Gravitation Part II

1. Gravitation Part II Newton’s Laws of Motion and Gravity

2. Isaac Newton (1642-1727) • Kepler's three laws were empirical, i.e. purely based on observations. Described how planets moved, but no fundamental reason why they work. • Newton developed physical understanding of Kepler's laws. Began to explain how everything works. Led to refinement and generalization of Kepler’s laws.

3. Newton's physics accomplishments • Foundation of modern physics. Replaced empirical descriptions with fundamental, physical explanations. • Laws of motion of objects on the Earth also pertains to objects in the heavens. Newton’s laws are universal. Study of motion produced by applied forces is called mechanics. • Unified all motions into three simple laws.

4. Some definitions • Speed measures how fast something is moving, for instance, car moving 100 km/hr. A scalar. • Velocity measures speed in a specified direction, for instance, car moving 100 km/hr north. A vector. • Acceleration is the rate at which velocity changes. Can be change in speed, direction, or both. A vector.

5. Important note: • Uniform circular motion is accelerated motion! • Direction of velocity is changing => acceleration Question: How many accelerators does your car have?

6. Newton’s Laws of Motion Built on Galileo’s ideas from his experiments Newton’s first law of motion (law of inertia): A body remains at rest, or moves in a straight line at a constant speed, unless acted upon by a net outside force. Implication: since planets do not have straight line motion, there must be a force acting on them!

7. Newton’s second law of motion: The acceleration of an object is proportional to the net outside force acting on the object, and inversely proportional to its mass. or Acceleration in same direction as applied force. Force is also a vector. Unit of force in mkssystem is the Newton (N). Must have same units as ma, so 1 N = 1 kg m s-2.

8. Newton’s third law of motion: Whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first body. Forces are interactions and act in simultaneous pairs. Implication: a planet and the Sun exert equal and opposite forces on each other. Question: Then why doesn't the Sun circle the planets?

9. What’s the nature of the force of gravity? Planets near the Sun move at high speed, and need strong force from the Sun. Planets in larger orbits need weaker pull from the Sun to stay in orbit.

10. The force of gravity Newton’s law of universal gravitation • Gravity is an attractive, universal and mutual force • Two bodies attract each other with a force that is directly proportional to the mass of each body and inversely proportional to the square of the distance between them. • Newton derived this from his three laws of motion and • measurements of accelerations. It does not depend on colors, shapes or compositions of objects.

11. In equation form: where: F = gravitational force between 2 objects, m1 = mass of first object, m2 = mass of second object, r = distance between objects, G = Newton’s gravitational constant, = 6.67 x 10-11 N m2/kg2

12. Your weight is the force of gravity between you and the Earth: Your mass does not change in different gravities, but your weight does. So mks units of weight are N (in Imperial units, pounds)

13. The mass of the Earth • Acceleration of gravity at surface of Earth (usually written g) measured to 9.8 m/s2 • Radius of the Earth measured to be 6378 km • Since => Newton's law of gravity provides us with a way to estimate masses of objects, using their motions! mearth m2 rearth2 = G = m2 a

14. Newton used the motion of the Moon to test universality of his law of gravitation. Fact: Earth-Moon distance is about 60 Earth radii, so acceleration of the Moon toward Earth should be 1/602, or 1/3600 as much as on object dropping on Earth’s surface. Is it? For motion in a circle, the centripetal (“directed toward the center”) acceleration a is

15. What’s the Moon’s speed? Distance = rate x time, so Speed = distance traveled in an orbit  time for one orbit: where 2r is circumference of orbit, and P is the sidereal period of the Moon. P = 27.3 days x 24 hr/day x 60 min/hr x 60 s/min = 2.36 x 106 s

16. The radius of the Moon’s orbit is 3.84 x 108 m, so = 1.02 x 103 m/s The Moon’s centripetal acceleration is then which is indeed (9.8 m/s2)/3600. Nice proof that it’s the same force pulling on the Moon that causes objects near Earth’s surface to fall, and that it must vary as the inverse square of the separation!

