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The Law of Universal

Gravitation

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A little background… Legend has it that Sir Isaac Newton was

struck on the head by a falling apple while napping under a tree. This prompted Newton to imagine that all bodies in the universe are attracted to each other in the same way that the apple was attracted to the Earth.

Newton analyzed astronomical data on the motion of the Moon around the Earth and stated that the law of force governing the motion of the planets has the same mathematical form as the force law that attracts the falling apple to the Earth.

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Newton’s Law of Universal Gravitation

every particle in the universe attracts every other particle with a force The force is directly proportional to the

product of their masses The force is inversely proportional to the

square of the distance between them.

So, if the particles have masses m1 and m2 and are separated by a distance d, the magnitude of the gravitational force is:

221

d

mmGF

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A closer look… G is the universal gravitational

constant, which has been measured experimentally as 6.67 x 10-11 .

The distance, d, between m1 and m2 is measured from the center of m1 to the center of m2.

2

2

kg

mN

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Remember Third Law?

By Newton’s third law, the magnitude of the force exerted by m1 on m2 is equal to the force exerted by m2 on m1, but opposite in direction. These gravitational forces form an action-reaction pair.

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Properties of the Gravitational Force:

The gravitational force acts as a long range force, which exists between two particles, regardless of the medium that separates them.

The force varies as the inverse square of the distance between the particles. This means force decreases rapidly with

increasing distance between the particles. The force is directly proportional to the mass

of each particle. This means, a larger mass will yield a larger force

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Weight and Gravitational Force Weight was previously defined as FW = m·g,

where g is the magnitude of the acceleration due to gravity. So,

The object’s mass, m, cancels out, giving us (surface gravity). Or, the gravitational acceleration experienced when above the surface of earth:

2d

mMGgm p

2d

MGg

p

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Formula Summary

221

d

mmGF

2d

MGg p

ME = 5.97 x 1024 kgrE = 6.37 x 106 mG = 6.67 x 10-11 Nm2/kg2

5280 ft = 1 mile 3.28 ft = 1 m

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Try calculating these…

6.67 x 10-11 x 2000 =6.67 x 10-11 x 52 =6.67 x 10-11 x 5.98 x 1024 =

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Example 1) Two masses, 25000 kg and 80000 kg, are separated by 200 meters. What is the force of attraction between them?

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Example 2) What is the force of attraction between a 2000 g mass and a 4000 g mass if their centers are 15 cm apart?