

Lesson 1: Motion Characteristics for Circular MotionThe Centripetal Force Requirement Mathematics of Circular Motion 
Lesson 1: Motion Characteristics for Circular MotionMathematics of Circular MotionThere 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. The acceleration of an object moving in a circle can be determined by either two of the following equations. 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 (F_{net}) 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 outward 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: The equations in the middle (above) and on the right (above) are derived from the equation on the left by the substitution of the expressions for acceleration. This set of circular motion equations can be used in two ways:
These two ways are illustrated below.


The solution of this problem begins with the identification of the known and requested information.
Known Information: m = 900 kg 
Requested Information: a = ???? 
To determine the acceleration of the car, use the equation a = v^{2 }/ R. The solution is as follows:
a = (10.0 m/s)^{2} / (25.0 m)
a = (100 m^{2}/s^{2}) / (25.0 m)
a = 4 m/s^{2}
To determine the net force acting upon the car, use the equation F_{net} = m•a. The solution is as follows.
F_{net} = (900 kg) • (4 m/s^{2})
F_{net} = 3600 N

The solution of this problem begins with the identification of the known and requested information.
Known Information: m = 95.0 kg 
Requested Information: v = ???? 
To determine the speed of the halfback, use the equation v = d / t where the d is onefourth of the circumference and the time is 2.1 s. The solution is as follows:
v = (0.25 • 2 • pi • R) / t
v = (0.25 • 2 • 3.14 • 12.0 m) / (2.1 s)
v = 8.97 m/s
To determine the acceleration of the halfback, use the equation a = v^{2} / R. The solution is as follows:
a = (8.97 m/s)^{2} / (12.0 m)
a = (80.5 m^{2}/s^{2}) / (12.0 m)
a = 6.71 m/s^{2}
To determine the net force acting upon the halfback, use the equation F_{net} = m•a. The solution is as follows.
F_{net} = (95.0 kg)*(6.71 m/s^{2})
F_{net} = 637 N
In Lesson 2 of this unit, circular motion principles and the above mathematical equations will be combined to explain and analyze a variety of realworld motion scenarios including amusement park rides and circulartype motions in athletics.
1. Anna Litical is practicing a centripetal force demonstration at home. She fills a bucket with water, ties it to a strong rope, and spins it in a circle. Anna spins the bucket when it is halffull of water and when it is quarterfull of water. In which case is more force required to spin the bucket in a circle? Explain using an equation as a "guide to thinking."
2. A Lincoln Continental and a Yugo are making a turn. The Lincoln is four times more massive than the Yugo. If they make the turn at the same speed, then how do the centripetal forces acting upon the two cars compare. Explain.
3. The Cajun Cliffhanger at Great America is a ride in which occupants line the perimeter of a cylinder and spin in a circle at a high rate of turning. When the cylinder begins spinning very rapidly, the floor is removed from under the riders' feet. What affect does a doubling in speed have upon the centripetal force? Explain.
4. Determine the centripetal force acting upon a 40kg child who makes 10 revolutions around the Cliffhanger in 29.3 seconds. The radius of the barrel is 2.90 meters.
Lesson 1: Motion Characteristics for Circular Motion
 Speed and Velocity
 Acceleration
 The Centripetal Force Requirement
 The Forbidden FWord
 Mathematics of Circular Motion
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