There is one other major law due to Newton that will be used in this course and this is hisfamous Law of Universal Gravitation. It deals with the force between any two massive objects.We will use the Law of Universal Gravitation together with Newton's Laws of Motion to discuss avariety of problems involving the motion of large objects like the Earth moving in orbit about theSun as well as small objects like the famous apple falling from a tree. Also it will be shown thatNewton's 3rd Law of Motion follows as a consequence of Newton's Law of Universal Gravitation.
Example: The Acceleration of Gravity on the Moon's Surface You could take the mass m to the Moon and the mass would remain the same but the force of gravity would be less. The force of gravity would be W=m g where g is the acceleration of gravity on the Moon and also the force of gravity acting on the mass m is given by the Law of Gravitation W=F= G m M R2 where M is the mass of the Moon. Since both ways of calculating the gravitational force should yield the same answer if follows m g = G m M R2 and canceling m which appears on both sides we get g on the Moon g = G M R2 The mass of the Moon is M=7.35 1022 kg and the Moon's radius is R=1738000. meters so M=7.35 1022 kg R = 1 738 000.; M = 7.35 * 1022; G = 6.67 * 10-11; g =
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G M R2 1.62298 So the acceleration of gravity on the Moon is g=1.6 m/s2 which is quite a bit smaller than g=9.8 m/s2 on Earth. 1.6 9.8 0.163265 So g=0.16 g or on the Moon the acceleration of gravity is about 16% that on Earth. | (7) (8) 8. Newton's Law Gravitation Rev.nb 15 A Numerical Example: Suppose an object has mass of 60 kg. What is its weight on Earth? m = 60.; g = 9.8; W = m g 588. So on Earth the weight is 588 Nt. What is the weight on the Moon? m = 60.; g = 1.6; W = m g 96. So on the Moon the weight is 96 Nt. which is quite a bit less than 588 Nt. | 16 8. Newton's Law Gravitation Rev.nb
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Non-Inertial Reference Frames or Coordinate Systems. Sometimes it is useful to use a coordinate system for which Newton's 1st Law or the law of inertia does NOT hold. These coordinate systems are non-inertial frames of reference. If you use Newton's 2nd law in a non-inertial coordinate system then you must include a correction. A moving ordinary elevator is an example of a non-inertial reference frame. First consider an elevator at rest with respect to the ground and a mass M inside and on the floor of the elevator. Also there is an ordinary bathroom scale between the mass M and the floor of the elevator. Later M=5 kg as a numerical example. y The Elevator Y M x X
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The Ground There are two coordinate system one of which is fixed to the ground and is called X, Y. The other coordinate system is fixed to the elevator and is called x, y. The basic idea is compare how an observer fixed to the X, Y coordinate system would view things in comparison to an observer in the x,y coordinate system. The walls of the elevator are clear glass or plastic so the observer X, Y outside can read the bathroom scale even though it is inside the elevator. Both observer X, Y and x,y read the same number off of the bathroom scale. | 8. Newton's Law Gravitation Rev.nb 17
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