Unit 9: Circular Motion and Gravitation
Problem Set C
Overview:
Problem Set C targets your ability to use mathematical formulas associated with circular and satellite motion to analyze the motion of objects and satellites moving in circles. In a sense, the problems of this set serve as a review of the mathematical principles practiced in Sets A and B. The only new mathematical principles introduced in this set pertain to Kepler's third law of planetary motion. A summary of this law and the other main ideas are presented below. At the end of the discussion, there is a table which lists all relevant formulas from the unit.
Before Newton proposed his law of universal gravitation, Johannes Kepler had mathematically analyzed available data regarding the motion of planets about the Sun and developed three generalizations or laws to describe their motion. His third law  the law of harmonies  stated that the ratio of period (T) squared to the radius (R cubed for every planet is the same value regardless of the planet's mass or distance from the Sun. As expressed by Kepler
While this observation was unique and perhaps a bit surprising, there was no explanation of why such a relationship existed. Newton was later able to use his universal gravitation law to explain this observation. Combining the universal gravitational equation with the equation for centripetal force, Newton derived an equation which related the T^{2}/R^{3} for any planet to the mass of the Sun. The result of the derivation is
where M_{Sun} is the mass of the Sun and G is 6.67x 10^{11} N•m^{2}/kg^{2}. Because Newton's equation was derived from equations that are applicable to any object acted upon by gravity alone as it orbited a central body, they apply to any satellite motion situation. That is, the moons of Jupiter orbiting Jupiter ought to exhibit this same characteristic of having the same T^{2}/R^{3} ratio. The satellites orbiting the Earth ought to exhibit this same characteristic of having the same T^{2}/R^{3} ratio. It is universal. In fact, the above equation can be written in more general terms as
where M_{central} is the mass of the central body about which the satellite orbits. The above equation applies to all satellite motion situations and relates orbital period, orbital radius and the mass of the central body.
Objects moving in circles have a speed which is equal to the distance traveled per time of travel. The distance around a circle is equivalent to a circumference and calculated as 2•pi•R where R is the radius. The time for one revolution around the circle is referred to as the period and denoted by the symbol T. Thus the average speed of an object in circular motion is given by the expression 2•pi•R / T. Often times the problem statement provides the rotational frequency in revolutions per minute or revolutions per second. Each revolution around the circle is equivalent to a circumference of distance. Thus, multiplying the rotational frequency by the circumference allows one to determine the average speed of the object.
The acceleration of objects moving in circles is based primarily upon a direction change. The actual acceleration rate is dependent upon how rapidly the direction is being change and is directly related to the speed and inversely related to the radius of the turn. It ends up that the acceleration is given by the expression v^{2} / R where v is the speed and R is the radius of the circle.
The equations for average speed (v) and average acceleration (a) are summarized below.


A successful mathematical analysis of objects moving in circles is heavily dependent upon a conceptual understanding of the direction of the acceleration and net force vectors. Movement along a circular path requires a net force directed towards the center of the circle. At every point along the path, the net force must be directed inwards. While there may be an individual force pointing outward, there must be an inward force which overwhelms it in magnitude and meets the requirement for an inward net force. Since net force and acceleration are always in the same direction, the acceleration of objects moving in circles must also be directed inward.
The last half of this problem set includes several problems in which a force analysis is conducted upon an object moving in circular motion. The goal of the analysis is either to determine the magnitude of an individual force acting upon the object or to use the values of individual forces to determine an acceleration. Like any force analysis problem, these problems should begin with the construction of a freebody diagram showing the type and direction of all forces acting upon the object. From the diagram, an F_{net} = m•a equation can be written. When writing the equation, recall that the F_{net} is the vector sum of all the individual forces. It is best written by adding all forces acting in the direction of the acceleration (inwards) and subtracting those which oppose it. Two examples are shown in the graphic below.
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 in understanding the conceptual and mathematical nature of circular and satellite motion.
Speed and Velocity  Centripetal Acceleration  Mathematics of Circular Motion
Law of Universal Gravitation  Value of g  Kepler's Three Laws  Mathematics of Satellite Motion
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