Presentation Transcript

1. 7.3 Universal Gravitation - objectives Newton's Law of Universal Gravitation Cavendish and the Value of G The Value of g
2. The Apple, the Moon, and the Inverse Square Law In the early 1600's, German mathematician and astronomer Johannes Kepler developed three laws to describe the motion of planets about the sun. However, there was no accepted explanation for why such paths existed. Newton was troubled by the lack of explanation for the planet's orbits. Newton knew that for the motion of the moon in a circular path required that there be an inward component of force. However, the nature of such a force - its cause and its origin - bothered Newton for some time.
3. And according to legend, a breakthrough came at age 24 in an apple orchard in England. Clearly, it was Newton's ability to relate the cause for heavenly motion (the orbit of the moon about the earth) to the cause for Earthly motion (the falling of an apple to the Earth) that led him to his notion of universal gravitation.
4. Newton's reasoning Suppose a cannonball is fired horizontally from a very high mountain in a region devoid of air resistance. In the presence of gravity, the cannonball would drop and fall to Earth. Now suppose that the cannonball is fired horizontally again with a greater speed. In this case, the cannonball would travel further before striking the ground Now suppose that there is a speed at which the cannonball could be fired such that the trajectory of the falling cannonball matched the curvature of the earth, then the cannonball would fall around the earth instead of into it.
5. And then at even greater launch speeds, a cannonball would once more orbit the earth, but in an elliptical path, like the planets The motion of the cannonball orbiting to the earth under the influence of gravity is similar to the motion of the moon orbiting the Earth. And if the orbiting moon can be compared to the falling cannonball, it can even be compared to a falling apple. The same force that causes objects on Earth to fall to the earth also causes objects in the heavens to move along their circular and elliptical paths.
6. It was known at the time, that the force of gravity causes earthbound objects (such as falling apples) to accelerate towards the earth at a rate of 9.81 m/s2. And it was also known that the moon accelerated towards the earth at a rate of 0.00272 m/s2. If the same force that causes the acceleration of the apple to the earth also causes the acceleration of the moon towards the earth, then there must be a plausible explanation for why the acceleration of the moon is so much smaller than the acceleration of the apple. What is it about the force of gravity that causes the more distant moon to accelerate at a rate of acceleration that is approximately 1/3600 the acceleration of the apple?
7. Newton knew that the force of gravity must somehow be "diluted" by distance. The riddle is solved by a comparison of the distance from the apple to the center of the earth with the distance from the moon to the center of the earth. The moon in its orbit about the earth is approximately 60 times further from the earth's center than the apple is. The mathematical relationship becomes clear. The force of gravity between the earth and any object is inversely proportional to the square of the distance that separates that object from the earth's center. The moon, being 60 times further away than the apple, experiences a force of gravity that is 1/(60)2 times that of the apple. The force of gravity follows an inverse square law.
8. Inverse square law. The relationship between the force of gravity (Fgrav – force between the earth and any other object) and the distance that separates their centers (d) can be expressed by the following relationship The force of gravity is inversely related to the square of the distance. This mathematical relationship is sometimes referred to as an inverse square law.
9. Fg r Relationships in the equation The inverse square law suggests that the force of gravity acting between any two objects is inversely proportional to the square of the separation distance between the object's centers. If the separation distance is increased by a factor of 2, then the force of gravity is decreased by a factor of four (22). And if the separation distance is increased by a factor of 3, then the force of gravity is decreased by a factor of nine (32).
