Chapter 12 Gravitation C. apple mg earth m1m1 m2m2 F 12 F 21 r In chapter 2 we saw that close to the surface of the earth the gravitational force F g is: a. Constant in magnitude F g = mg and b. Is directed towards the center of the earth In this chapter we will give the general form of the gravitational force between two masses m 1 and m 2. This force explains in detail the motion of the planets around the sun and of all celestial bodies. This is a law truly at a cosmic level(12-1)

Final Exam: Friday 5/2/03 8:00-11:00 am Rm. 215 NSC 3 long problems from chapters 9,10,11,12, and 13 6 mini problems from chapters 1-13 Bring with you: 2 pens 1 page with equations (both sides) Your ID

Celestial objects are divided into two categories: 1.Stars They have fixed positions with respect to each other That is the reason why we can group hem into constellations that maintain the same shape 2. Planets They follow complicated paths among the stars An example is given in the picture. Note: All stars rotate around the star polaris every 24 hours (12-2)

Polaris Celestial sphere Rotation Axis of the Celestial sphere earth N S Star Geocentric System 1. The earth is at the center of the universe 2.The stars are fixed on a sphere (known as the “celestial sphere”) which rotates about its axis every 24 hours. The axis connects the center of the earth with the star polaris. All stars move on circular paths on the celestial sphere and complete a rotation every 24 hours (12-3)

Ptolemaic System Ptolemy in the 2 nd century AD used the following geocentric model to describe the complicated motions of the planets. The planets and the sun move on small circular paths called the epicycles. The centers of the epicycles move around the earth on larger circles called deferents Ptolemy’s system gives a reasonable description of the motion of the planets and it was accepted for 1400 years (12-4)

The Heliocentric System In 1543 Copernicus introduced the heliocentric system. According to this scheme the sun is at the center of the solar system. The planets and the earth rotate about the sun on circular orbits. The immobility of the stars was ascribed to their great distance. The heliocentric system was not accepted for almost a century (12-5)

Nicolaus Copernicus ( ) He discussed his ideas in the book: De Revolutionibus Orbitum Coelestium published after his death

Thycho Brache in Denmark constructed sophisticated astronomical instruments and studied in detail the motion of the planets with a accuracy of ½ minute (1 minute = 1/60 degree) Brache died in 1601 before he had a chance to analyze his data. This task was carried out by his assistant Johannes Kepler for the next 20 years. His conclusions are summarized in the form of three laws that bear his name (Kepler’s laws) (12-6)

Kepler’s first law Planets move on elliptical paths (orbits) with the sun at one focus (12-7) Kepler’s second law During equal time intervals the vector r that points from the sun to a planet sweeps equal areas tt tt AA AA

Sun P1P1 P2P2 R1R1 R2R2 Kepler’s third law If T is the time that it takes for a planet to complete one revolution around the Sun and R is half the major axis of the ellipse then: C is constant for all planets of the solar system If the planet moves on a circular path then R is simply the orbit radius. For the two planets in the figure Kepler’s third law can be written as: (12-8)

Isaac Newton: Newton had formulated his three laws of mechanics. It was natural to check and see if they apply beyond the earth and be able to explain the motion of the planets around the sun. Newton after studying Kepler’s laws came to the following conclusions: 1. Kepler’s second law implies that the attractive force F exerted by the sun on the planets is a central force 2. If he assumed that the magnitude of the attractive force F has the form: (k is a constant) and applied Newton’s second law and calculus (which he had also discovered) he got orbits that are conic sections 3. A force gave Kepler’s third law: (12-9)

Conic Section is one of the four possible curves (circle, ellipse, parabola, hyperbola) we get when we cut the surface of a cone with a plane, as shown in the figure below (12-10)

Kepler’s third law for circular orbits Sun P R m v F M (12-11)

m1m1 m2m2 F 12 F 21 r Newton’s Law of Gravitational Attraction (12-12)

Revisit Kepler’s third law Sun P R m v F M (12-13)

m1m1 m2m2 F 12 F 21 r The gravitational constant G was measured in 1798 by Henry Cavendish. He used a balance with a quartz fiber. In order to twist a quartz fiber by an angle  one has to exert a torque  = c  (this is very similar to the spring force F = kx). The constant c can be determined easily (12-14)

(12-15) F r F /2 r

C. apple mg earth R M (12-16) m

C   R R Sun’s rays s Alexandria Syene well   h Ground in Alexandria Sun’s rays Erathosthene’s stick (12-17)

Variation of g with height h C R A B m m M (12-18)

Note: As h increases, g(h) decreases A B m m R C (12-19)

Gravitational Potential close to the surface of the earth In chapter 7 we saw that the potential U of the gravitational force close to the surface of the earth is: U = mgh mg h m floor U = 0 Note 1: Close to the surface of the earth the gravitational force is constant and equal to mg Note 2: The point at which U = 0 can be chosen arbitrarily we will now remove the restriction that m is close to the surface of the earth and determine U (12-20)

rMm.. m F O x-axis path M x dx Gravitational Potential U (12-21)

Escape velocity is the minimum speed with which we must launch an object from the surface of the earth so that it leaves the earth for ever m M r=  v = 0 R M m veve Before After (12-22)

Example (12-2) page 327 An object of mass m moves on a circular orbit of radius r around a planet of mass M. Calculate the energy E = K + U (12-23)

Gravitational force between extended spherical objects. M m r R C. M'M' m r C (12-24)

M mM m r M m r If m is outside the shell the gravitational force F is as if all the mass M of the shell is concentrated at its center and all the mass m of the sphere is concentrated at it center If m is anywhere inside the shell then the gravitational force between the shell and the sphere is zero ! Special case: The sphere and the shell are co-centric (12-25)