AQA A-level Physics Newton's law of gravitation
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A gravitational field is the region of space surround a mass, in when a body is placed, it feels an attractive force. This force is a gravitational pull. By Newton's Third Law, the force exerted by mass on the body is equal to the force exerted by the body on the mass. By Newton's Law of Gravitation, this can be found using the following formula: F = GmM/r².
Newton's Law of Gravitation is an inverse square law. This means that the gravitational force is directly proportional to the mass of the objects and inversely proportional to the square of the distance between their centers of mass. Gravitational field strength is defined as the gravitational force exerted on a body placed in a gravitational field per unit mass and it is given in m/s². It can be calculated using g = GM/r², where M is the mass of the object generating the field, or g = F/m, where m is the mass of the object placed in the gravitational field. To derive Newton's Law of Gravitation, let M be the mass of the object generating the gravitational field and m be the mass of the object placed in it. By Newton's Second Law, F = mg, so gravitational force is proportional to the mass of the object. By Newton's Third Law, there must be an equal and opposite force acting on the object generating the field, so, combined with Newton's Second Law, F = Mg, so gravitational force is also proportional to the mass of the object. Consider the Moon orbitting the Earth. The gravitational force on the Moon is what constitutes as the centripetal force. Thus, Gravitational Force = Centripetal Force = mv²/r. For an object in circular motion, v = 2πr/T. Substituting this gives F = 4π²r²m/rT², which simplifies to F = 4π²rm/T² and can be rearranged to T²F/r = 4π²m. Kepler's Third Law states that T²/r³ = constant, so for this to hold, F must be inversely proportional to r². Putting all the proportionalities together, F must be proportional to mM/r², so F = GmM/r² where G is a constant.
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