10. Check Your Understanding 1 . Suppose that two objects attract each other with a gravitational force of 16 N. If the distance between the two objects is doubled, what is the new force of attraction between the two objects? 2. Suppose that two objects attract each other with a gravitational force of 16 N. If the distance between the two objects is tripled, then what is the new force of attraction between the two objects? 3. Suppose that two objects attract each other with a gravitational force of 16 N. If the distance between the two objects is reduced in half, then what is the new force of attraction between the two objects? 4. Suppose that two objects attract each other with a gravitational force of 16 N. If the distance between the two objects is reduced by a factor of 5, then what is the new force of attraction between the two objects? 4 N 16/9 N 64 N 400 N
11. Example 1 An astronaut weighs 8.00 × 102 N on the surface of Earth. What is the weight of the astronaut 6.37 × 106 meters above the surface of Earth? 0.00 N 2.00 × 102 N 1.60 × 103 N 3.20 × 103 N
12. Example 2 As a meteor moves from a distance of 16 Earth radii to a distance of 2 Earth radii from the center of Earth, the magnitude of the gravitational force between the meteor and Earth becomes 1/8  as great 8 times as great 64 times as great 4 times as great
13. Newton's Law of gravitation is Universal Consider Newton's famous equation Fnet = m • a Newton knew that the force that caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple's downward acceleration also causes the earth's upward acceleration (Newton's third law), that force must also depend upon the mass of the earth. So for Newton, the force of gravity acting between the earth and any other object is directly proportional to the mass of the earth, directly proportional to the mass of the object, and inversely proportional to the square of the distance that separates the centers of the earth and the object. Newton's law of universal gravitation is about the universality of gravity. ALL objects attract each other with a force of gravitational attraction.
14. What is Gravity? Attractive force between two objects with mass Strength of attraction depends on: Mass of each object (in kilograms) Distance between the objects (in meters) G represent Gravitational Constant G = 6.67 x 10-11 N∙m2/kg2
15. 3000 m 6 x 105 kg Fg = 1.78 x 10-6 N 4 x 105 kg
16. Gravitational Proportions The force of gravity is DIRECTLYproportional to the masses of the objects attracted to one another If the mass of BOTH objects is doubled, the force of gravity is… If the mass of one object is doubled, the force of gravity is… DOUBLED QUADRUPLED
17. Gravitational Proportions The force of gravity is INVERSELYproportional to the SQUARE of the distance between the objects attracted to one another INVERSE SQUARE LAW If the distance between the objects is doubled, the force of gravity is… If the distance between the objects is tripled, the force of gravity is… 1/4th as GREAT 1/9th as GREAT
18. Example 3 Determine the force of gravitational attraction between the earth (m = 5.98 x 1024 kg) and a 70-kg physics student if the student is standing at sea level, a distance of 6.38 x 106 m from earth's center. Fgrav = mg = (70 kg)(9.8 m/s2) = 686 N
19. The Universality of Gravity 686 N 3.27 • 10-7 N 4.67 • 10-9 N 1823 N
20. The Value of g Fgrav = m∙g We can use the two equations above to derive an equation for the value of g. The acceleration of gravity is dependent upon the mass of the earth (approx. 5.98x1024 kg) and the distance (d) that an object is from the center of the earth. The acceleration of gravity is location dependent.
21. Note that g is inversely proportional to the distance squared – inverse square law.
22. The table below shows the value of g at various locations from Earth's center. The table below shows the value of g at various locations from Earth's center. 7.33 3.08 1.49 0.13
23. Using this equation, the following acceleration of gravity values can be calculated for the various planets. 3.75 26.0
24. G ME g = RE2 Example - Calculating the mass of the Earth Knowing G, and the radius of the Earth, RE = 6.37 x 106 m, we can now actually calculate the mass of the Earth. 9.81 m/s2 = 6.67x10-11 N∙m2/kg2(ME)/(6.37 x 106 m)2 ME= 5.98 ·1024kg
25. Gravity is a field force Gravitational field – a region in space where an object would experience a gravitational force. Every mass is surrounded by a gravitational field. As the distance from the Earth increases, the strength of gravitational force decreases. As the distance from the Earth increases, the arrows are further apart and the length of arrows are shorter, indicating the strength of the gravitational force decreases.
26. G ME g = RE2 g = Fg/m or The gravitational field around Earth Gravitational field strength is a vector quantity. Its direction is directed toward the center of Earth, or normal to Earth’s surface. Its magnitude at a pointequals the force per unit mass at that point. The concentration of the field lines increases as the distance from Earth decreases.
27. example The weight of an object was determined at five different distances from the center of Earth. The results are shown in the table below. Position A represents results for the object at the surface of Earth. What is the approximate mass of the object? 100 kg
28. Example Earth’s mass is approximately 81 times the mass of the Moon. If Earth exerts a gravitational force of magnitude F on the Moon, what is the magnitude of the gravitational force of the Moon on Earth?