17. Newton’s form of Kepler’s first law Since neither mass can be stationary. Hence, Sun cannot sit at one focus. Newton found center of mass is at the focus. = m1 a1 = m2a2 If distances of m1 and m2 to C of M are r1 and r2, then m1r1 = m2r2 r1 and r2 change over orbit, but this relation always holds. Both objects have the same period.

18. Center of mass applet Hence, Sun wobbles due to pull from all the planets. Each has different acceleration, period, eccentricity. Sun motion applet

19. Newton’s form of Kepler’s third law Newton found that Kepler’s third law (P2 a3) needed to be modified – into a general and very useful equation. Newton’s form of Kepler’s third law: with P in seconds, a in meters, masses in kg. Now a is the mean separation of the objects over their orbit. Q: Why did Kepler miss the term with the masses?

20. Example: By measuring period and mean separation of a moon orbiting a planet, you can calculate the sum of masses of satellite and its planet, which is approximately the planet mass. Worked example: Imagine the Sun is suddenly replaced by a star of four times its mass. If the Earth’s orbit stays the same size, so the mean separation is still 1AU, what would be the Earth’s year?

21. For the Earth-Sun system , so this simplifies to approximately If the mass of the Sun increases by a factor of four:

22. Note: if P in years, a in AU, and masses in MSun , it is still the case that Mean separation is 1 AU in both cases, so P2 is smaller by a factor of ¼, so P must be ½ year.

23. In Newton’s time, actual distances of planets from Sun were just starting to be measured. One technique is “Earth-baseline parallax”. “Earth-baseline” or “diurnal” parallax uses telescopes on either side of Earth to measure planet distances. Use small angle formula: so First attempted by Cassini for Mars in 1672. Got answer 20% too large. Refined by Halley (1761) using parallax of Venus transit.

25. The Copernican model • Sun in center, circular orbits w/ uniform speed, rotating Earth • Objections: orbital, and rotational speed higher than anyone could imagine. • No parallax effects observed? • Against religious doctrine

26. Reminder parallax • Even larger effect if measured at two different sides of the Sun • The stars were so far away, that the resulting parallax angles were much smaller than anticipated

27. Johannes Kepler (1571 – 1630) analyzed Tycho’s accurate planetary position data.

28. Kepler's first law The orbit of a planet around the Sun is an ellipse, with the Sun at one focus. • Examples: planetary orbit with smallest eccentricity is Venus (e = 0.007). Largest is Mercury (e = 0.206). For Earth, e = 0.017. • Knowing a and e specifies the ellipse: e=c/a

29. Kepler's second law A line joining a planet and the Sun sweeps out equal area at equal intervals of time. In other words, a planet moves faster near the Sun, and slower farther away. Perihelion = point in orbit closest to the Sun Aphelion = point in orbit farthest from the Sun

30. Time it takes planet to go from point A to point B is the same as it takes to go from point C to point D.

31. [Advanced note: this law is a direct consequence of conservation of angular momentum] We can calculate the distance from a planet to the Sun at perihelion and aphelion. Dperi = a – c = a – ae = a(1 - e) Dap = a + c = a +ae = a(1 + e)

32. Kepler's third law The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit. The larger the planet’s orbit, the longer it takes the planet to go around the Sun. P2 a3 P2 = a3, when P is measured in years, and a is measured in AU

33. Example: For Earth, P = 1 year, a = 1 AU, P2 = 1 x 1 = 1, and a3 = 1 x 1 x 1 = 1. More interesting example: For Venus, we know a = 0.72 AU. What is P? P2 = a3, so = 0.61 years.

34. Check your Appendix 1 - is Kepler's 3rd law valid for all planets? • Kepler had NO IDEA WHY THESE LAWS WORKED. That took Newton!

35. Doesn’t work at Regener, use UNL or Cornell