2. Gravity Newton’s Law of Gravitation Kepler’s Laws of Planetary Motion Gravitational Fields

3. Newton’s Law of Gravitation m1m1 m2m2 r There is a force of gravity between any pair of objects anywhere. The force is proportional to each mass and inversely proportional to the square of the distance between the two objects. Its equation is: F G = G m 1 m 2 r 2r 2 The constant of proportionality is G, the universal gravitation constant. G = 6.67 · 10 -11 N·m 2 / kg 2. Note how the units of G all cancel out except for the Newtons, which is the unit needed on the left side of the equation.

4. Gravity Example F G = G m 1 m 2 r 2r 2 How hard do two planets pull on each other if their masses are 1.23  10 26 kg and 5.21  10 22 kg and they 230 million kilometers apart? This is the force each planet exerts on the other. Note the denominator is has a factor of 10 3 to convert to meters and a factor of 10 6 to account for the million. It doesn’t matter which way or how fast the planets are moving. (6.67 · 10 -11 N·m 2 / kg 2 ) (1.23 · 10 26 kg) (5.21 · 10 22 kg) = (230 · 10 3 · 10 6 m) 2 = 8.08 · 10 15 N

5. Inverse Square Law  Newton’s Law of Universal Gravitation is often called an inverse square law, since the force is inversely proportional to the square of the distance.

6. An Inverse-Square Force

7. Calculating the mass of the Earth Knowing G, we can now actually calculate the mass of the Earth. All we do is write the weight of any object in two different ways and equate them. Its weight is the force of gravity between it and the Earth, which is F G in the equation below. M E is the mass of the Earth, R E is the radius of the Earth, and m is the mass of the object. The object’s weight can also be written as mg. = F G = G m 1 m 2 r 2r 2 G M E m R E 2 = m g The m’s cancel in the last equation. g can be measured experimentally; Cavendish determined G ’s value; and R E can be calculated at 6.37 · 10 6 m (see next slide). M E is the only unknown. Solving for M E we have: g R E 2 G M E = = 5.98 · 10 24 kg

8. Kepler's Laws There are three laws that Johannes Kepler formulated when he was studying the heavens THE LAW OF ORBITS - "All planets move in elliptical orbits, with the Sun at one focus.” THE LAW OF AREAS - "A line that connects a planet to the sun sweeps out equal areas in the plane of the planet's orbit in equal times, that is, the rate dA/dt at which it sweeps out area A is constant.” THE LAW OF PERIODS - "The square of the period of any planet is proportional to the cube of the semi major axis of its orbit."

9. Kepler’s First Law Planets move around the sun in elliptical paths with the sun at one focus of the ellipse. An ellipse has two foci, F 1 and F 2. For any point P on the ellipse, F 1 P + F 2 P is a constant. The orbits of the planets are nearly circular (F 1 and F 2 are close together), but not perfect circles. A circle is a an ellipse with both foci at the same point--the center. Comets have very eccentric (highly elliptical) orbits. F1F1 F2F2 Sun Planet P

10. Kepler’s Second Law Sun While orbiting, a planet sweep out equal areas in equal times. C D A B The blue shaded sector has the same area as the red shaded sector. Thus, a planet moves from C to D in the same amount of time as it moves from A to B. This means a planet must move faster when it’s closer to the sun. For planets this affect is small, but for comets it’s quite noticeable, since a comet’s orbit is has much greater eccentricity. (proven in advanced physics)

11. Kepler’s Third Law The square of a planet’s period is proportional to the cube of its mean distance from the sun: T 2  R 3 Assuming that a planet’s orbit is circular (which is not exactly correct but is a good approximation in most cases), then the mean distance from the sun is a constant--the radius. F is the force of gravity on the planet. F is also the centripetal force. If the orbit is circular, the planet’s speed is constant, and v = 2  R / T. Therefore, Sun Planet F R M m G M m R 2R 2 m v 2 R = G M R 2 m [2  R / T] 2 R = Cancel m’s and simplify: 4 2 R4 2 R T 2 = Rearrange: G MG M T 2 4  2 = R3R3 Since G, M, and  are constants, T 2  R 3.

12. 1 year 365 days Third Law Example (cont.) What is Jupiter’s orbital speed? answer: Since it’s orbital is approximately circular, and it’s speed is approximately constant: 2  (5.2) (93 · 10 6 miles) v = d t = 11.9 years Jupiter is 5.2 AU from the sun (5.2 times farther than Earth is). · · 1 day 24 hours  29,000 mph. Jupiter’s period from last slide This means Jupiter is cruising through the solar system at about 13,000 m/s ! Even at this great speed, though, Jupiter is so far away that when we observe it from Earth, we don’t notice it’s motion. Planets closer to the sun orbit even faster. Mercury, the closest planet, is traveling at about 48,000 m/s !

13. Third Law Practice Problem Venus is about 0.723 AU from the sun, Mars 1.524 AU. Venus takes 224.7 days to circle the sun. Figure out how long a Martian year is. answer: 686 days

14. The Gravitational Field  During the 19th century, the notion of the “field” entered physics (via Michael Faraday).  Objects with mass create an invisible disturbance in the space around them that is felt by other massive objects - this is a gravitational field.

15. Gravitational Field Strength  To measure the strength of the gravitational field at any point, measure the gravitational force, F, exerted on any “test mass”, m.  Gravitational Field Strength, g = F/m

16. Gravitational Force  If g is the strength of the gravitational field at some point, then the gravitational force on an object of mass m at that point is F grav = mg.  If g is the gravitational field strength at some point (in N/kg), then the free fall acceleration at that point is also g (in m/s 2 ).

17. Gravitational Field Inside a Planet  If you are located a distance r from the center of a planet:  all of the planet’s mass inside a sphere of radius r pulls you toward the center of the planet.  All of the planet’s mass outside a sphere of radius r exerts no net gravitational force on you.

18. Gravitational Field Inside a Planet  Half way to the center of the planet, g has one-half of its surface value.  At the center of the planet, g = 0 N/kg.

19. Early Astronomers In the 2 nd century AD the Alexandrian astronomer Ptolemy put forth a theory that Earth is stationary and at the center of the universe and that the sun, moon, and planets revolve around it. Though incorrect, it was accepted for centuries. In the early 1500’s the Polish astronomer Nicolaus Copernicus boldly rejected Ptolemy’s geocentric model for a heliocentric one. His theory put the sun stated that the planets revolve around the sun in circular orbits and that Earth rotates daily on its axis. In the late 1500’s the Danish astronomer Tycho Brahe made better measurements of the planets and stars than anyone before him. The telescope had yet to be invented. He believed in a Ptolemaic-Coperican hybrid model in which the planets revolve around the sun, which in turn revolves around the Earth.

20. Early Astronomers In the late 1500’s and early 1600’s the Italian scientist Galileo was one of the very few people to advocate the Copernican view, for which the Church eventually had him placed under house arrest. After hearing about the invention of a spyglass in Holland, Galileo made a telescope and discovered four moons of Jupiter, craters on the moon, and the phases of Venus. The German astronomer Johannes Kepler was a contemporary of Galileo and an assistant to Tycho Brahe. Like Galileo, Kepler believed in the heliocentric system of Copernicus, but using Brahe’s planetary data he deduced that the planets move in ellipses rather than circles. This is the first of three planetary laws that Kepler formulated based on Brahe’s data. Both Galileo and Kepler contributed greatly to work of the English scientist Sir Isaac Newton a generation later.

21. SIR ISSAC NEWTON

22. THANK YOU  Created By – Prof. Swapan Kumar Gupta  SHD College,Pathankhali