Newton’s Law of Universal Gravitation

1. Newton’s Law of Universal Gravitation • According to the Greeks, objects have a built-in desire to fall. They fall until they reach a lowest energy point. Here they are very stable. • According to Galileo and Newton, a force called gravity exists. It is an attractive force between the Earth and other objects or between any two objects. • Newton studied the motion of the planets and the moon. He wondered what kept the moon in its orbit around the Earth and what kept the Earth in orbit around the sun. • The idea that gravity extends throughout the universe is credited to Newton who is said to have thought of it when an apple fell on his head.

2. The Falling Moon • Newton compared the falling apple to the falling moon. • Remember that a projectile will move in a straight line unless acted upon by a force. Here the force is gravitational attraction of Earth. This is a centripetal force so the object is accelerated to the center of the Earth. The moon follows the straight line path but is pulled down as it travels over. • The moon falls in the sense that it falls below the straight line path that inertia would carry it on if no forces were acting on it. • He used a cannonball example to prove his point. In his thought experiment, he fires the cannonball with ever-increasing velocity. The cannonball eventually would have tangential velocity sufficient to carry it around the earth.

3. The Falling Moon • Newton tested his hypothesis by reasoning that the mass of an object should not affect how far it falls. • How far an object falls should only relate to its distance from Earth’s center. • In fact, it is related to the square of the distance from Earth’s center. • The moon accelerates to the Earth at about 1/3600 g. • Even though this acceleration is very small, remember that the force of Earth-on-moon and moon-on-Earth.

4. The Falling Earth • Why does the earth not crash into the sun? HINT: Does the Earth move? What does it have? • Which attraction is greater, the sun for the earth or the earth for the sun? HINT: One of Newton’s laws addresses this. • If there is an attraction for all objects, why do we not feel gravitated towards large buildings and other massive objects? HINT: Are we very massive compared to the Earth?

5. Newton’s Law of Universal Gravitation • Newton’s Law states that every object attracts every other object with a force that is directly proportional to the mass of each object. • He also deduced that the force decreases as the square of the distance between the objects increases. • F = Gm1m2/d2, where G is the universal gravitation constant, 6.67 x 10-11Nm2/kg2. m1 is the mass of one object and m2 is the mass of a second object. d is the distance between their centers. • The gravitation constant was measured by Henry Cavendish.

8. Khan Academy on Universal Gravitation • http://www.khanacademy.org/science/physics/mechanics/v/introduction-to-newton-s-law-of-gravitation • http://www.khanacademy.org/science/physics/mechanics/v/gravitation--part-2 • http://www.khanacademy.org/science/physics/mechanics/v/viewing-g-as-the-value-of-earth-s-gravitational-field-near-the-surface

9. Newton’s Law of Universal Gravitation • The force between you and any object is usually very small. The force of attraction between you and the earth is _____. • Your weight depends on your distance from the center of the earth. The closer you are to the center, the smaller will be your weight. This is due to the change in the mass and radius of the planet. • What happens to the Fg if • The mass of the planet doubles? • The mass of you doubles? • The distance between you and the center of the planet doubles? • The distance between you and the center of the planet cuts in half? • Your mass doubles and the planet’s radius cuts in half? • Cavendish went so far as to mass the earth. Its mass is 5.98 x 1024kg.

10. Newton’s Law of Universal Gravitation • The distance that an object is from the center of the Earth affects its acceleration due to gravity. Earth’s radius is 6.38 x 106m. That is the average radius. If one is on a mountain that is very high, its height must be taken into consideration. • That means that you must add the radius of the Earth to the distance an object is above the surface. • If you are 2 Earth radii high, then your distance is 3 Earth radii.

11. Gravitational Interactions • A force field exerts a force on objects in its vicinity. That means that technically an object interacts with the field exhibited by an object and not the object itself. • A field is represented by field lines. Where the lines are closer together, the field is stronger. They extend in all directions. • A gravitational field for a planet is represented by vectors which point to the center of mass. Remember gravity is only attractive.

12. Weight and Weightlessness • Suppose you weighed in an elevator. • What would be your weight if the elevator accelerated downward? • What would be your weight if the elevator accelerated upward? • What would be your weight if the elevator was not accelerating? • What would be your weight if the elevator cable broke and the elevator fell freely? • Weight then is the force that you exert against a support. • Weightlessness then becomes the absence of a supporting force.

13. History • Tycho Brahe spent his life accurately predicting astronomical events. • He believed that the Earth was the center of the universe. • His protege’ Johannes Kepler believed that the sun was the center of the universe. He formulated 3 laws based on his observations of the motions of the planets.

14. Kepler’s Laws • First Law: The paths of the planets are ellipses with the center of the sun at one focus. • Second Law: An imaginary line from the sun to a planet sweeps out equal areas in equal time intervals. Thus planets move fastest when they are closest to the sun. • Third Law: The ratio of the squares of the periods of any two planets revolving about the sun is equal to the ratio of the cubes of their average distances from the sun. T12/ T22 = r13/ r23