Chapter 5 Circular Motion; Gravitation

© 2004 Pearson Education Inc., publishing as Addison- Wesley The Force of Gravity What is the universal law of gravitation? What types of orbits are possible according to the law of gravitation? How can we determine the mass of distant objects? Our goals for learning:

5-6 Newton’s Law of Universal Gravitation If the force of gravity is being exerted on objects on Earth, what is the origin of that force? Newton’s realization was that the force must come from the Earth. He further realized that this force must be what keeps the Moon in its orbit.

5-6 Newton’s Law of Universal Gravitation The gravitational force on you is one-half of a Third Law pair: the Earth exerts a downward force on you, and you exert an upward force on the Earth. When there is such a disparity in masses, the reaction force is undetectable, but for bodies more equal in mass it can be significant.

© 2004 Pearson Education Inc., publishing as Addison-Wesley Newton ’ s Universal Law of Gravitation Isaac Newton discovered that it is gravity which plays the vital role of determining the motion of the planets - concept of action at a distance

Questions If the planets are orbiting the sun, what force is keeping them in orbit? What force keeps the moon in its orbit? Could the force of gravity be universal?

Newton’s Law of Universal Gravitation Any two objects attract each other with a gravitational force, proportional to the product of their masses and inversely proportional to the square of the distance between them. The force acts in the direction of the line connecting the centers of the masses.

© 2004 Pearson Education Inc., publishing as Addison-Wesley Newton ’ s Universal Law of Gravitation Between every two objects there is an attractive force, the magnitude of which is directly proportional to the mass of each object and inversely proportional to the square of the distance between the centers of the objects.

Change of Gravitational Force with Distance Law of universal gravitation is known as an inverse square law.

© 2004 Pearson Education Inc., publishing as Addison-Wesley Newton ’ s Universal Law of Gravitation G=6.67 x m 3 /(kg s 2 )

© 2004 Pearson Education Inc., publishing as Addison-Wesley How does the acceleration of gravity depend on the mass of a falling object? It does not. All falling objects fall with the same acceleration (on a particular planet). Now see why… F = ma and on Earth acceleration due to gravity denoted “ g ” so F=mg or g=F/m If mass of earth is M 1 then F g =GM 2 /d 2

5-6 Newton’s Law of Universal Gravitation Therefore, the gravitational force must be proportional to both masses. By observing planetary orbits, Newton also concluded that the gravitational force must decrease as the inverse of the square of the distance between the masses. In its final form, the Law of Universal Gravitation reads: where (5-4)

Henry Cavendish’s experiment determined the proportionality constant G in

5-6 Newton’s Law of Universal Gravitation The magnitude of the gravitational constant G can be measured in the laboratory. This is the Cavendish experiment.

5-7 Gravity Near the Earth’s Surface; Geophysical Applications Now we can relate the gravitational constant to the local acceleration of gravity. We know that, on the surface of the Earth: Solving for g gives: Now, knowing g and the radius of the Earth, the mass of the Earth can be calculated: (5-5)

5-7 Gravity Near the Earth’s Surface; Geophysical Applications The acceleration due to gravity varies over the Earth’s surface due to altitude, local geology, and the shape of the Earth, which is not quite spherical.

Problem 1 Two spheres of mass 35kg are 30m apart. A)What force does one exert on the other? B)If the mass of one is tripled and the radius is quadrupled how does the force change?

Problem 2 Two spheres of equal mass have a force of gravity of 7x10 -9 N exerted on each other. If the distance between them is 7m, find the mass.

Problem 3 Find the value of the gravitational acceleration g. The mass of the Earth is 6.0 x kg. The radius of the Earth is 6.38 x 10 6 m.

Homework – Chapter 5 28, 29, 30, 33, 36, 41 Kahoot