Unit 9: Circular Motion and Gravitation
Problem Set B
Overview:
Problem Set B targets your ability to use the Newton's laws of universal gravitation and the other equations derived from it in order to analyze the motion of satellites and celestial objects. A number of concepts and mathematical formulas will be of importance in your completion of this set. These concepts and formulas are described below.
Planets and satellites (both manmade satellites and the natural satellites commonly referred to as moons) move along paths that are nearly circular. Their motion can be described by the circular motion equations used in previous parts of this unit. Their orbital period (T) is simply the time to make one revolution or orbit. Their orbital speed (v) is simply the ratio of distance traveled (d) per time of travel (t). And their acceleration (a) is expressed in the same manner as the acceleration of any object moving in a circle. These three quantities are related by the following equations.


Orbiting satellites are simply projectiles  objects upon which the only force is gravity. The force which governs their motion is the the force of gravitational attraction to the object which is at the center of their orbit. Planets orbit the sun as a result of the gravitational force of attraction to the sun. Natural moons orbit planets as a result of the gravitational force of attraction to the planet. Gravitation is a force which acts over large distances in such a manner that any two objects with mass will attract. Newton was the first to propose a theory to describe this universal mass attraction and to express it mathematically. The law, known as the law of universal gravitation states that the force of gravitational attraction is directly proportional to the product of the masses and inversely proportional to the square of the separation distance between their centers. In equation form,
where m_{1} and m_{2} are the masses of the attracting objects (in kg) and d is the separation distance as measured from object center to object center (in meters) and G is the proportionality constant (sometimes referred to as the universal gravitation constant). The value of G is 6.67x 10^{11} N•m^{2}/kg^{2}.
Since orbiting satellites are acted upon solely by the force of gravity, their acceleration is the acceleration due to gravity (g). On earth's surface, this value was 9.8 m/s^{2}. For locations other than Earth's surface, there is a need for an equation which expresses g in terms of relevant variables. The acceleration of gravity depends upon the mass of the object which is at the center of the orbit (M_{central}) and the separation distance from that object (d). The equation which relates these two variables to the acceleration of gravity is derived from Newton's law of universal gravitation. The equation is
where G is 6.67x 10^{11} N•m^{2}/kg^{2}.
The speed required of a satellite to remain in an orbit about a central body (planet, sun, other star, etc.) is dependent upon the radius of orbit and the mass of the central body. The equation expressing the relationship between these variables is derived by combining circular motion definitions of acceleration with Newton's law of universal gravitation. The equation is
where M_{central} is the mass of the central body about which the satellite orbits, R is the radius of orbit and G is 6.667x 10^{11} N•m^{2}/kg^{2}.
For the general motion of an object in a circle, the period is related to the radius of the circle and the speed of the object by the equation v = 2 • pi • R / T. In the case of an orbiting satellite, this equation for speed can be equated with the equation for the orbital speed derived from universal gravitation to derive a new equation for orbital period. The result of the derivation is
where M_{central} is the mass of the central body about which the satellite orbits, R is the radius of orbit and G is 6.67x 10^{11} N•m^{2}/kg^{2}. Expressed in this manner, the equation shows that the ratio of period squared to the radius cubed for any satellite that is orbiting a central body is the same regardless of the nature of the satellite or the radius of its orbit. This principle is consistent with as Kepler's third law of planetary motion.
Summary of Mathematical Formulas
One difficulty a student may encounter in this problem set is the confusion as to which formula to use. The table below provides a useful summary of the formulas pertaining to circular motion and satellite motion. In the table, many of the formulas were derived from other equations. Thus, there will often be more than one means of determining an unknown quantity. In approaching these problems, it is suggested that you practice the usual habits of an effective problemsolver; identify known and unknown quantities in the form of the symbols of physics formulas, plot out a strategy for using the knowns to solve for the unknown, and then finally perform the necessary algebraic steps required for the solution.
To calculate ... 
... use the equation(s): 
(v) 
v = 2 • pi • R / T v = SQRT (G • M_{central} / R) for satellites only 
(a) 
a = v^{2} / R or a = F_{net} / m a = g = G • M_{central} / d^{2} for satellites only 
(F_{net}) 
F_{net} = m • aorF_{net} = m • v^{2} / R F_{net} = F_{grav }= G • m_{sat} • M_{central} / d^{2} for satellites only 
(T) 
T = 2 • pi • R / v T^{2} = 4 • pi^{2} / (G • M_{central }) • R^{3 }for satellites only 
Additional Readings/Study Aids:
The following pages from The Physics Classroom tutorial may serve to be useful to understanding Newtons law of universal gravitation and the mathematics of satellite motion.
Law of Universal Gravitation  Value of g  Kepler's Three Laws  Mathematics of Satellite Motion
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