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Chapter 7: Gravitation

PHYSICS Principles and Problems. Chapter 7: Gravitation. Gravitation. CHAPTER 7. BIG IDEA. Gravity is an attractive field force that acts between objects with mass. Planetary Motion and Gravitation. SECTION 7.1. MAIN IDEA

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Chapter 7: Gravitation

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  1. PHYSICS Principles and Problems Chapter 7: Gravitation

  2. Gravitation CHAPTER7 BIG IDEA Gravity is an attractive field force that acts between objects with mass.

  3. Planetary Motion and Gravitation SECTION7.1 MAIN IDEA The gravitational force between two objects is proportional to the product of their masses divided by the square of the distance between them. • What is relationship between a planet’s orbital radius and period? • What is Newton’s law of universal gravitation, and how does it relate to Kepler’s laws? • Why was Cavendish’s investigation important? Essential Questions

  4. New Vocabulary Kepler’s first law Kepler’s second law Kepler’s third law Gravitational force Law of universal gravitation Planetary Motion and Gravitation SECTION7.1 • Review Vocabulary • Newton’s third Law states all forces come in pairs and that the two forces in a pair act on different objects, are equal in strength and are opposite in direction • Period(T) : • The amount of time required for an object to repeat one complete cycle of motion (1 lap)

  5. Planetary Motion and Gravitation SECTION7.1 Kepler’s Laws • Kepler discovered the laws that describe the motions of every planet and satellite. • Kepler’s first law states that the paths of the planets are ellipses, with the Sun at one focus. Click image to view the movie.

  6. Planetary Motion and Gravitation SECTION7.1 Kepler’s Laws • Kepler found that the planets move faster when they are closer to the Sun and slower when they are farther away from the Sun. • Kepler’s second law states that an imaginary line from the Sun to a planet sweeps out equal areas in equal time intervals. Click image to view the movie.

  7. Planetary Motion and Gravitation SECTION7.1 Kepler’s Laws • Kepler also found that there is a mathematical relationship between periods of planets and their mean distances away from the Sun.

  8. Planetary Motion and Gravitation SECTION7.1 Kepler’s Laws • Kepler’s third law states that the square of the ratio of the periods of any two planets revolving about the Sun is equal to the cube of the ratio of their average distances from the Sun. Click image to view the movie.

  9. Planetary Motion and Gravitation SECTION7.1 Kepler’s Laws • Match Kepler’s Laws with the correct example: • The distance from Earth to the Sun changes throughout the year. • If you know the periods of Earth and Mars, as well as the Earth’s radius, then you can calculate the radius of Mars. • Earth moves faster when it is closer to the Sun. 1st 3rd 2nd

  10. Planetary Motion and Gravitation SECTION7.1 Compare the distances traveled from point 1 to point 2 and from point 6 to point 7 in the figure below. Through which distance would Earth be traveling fastest? The distance between points 1 and 2 is longer than the distance between points 6 and 7. Earth is closer to the Sun and travels faster between points 1 and 2 than between points 6 and 7.

  11. Planetary Motion and Gravitation SECTION7.1 Kepler’s Laws • Thus, if the periods of the planets are TA and TB, and their average distances from the Sun are rA and rB, Kepler’s third law can be expressed as follows: • The squared quantity of the period of planet A divided by the period of planet B, is equal to the cubed quantity of planet A’s average distance from the Sun divided by planet B’s average distance from the Sun.

  12. Planetary Motion and Gravitation SECTION7.1 Kepler’s Laws • The first two laws apply to each planet, moon, and satellite individually. • The third law, however, relates the motion of several objects about a single body.

  13. Planetary Motion and Gravitation SECTION7.1 Callisto’s Distance from Jupiter Galileo measured the orbital sizes of Jupiter’s moons using the diameter of Jupiter as a unit of measure. He found that lo, the closest moon to Jupiter, had a period of 1.8 days and was 4.2 units from the center of Jupiter. Callisto, the fourth moon from Jupiter, had a period of 16.7 days. Using the same units that Galileo used, predict Callisto’s distance from Jupiter.

  14. Planetary Motion and Gravitation SECTION7.1 Callisto’s Distance from Jupiter Step 1: Analyze and Sketch the Problem • Sketch the orbits of Io and Callisto.

  15. Planetary Motion and Gravitation SECTION7.1 Callisto’s Distance from Jupiter Label the radii. Known: TC = 16.7 days TI = 1.8 days rI = 4.2 units Unknown: rC = ?

  16. Planetary Motion and Gravitation SECTION7.1 Callisto’s Distance from Jupiter Step 2: Solve for the Unknown

  17. Planetary Motion and Gravitation SECTION7.1 Callisto’s Distance from Jupiter Solve Kepler’s third law for rC.

  18. Planetary Motion and Gravitation SECTION7.1 Callisto’s Distance from Jupiter (cont.) Substitute rI = 4.2 units, TC = 16.7 days, TI = 1.8 days in:

  19. Planetary Motion and Gravitation SECTION7.1 Callisto’s Distance from Jupiter (cont.) Step 3: Evaluate the Answer

  20. Planetary Motion and Gravitation SECTION7.1 Callisto’s Distance from Jupiter (cont.) Are the units correct? rC should be in Galileo’s units, like rI. Is the magnitude realistic? The period is large, so the radius should be large.

  21. Planetary Motion and Gravitation SECTION7.1 Callisto’s Distance from Jupiter (cont.) Step 1: Analyze and Sketch the Problem Sketch the orbits of Io and Callisto. Label the radii. Step 2: Solve for the Unknown Solve Kepler’s third law for rC. Step 3: Evaluate the Answer The steps covered were:

  22. Planetary Motion and Gravitation SECTION7.1 Newton’s Law of Universal Gravitation • Newton found that the magnitude of the force, Fg, on a planet due to the Sun varies inversely with the square of the distance, r, between the centers of the planet and the Sun. • That is, F is proportional to 1/r2. The force, F, acts in the direction of the line connecting the centers of the two objects.

  23. Planetary Motion and Gravitation SECTION7.1 Newton’s Law of Universal Gravitation (cont.) • The sight of a falling apple made Newton wonder if the force that caused the apple to fall might extend to the Moon, or even beyond. • He found that both the apple’s and the Moon’s accelerations agreed with the 1/r2 relationship.

  24. Planetary Motion and Gravitation SECTION7.1 Newton’s Law of Universal Gravitation (cont.) • According to his own third law, the force Earth exerts on the apple is exactly the same as the force the apple exerts on Earth. • The force of attraction between two objects must be proportional to the objects’ masses, and is known as the gravitational force.

  25. Planetary Motion and Gravitation SECTION7.1 Newton’s Law of Universal Gravitation (cont.) • The law of universal gravitation states that objects attract other objects with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. • The gravitational force is equal to the universal gravitational constant, times the mass of object 1, times the mass of object 2, divided by the square of the distance between the centers of the objects.

  26. Planetary Motion and Gravitation SECTION7.1 Newton’s Law of Universal Gravitation (cont.) • According to Newton’s equation: • F is inversely related to the square of the distance (r). • F is directly proportional to the product of the two masses. F

  27. Planetary Motion and Gravitation SECTION7.1 Universal Gravitation and Kepler’s Third Law • Newton stated his law of universal gravitation in terms that applied to the motion of planets around the Sun. This agreed with Kepler’s third law and confirmed that Newton’s law fit the best observations of the day.

  28. Planetary Motion and Gravitation SECTION7.1 Universal Gravitation and Kepler’s Third Law (cont.) • Consider a planet orbiting the Sun. Newton's second law of motion, Fnet = ma, can be written as Fnet = mpac.

  29. Planetary Motion and Gravitation SECTION7.1 Universal Gravitation and Kepler’s Third Law (cont.) • In the equation on the previous slide, Fnet is the gravitational force, mp is the planet’s mass, and ac is the centripetal acceleration of the planet. • For simplicity, assume circular orbits.

  30. Planetary Motion and Gravitation SECTION7.1 Universal Gravitation and Kepler’s Third Law (cont.) • Recall from your study of circular motion, that for a circular orbit, ac = 4π2r/T2. This means that Fnet = mpac may now be written as Fnet = mp4π2r/T2. • In this equation, T is the time required for the planet to make one complete revolution around the Sun.

  31. Planetary Motion and Gravitation SECTION7.1 Universal Gravitation and Kepler’s Third Law (cont.) • In the equation Fnet = mp4π2r/T2, if you set the right side equal to the right side of the law of universal gravitation, you arrive at the following result:

  32. Planetary Motion and Gravitation SECTION7.1 Universal Gravitation and Kepler’s Third Law (cont.) • The period of a planet orbiting the Sun can be expressed as follows. • The period of a planet orbiting the Sun is equal to 2π times the square root of the orbital radius cubed, divided by the product of the universal gravitational constant and the mass of the Sun.

