa arabola or hyperbola. Circles, ellipses, parabolas, and hyperbolas are called conic sections.

27 To remain in a circular orbit just above the Earth s surface, a satellite must have v = 7.9 km/sec. To attain an open orbit, a satellite must reach at least 11. km/sec.

28 Some ex tra math: First cosmic velocity: mv r G Mm r GM E v 1st R E v 7.9 km/sec 1st Second (escape) cosmic velocity: mv Mm G r v nd GM E R E v 11. nd km/sec v 1st

29 Kepler s Second Law: A line from a planet to the sweeps out equal areas equal time intervals. Sun in Newton s revision: Angular momentum is conserved.

30 The product of the orbital speed (v) and the distance from the center of masss (r) is constant: v x r = const As r increases, v must tdecrease.

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32 P a P a 3 orbital period (in years) semi imajor axis (in A.U.) Kepler s Third Law: Newton 's P a revision ii : 4 3 P a G( M m) orbit tal period id(in seconds) d) semimajor axis (in meters) G universal constant of gravitation M mas ss of one object (in kilograms) m mass of other object (in kilograms)

33 Some ext tra math: Newton's revision at work : P For the Earth's orbit M a G( M m) kg (mass of the Sun) a m (1AU) A.U.) G P -11 m / kg sec 7 sec!

34 Kepler s third law appli ies only to objects orbiting the Sun. Newton s revision applies to all pairs of object orbiting each other. 4 a 3 M m GP Newton s revision can be used to find masses of distant objects (e.g., binary stars).

35 Kepler described how planets move; Newton explained why they move that way. Kepler s laws result naturally from Newton s laws of motion and Newton s law of gra avity. Kepler s laws of pl lanetary motion, as modified by Newton, are UNIVERSAL!

36 Tides are caused by the difference between the Moon s gravitational force on different side es of the Earth. Time between high tides = 1 hours, 5 min Time between moonris ses = 4 hours, 50 min

37 The gravitational force be etween two objects decreases as the distance between them increases. The Moon s gravitational pull on an object will be 7% greater on the closer side of the Earth than on the further side.

38 If the Moon s pull were constant, then Earthwouldbe undistorted. After subtracting average pull, Earth is stretched t in Moon s direction. Result: TWO tidal bulges, on opposite sides of Earth.

39 Why do we notice tides at the seashore? Rock is stiff: Tidal bulges in rock are only 0.3 m high. Water is fluid: Tidal bulges in water are 1 meter high. Water bulges rise above rock bulges.

40 The Sun also creates tides on Earth. High tides are highest when Sun, Earth and Moon line up (called spring tide ). High tides are lowest when Sun, Earth & Moon are at right angles ( neap tide ).

41 Tides forces are sl lowing the Earth s rotation and enlarging the Moon s orbit. The ocean s tidal bulges press down on the ocean floor. Friction robs energy from Earth s rotation and uses it to heat the ocean. This process is known as tidal breaking. (Think of the tidal bulges as brake pads!) The length of the day is ncreasing by 0.00 seconds per centu ury.

42 Moon creates tidal bulges on Earth: Earth creates BIGGER bulges on Moon. The Moon has already undergone tidal braking. The Moon s rotation has slowed to the point where rotation period equals the orbital period. This is why the Moon alwa ays keeps the same side turned to us. Google Moon

43 Friction between the tidal bulges and ocean floor drags the bulges in the direction of the Earth s rotation. Bulges lead Moon by about 10 degrees. The leading bulge steadily tugs the Moon into a larger orbit. The average Earth-Moon distance is increasing by 4 meters per century.

44 How do we know the distanc ce is increasing? Measure the distance to the moon with great accuracy and watch it change! Shoot the moon with a laser watch it bounce off and

45 How do we know the distanc ce is increasing? Measure the distance to the moon with great accuracy and watch it change! Shoot the moon with a laser watch it bounce off and

46 Few closing questions: 1) If the Sun was twice as massive, how long would be the year on Earth? ) Is the center of mass of the Solar System inside or outside the Sun? 3) How long will it take for a day on Earth thto double in duration? 4) How long will it take for the Moon to double its distance from Earth?