Circular Motion and Planetary Motion

Suppose that you were driving a car with the steering wheel turned in such a manner that your car followed the path of a perfect circle with a constant radius. And suppose that as you drove, your speedometer maintained a constant reading of 10 mi/hr. In such a situation as this, the motion of your car could be described as experiencing uniform circular motion. Uniform circular motion is the motion of an object in a circle with a constant or uniform speed.
Calculation of the Average Speed
Average speed = distance / time = circumference / time
The circumference of any circle can be computed using from the radius according to the equation
Circumference = 2*pi*Radius
Combining these two equations above will lead to a new equation relating the speed of an object moving in uniform circular motion to the radius of the circle and the time to make one cycle around the circle (period).
The Direction of Velocity Vector As the object rounds the circle, the direction of the velocity vector is different than it was the instant before. So while the magnitude of the velocity vector may be constant, the direction of the velocity vector is changing. The best word that can be used to describe the direction of the velocity vector is the word tangential. The direction of the velocity vector at any instant is in the direction of a tangent line drawn to the circle at the object's location. (A tangent line is a lone which touches a circle at one point but does not intersect it.) The diagram at the right shows the direction of the velocity vector at four different point for an object moving in a clockwise direction around a circle. While the actual direction of the object (and thus, of the velocity vector) is changing, it's direction is always tangent to the circle.
Acceleration
An accelerating object is an object which is changing its velocity. And since velocity is a vector which has both magnitude and direction, a change in either the magnitude or the direction constitutes a change in the velocity. For this reason, it can be safely concluded that an object moving in a circle at constant speed is indeed accelerating. It is accelerating because the direction of the velocity vector is changing.It is calculated using the following equation:
Ave. acceleration = velocity / time = vf-vi / t
where vi represents the intial velocity and vf represents the final velocity after some time of t. The numerator of the equation is found by subtracting one vector (vi) from a second vector (vf).
The Centripetal Force Requirement
An object moving in a circle experiences an acceleration. Even in moving around the perimeter of the circle with a constant speed, there is still a change in velocity and subsequently an acceleration. this acceleration is directed towards the center of the circle. And in accord with Newton's second law of motion, an object which experiences an acceleration must also be experiencing a net force. the direction of the net force is in the same direction as the acceleration. So for an object moving in a circle, there.must be an inward force acting upon it in order to cause its inward acceleration. This is sometimes referred to as the centripetal force requirement. The word centripetal means center-seeking. For object's moving in circular motion, there is a net force acting towards the center which causes the object to seek the center.
Centrifugal force An object traveling in a circle behaves as if it is experiencing an outward force. This force, known as the centrifugal force, depends on the mass of the object, the speed of rotation, and the distance from the center. The more massive the object, the greater the force; the greater the speed of the object, the greater the force; and the greater the distance from the center, the greater the force.
Mathematics of Circular Motion
There are three mathematical quantities which will be of primary interest to us as we analyze the motion of objects in circles. These three quantities are speed, acceleration and force. The speed of an object moving in a circle is given by the following equation.
Average Speed = distance / time = 2 + pie * R / T
where R represents radius
T represents period
The acceleration of an object moving moving in a circle can be determined by either two of the following equations.
Acceleration = v2 / R Acceleration = 4 * pie2 * R / T2
where v represents speed
R represents radius
T represents period
The equation on the right (above) is derived from the equation on the left by the substitution of the expression for speed.
The net force (Fnet) acting upon an object moving in circular motion is directed inwards. While there may by more than one force acting upon the object, the vector sum of all of them should add up to the net force. In general, the inward force is larger than the oUPNard force (if any) such that the outward force cancels and the unbalanced force is in the direction of the center of the circle. The net force is related to the acceleration of the object (as is always the case) and is thus given by the following three equations:
Fnet = m * a Fnet = m * v2 / R Fnet = m * 4 * pie2 * R / T2
where m represents mass
v represents speed
R represents radius
T represents period
The equations in the middle (above) and on the right (above) are derived from the equation on the left by the left by the substitution of the expressions for acceleration.