Newton’s Laws of Motion and Universal Gravitation

           Orbit Meccanics:

Newton’s laws of motion describe the relationship between the motion of a particle and the forces acting on it. The first law states that if no forces are acting, a body at rest will remain at rest, and a body in motion will remain in motion in a straight line. Thus, if no forces are acting, the velocity (both magnitude and direction) will remain constant.

The second law tells us that if a force is applied there will be a change in velocity, i.e. an acceleration, proportional to the magnitude of the force and in the direction in which the force is applied. This law may be summarized by the equation

F = ma   (1)

where F is the force, m is the mass of the particle, and a is the acceleration.

The third law states that if body 1 exerts a force on body 2, then body 2 will exert a force of equal strength, but opposite in direction, on body 1. This law is commonly stated, “for every action there is an equal and opposite reaction”. In his law of universal gravitation, Newton states that two particles having masses m1 and m2 and separated by a distance r are attracted to each other with equal and opposite forces directed along the line joining the particles. The common magnitude F of the two forces is

F = G* ((m1*m2)/r2 )   (2)

where G is an universal constant, called the constant of gravitation, and has the value 6.67259×10-11 N-m2/kg2.

Let’s now look at the force that the Earth exerts on an object. If the object has a mass m, and the Earth has mass M, and the object’s distance from the center of the Earth is r, then the force that the Earth exerts on the object is GmM /r2 . If we drop the object, the Earth’s gravity will cause it to accelerate toward the center of the Earth. By Newton’s second law (F = ma), this acceleration g must equal (GmM /r2)/m, or

g =  ((G*M)/r2 )   (3)

At the surface of the Earth this acceleration has the valve 9.80665 m/s2 (32.174 ft/s2). Many of the upcoming computations will be somewhat simplified if we express the product GM as a constant, which for Earth has the value 3.986005×1014 m3/s2 . The product GM is often represented by the Greek letter μ.

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