Universal Gravitation

What is Gravity? The Falling Apple Gravity is the way in which masses communicate with each other. Every mass in the universe reaches out to attract every other one, and every mass feels an attraction from every other one. The Falling Apple Newton knew that: Without an outside force, moving objects continue to move at constant speed in a straight line. If an object undergoes a change in speed or direction, then a force is responsible.

The Falling Moon The moon is falling toward Earth, just as the apple is. Newton hypothesized that the moon was simple a projectile circling Earth under the attraction of gravity. He compared motion of the moon to a cannonball fired from a top of high mountain.                                                            

How far the moon falls and how far an apple at Earth’s surface falls, should relate only to their respective distance from Earth’s center. The moon was already known to be 60 times farther from the center of the Earth than an apple at Earth’s surface. In one second  the apple falls 5 m (exactly 4.9 m) Gravitational attraction to earth is “diluted” by distance  The moon should fall 1/60 of 1/60 (or 1/(60)2 ) of 4.9 m, so in one second the moon should fall 1.4 millimeters.

The Falling Earth Newton generalized his moon findings to all objects, and stated that all objects in the universe attract each other. ∙ The Earth was no longer the center of the Universe, not even the center of the solar system. ∙ The planets and the Earth orbit the sun in the same way the moon orbits the Earth. ∙ The planets continually “fall” around the sun in closed paths. Question: Since the moon is gravitationally attracted to the Earth, why does it not simply crash into Earth?

Newton’s Law of Universal Gravitation Newton did not discover gravity. What he discovered was that gravity is universal. Newton’s law of universal gravitation states: ∙ Every object attracts every object with a force that for any two objects is directly proportional to the mass of each object. ∙ This force decreases as the square of the distance between the centers of mass of the object increases. Greater the masses  Greater the force of attraction between them. The farther away the objects from each other  Less the force of attraction between them. m1 ∙ m2 F ~ d 2

The Universal Gravitational Constant The proportionality form or the law of universal gravitation can be expressed as an exact equation if G (universal gravitational constant) is introduced. m1 ∙ m2 F = G d 2 G is given by the magnitude of a force between two masses of 1 kilogram each, one meter apart: 0.000 000 000 0667 Newton (extremely weak force). G was first measure 150 years after Newton’s discovery of universal gravitation, by an English physicist, Henry Cavendish. G = F G = 6.67 x 10 -11 N∙m2/kg2

Gravity and Distance: The Inverse-Square Law We can understand how gravity is reduced with distance by considering an imaginary “butter gun” used for buttering toast. How thick will the butter be on each piece of toast as I increase the distance? When a quantity varies as the inverse square of its distance from it source, it follows an inverse-square law. This law apply not only to the spreading of butter from a butter gun, and the weakening of gravity with distance, but to all cases where the effect for a localized source spreads evenly throughout the surrounding space. More examples are light, radiation and sound.

The greater the distance from Earth’s center, the less an object will weigh. The gravitational influence of every object, however small or far, is excerted through all space. Question: Suppose that an apple at the top of a tree is pulled by Earth’s gravity with a force of 1 Newton. If the tree were twice as tall, would the force of gravity on the apple be only ¼ as strong? Explain.

Concept Summary ∙ The moon and other objects in orbit around Earth are actually falling toward Earth but have great enough tangential velocity to avoid hitting Earth ∙ According to Newton’s Law of universal gravitation, everything pulls on everything else with a force that depends upon the masses of the objects and the distance between their center of mass. ∙ The greater the masses, the greater is the force. ∙ The greater the distance, the smaller is the force ∙ Gravitation decreases according to the inverse-square law. The force of gravity weakens as the distance squared.