Teacher Notes


Teacher Notes
Publication No. 13785
Uniform Circular MotionInquiry Lab Kit for AP® Physics 1Materials Included In KitAlligator clips, 12 Additional Materials RequiredBalance, 0.1g precision (may be shared) Prelab Preparation
Safety PrecautionsThe very nature of the motion in this activity makes it potentially dangerous. Use caution when twirling the rubber stopper. This lab is best conducted outdoors, in an open gymnasium, or other large open area. Wear safety glasses during the experiment. Please follow all laboratory safety guidelines. Lab Hints
Teacher Tips
Further ExtensionsOpportunities for Inquiry The calculations for centripetal force in this lab assume the stopper is revolving in a horizontal plane at a right angle to the vertical, where the radius of the circle is the same as the measured length of the string. The force of gravity actually pulls the stopper at a slight downward angle, and the string sweeps across the surface of a cone. Design an experiment to measure the effect of centripetal force on the angle of an object in motion as a conical pendulum. Correlation to Next Generation Science Standards (NGSS)^{†}Science & Engineering PracticesPlanning and carrying out investigationsAnalyzing and interpreting data Using mathematics and computational thinking Constructing explanations and designing solutions Engaging in argument from evidence Disciplinary Core IdeasHSPS2.A: Forces and MotionHSETS1.B: Developing Possible Solutions Crosscutting ConceptsPatternsCause and effect Scale, proportion, and quantity Performance ExpectationsHSPS21. Analyze data to support the claim that Newton’s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration. Answers to Prelab Questions
Sample DataIntroductory Activity {13785_Data_Table_1}
Graph A
{13785_Data_Figure_1}
Graph B
{13785_Data_Figure_2}
The graphs show that the relationship between the velocity and the centripetal force is nonlinear, but the relationship between the square of the velocity and the force is linear. This verifies the equation for F_{c}. Changing Radius Data Mass of Stopper (m_{1}): 0.014 kg Mass of Washers + Clips (m_{2}): 0.111 kg {13785_Data_Table_2}
Changing Mass Data Radius: 0.5 m Mass of Washers + Clips (m_{2}): 0.111 kg {13785_Data_Table_3}
Analyze the Resuts Changing Radius As the radius decreased, the tangent velocity also decreased when the tension in the string and the mass of the stopper were kept constant. Since a_{c} = F/m, one would predict the acceleration of the stopper would remain constant. However, it appears that the centripetal force increased as the radius decreased, and if the force increased, the acceleration would also increase. An error analysis is needed to determine if the calculated centripetal force for each radius is the same within experimental error. The percent error between the theoretical tension in the string and the calculated centripetal force was 3.7% for both the largest and smallest radii (2.75% for the 0.5m radius). With a radius of 0.25 m, the angular velocity was quite fast, and measuring the time accurately for 20 revolutions was difficult. The range of time for 20 revolutions was greatest with the 0.25m radius, so an analysis for this experiment is shown. {13785_Data_Table_4}
Since the theoretical value for the tension in the string (1.09 N) and the calculated average centripetal force for each radius (1.05, 1.12, 1.13 N, respectively) fall between the values of 0.96 and 1.26 N, we can conclude that the centripetal force remains constant within experimental error. Changing Mass As the mass of the stopper increased, its tangent velocity decreased when the tension in the string and the radius were kept constant. Calculations of centripetal acceleration are shown below. When the mass of the stopper increased by a factor of 2, the centripetal acceleration decreased by the same factor. Stopper mass 0.007 kg: a_{c} = (79.2 m/s)^{2}/0.5 m = 158 m/s^{2} Stopper mass 0.014 kg: a_{c} = (39.9 m/s)^{2}/0.5 m = 79.8 m/s^{2} Stopper mass 0.026 kg: a_{c} = (21.9 m/s)^{2}/0.5 m = 43.8 m/s^{2} Answers to QuestionsGuidedInquiry Discussion Questions
ReferencesAP^{®} Physics 1: AlgebraBased and Physics 2: AlgebraBased Curriculum Framework; The College Board: New York, NY, 2014. 
Student Pages


Student PagesUniform Circular MotionIntroductionCan an object traveling at a constant speed be accelerating? It can if it is changing direction. An object traveling in a circle is accelerating as it constantly changes direction, even while maintaining a steady speed. Circular motion is manifested in many sports, amusement park rides, highway designs and even satellites and planets. Investigate the force that causes an object to accelerate as it moves in a circular path. Concepts
BackgroundCentripetal force is the “center seeking” force that makes an object move in a circle. According to Newton’s first law, when an object is in motion, it will remain in motion unless acted upon by an unbalanced force. This means an object will travel in a straight line at a constant speed as long as no outside force is acting on it. In order for an object to move in a circle, an inward force is needed. For example, imagine a rubber stopper being whirled around on the end of a string. The hand holding the string creates tension on the string that exerts an inward force (centripetal) on the rubber stopper (see Figure 1). If the string were to break, the stopper would fly outward in a straight line. The mathematical expression shown in Equation 1 for centripetal force is the same as for any other force, based on Newton’s second law of motion. {13785_Background_Figure_1}
According to Equation 1, a force will cause an object to accelerate. Therefore a centripetal force will pull an object toward the center of the circle causing a centripetal acceleration.
{13785_Background_Equation_1}}
where
F is the force (N) {13785_Background_Equation_2}
where
a is the centripetal acceleration If we substitute Equation 2 into Equation 1, centripetal force (F_{c}) can be expressed as Equation 3. {13785_Background_Equation_3}
Notice that in order to solve for the centripetal force using Equation 3, the mass and velocity of an object, as well as the radius of its circular path, must be known. The mass can easily be measured using a balance, and the radius can be measured with a meter stick. But how can the velocity of the object be measured? The typical equation for calculating the average speed of an object can be used to determine the velocity.
{13785_Background_Equation_4}
Now the question is, how can you find the distance around a circle? When an object makes one complete revolution, it travels a distance equal to the circumference of a circle, 2πr. The time it takes for one complete revolution around a circle is known as the period, T. Therefore, for objects moving in a circle, the velocity can be expressed as Equation 5.
{13785_Background_Equation_5}
where
v_{t} is the tangent velocity (m/s) Experiment OverviewThe purpose of this advanced inquiry lab is to determine the relationship between the velocity and centripetal force of an object moving in a circle. The investigation begins with an introductory activity to observe the motion of a rubber stopper rotating in a horizontal plane. The procedure provides a model for guidedinquiry design of experiments to determine what factors affect the centripetal acceleration of an object in circular motion. Sources of experimental error will be identified. MaterialsAlligator clip Prelab Questions
Safety PrecautionsThe very nature of the motion in this activity makes it potentially dangerous. Use caution when twirling the rubber stopper. This lab is best conducted outdoors, in an open gymnasium or other large open area. Wear safety glasses whenever anyone is conducting the lab in the area. Please follow all laboratory safety guidelines. ProcedureIntroductory Activity
Part B. Measuring the Force
Analyze the Results
{13785_Procedure_Equation_6}
where g is the acceleration of gravity Equation 6 provides the theoretical tension in the string, which can be compared to the experimental values of the centripetal force determined in Part B of the Introductory Activity. Prepare graphs of (a) velocity and (b) velocity squared (xaxis) versus the theoretical centripetal force in newtons (yaxis). Explain the shape of each graph. GuidedInquiry Design and ProcedureForm a working group with other students and discuss the following questions.
Student Worksheet PDF 