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**Universal Gravitation**

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Circular Orbits The moon moves in a nearly circular path around the Earth. The path is called an orbit The period is about 28 days The Earth and planets move in nearly circular paths around the sun. The moon and planets have a centripetal acceleration. A centripetal force - the law of action

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Central Force There is no normal or tension force affecting the planets. Like falling bodies that accelerate, the moon accelerates toward the earth. Gravity is a central force. orbital velocity centripetal acceleration Moon Earth

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**Inverse Square Distance**

The Earth’s radius is about rE = 6400 km. The centripetal acceleration is a constant. g = 9.8 m/s2. The distance from the Earth to the Moon is about 380,000 km = 60 rE . The centripetal acceleration of the moon is known from the period. a = v2 / r = (4p2 / T2) r a = m/s2 = g / 3600 If a falling object and the moon are acted on by the same force, the force gets weaker as the square of the distance. This is an inverse square law: a = C / r2.

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**Mass Dependence Newton’s law of action applies to orbits.**

Centripetal acceleration times mass is the force of gravity FEM = mM a = mM C / r2 Moon FEM a Earth

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Equal and Opposite Newton’s law of reaction also applies to the forces. The forces are equal and opposite FME = FEM mE K / r2 = mM C / r2 FEM Moon FME Earth

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**Gravitational Constant**

To make these equations agree Newton made the equation depend on the two masses. FEM = G mE mM / r2 Other planets also obey the same law. The gravitational force is universal. F = G m1 m2 / r2 The gravitational constant, G = 6.67 x Nm2/kg2. next

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