  33. Planetary Motion and Gravitation SECTION7.1 Universal Gravitation and Kepler’s Third Law (cont.) • Identify how the period of a planet varies with each factor below. • distance, r, of the planet from the Sun: • The period is proportional to the square root of the cube of the distance r. • the Sun’s mass, Ms: • The period is inversely proportional to the square root of the Sun’s mass, Ms.

  34. Planetary Motion and Gravitation SECTION7.1 Universal Gravitation and Kepler’s Third Law (cont.) • In the equation below, squaring both sides makes it apparent that this equation is Kepler’s third law of planetary motion: the square of the period is proportional to the cube of the distance that separates the masses.

  35. Planetary Motion and Gravitation SECTION7.1 Universal Gravitation and Kepler’s Third Law (cont.) • The factor 4π2/Gms depends on the mass of the Sun and the universal gravitational constant. Newton found that this derivative applied to elliptical orbits as well.

  36. Planetary Motion and Gravitation SECTION7.1 Universal Gravitation and Measuring Gravitation • Isaac Newton determined that there is a gravitational force between any objects that have mass, but Henry Cavendish was the fist scientist to measure the force.

  37. Planetary Motion and Gravitation SECTION7.1 Measuring the Universal Gravitational Constant Click image to view the movie.

  38. Planetary Motion and Gravitation SECTION7.1 Cavendish’s Apparatus Explain why the rod and sphere in Cavendish’s apparatus must be sensitive to horizontal forces. The amount of horizontal rotation of the rod is used to determine the force of attraction between the spheres.

  39. Planetary Motion and Gravitation SECTION7.1 Measuring the Universal Gravitational Constant (cont.) • Cavendish’s experiment often is called “weighing Earth,” because his experiment helped determine Earth’s mass. Once the value of G is known, not only the mass of Earth, but also the mass of the Sun can be determined. • In addition, the gravitational force between any two objects can be calculated using Newton’s law of universal gravitation.

  40. Planetary Motion and Gravitation SECTION7.1 Measuring the Universal Gravitational Constant (cont.) • The attractive gravitational force, Fg, between two bowling balls of mass 7.26 kg, with their centers separated by 0.30 m, can be calculated as follows:

  41. Planetary Motion and Gravitation SECTION7.1 Measuring the Universal Gravitational Constant (cont.) • On Earth’s surface, the weight of the object of mass m, is a measure of Earth’s gravitational attraction: Fg = mg. If mE is Earth’s mass and rE its radius, then: • This equation can be rearranged to get mE.

  42. Planetary Motion and Gravitation SECTION7.1 Measuring the Universal Gravitational Constant (cont.) • Using rE = 6.38×106 m, g = 9.80 m/s2, and G = 6.67×10−11 N·m2/kg2, the following result is obtained for Earth’s mass:

  43. Planetary Motion and Gravitation SECTION7.1 Measuring the Universal Gravitational Constant (cont.) • When you compare the mass of Earth to that of a bowling ball, you can see why the gravitational attraction between everyday objects is not easily observed. • Cavendish’s investigation determined the value of G, confirmed Newton’s prediction that a gravitational force exists between any two objects and helped calculate the mass of Earth.

  44. Section Check SECTION7.1 Which of the following helped calculate Earth’s mass? A. Inverse square law B. Cavendish’s experiment C. Kepler’s first law D. Kepler’s third law

  45. Section Check SECTION7.1 Answer Reason:Cavendish's experiment helped calculate the mass of Earth. It also determined the value of G and confirmed Newton’s prediction that a gravitational force exists between two objects.

  46. Section Check SECTION7.1 Which of the following is true according to Kepler’s first law? A. Paths of planets are ellipses with the Sun at one focus. B. Any object with mass has a field around it. C. There is a force of attraction between two objects. D. The force between two objects is proportional to their masses.

  47. Section Check SECTION7.1 Answer Reason:According to Kepler’s first law, the paths of planets are ellipses, with the Sun at one focus.

  48. Section Check SECTION7.1 An imaginary line from the Sun to a planet sweeps out equal areas in equal time intervals. This is a statement of: A. Kepler’s first law B. Kepler’s second law C. Kepler’s third law D. Cavendish’s experiment

  49. Section Check SECTION7.1 Answer Reason:According to Kepler’s second law, an imaginary line from the Sun to a planet sweeps out equal areas in equal time intervals.

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