Gravitation

“Gravity is a myth, the Earth sucks”
Unknown

 

NEWTON’s LAW of GRAVITATION

• All objects feel a force of attraction to each other known as the gravitational force.  The magnitude of this force between two “point” particles of masses m₁ and m₂ separated by a distance r₁₂ is given by the equation below

where G is the gravitational constant whose value is measured experimentally as 6.6726 x 10^-11 m³/kg.s².  G is believed to be a Universal constant whose value is the same between any two point objects at all time and all space.

•   Do not confuse “G” and “g”.  g (=9.8 m/s²) is the value of the acceleration due to gravity at the earth’s surface, it is not a universal constant.

•  For a uniform spherical body radius R, the gravitational force on a “point” mass distance r from the centre of the sphere can be calculated assuming all of the mass of the sphere at a radius less than r is concentrated at the centre of the sphere.  In so far as the earth is a uniform sphere, we may calculate the gravitational force due to the earth as if the mass of the earth were located at its centre.

•  The earth is not exactly a sphere, it bulges at the equator, nor is its density uniform.  In addition, its rotation causes the value of g to decrease slightly as one moves from the poles to the equator.  However, all of these effects are less than 1% of the nominal value of 9.81 m/s² at the earth’s surface. 

• The direction of the gravitational force is along the line joining the two point masses.  Each mass is attracted to the other with equal but opposite forces (given by the above equation) which comprise an action/reaction pair of forces.  Note that this means that if an object on the earth’s surface feels a gravitational force due to the earth, the earth feels an equal but opposite force.  The acceleration of the object will be “g”, but the acceleration of the earth will be undetectable because of the large mass of the earth (a = F/m).

• The fact that the value of G is so small means that the magnitude of the gravitational force between “normal” objects is very small.  The first measurement of the value of G was by Henry Cavendish     in 1798.    Having obtained the value of G, Cavendish was able to estimate the mass of the earth, in fact he titled his paper “weighing the earth”.  In this case the force on an object at the earth’s surface is given by,

• Away from the earth’s surface the acceleration due to gravity on an object depends on its height, h, above the surface, 

• Satellite Motion

As we have already seen an object is kept in uniform circular motion by a centripetal force acting towards the centre of the circle.  For satellites in circular orbits around the earth the centripetal force is the force of gravity and we may write

            But, for circular motion the period, T, and velocity, v, are related by

In other words there is a direct relationship between the period and radius of circular orbits.  This is apparent in the orbital motion of the planets around the Sun; as the distance from the Sun increases the period also increases.  This relationship is known as Kepler’s third law.

• Circular orbits are a special case of orbital motion which can be described by simple application of Newton’s Laws.  Elliptical, parabolic and hyperbolic orbits, much more common in nature, require more complex analysis.  It is interesting to note that projectile trajectories are actually elliptical orbits (see below).

"Far too noisy, my dear Mozart, far too many notes"
Archduke Ferdinand of Austria on Wolfgang Amadeus Mozart
 

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu