Materials Included In Kit

Additional Materials Required

Prelab Preparation

1. Cut the string into 1.4-m lengths. Each group of students will need one 1.4-m piece.
2. Wrap each glass handle tube with masking tape in a diagonal pattern from one end to the other, without covering the polished ends of the tube. The tape will contain any broken glass in the event the tube is dropped.

Safety Precautions

The 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

• This laboratory activity can be completed in two 50-minute class periods. It is important to allow time between the Introductory Activity and the Guided-Inquiry Activity for students to discuss and design the guided-inquiry procedures. Also, all student-designed procedures must be approved for safety before students are allowed to implement them in the lab. Prelab Questions may be completed before lab begins the first day.
• This activity requires a great deal of open space. If space is limited, students may work in groups of 3–4 rather than in pairs.
• For the guided-inquiry experiments, students may use the washers to provide the tension in the string or they may choose to use a heavier hanging weight, a fixed spring scale, or a force sensor if available.

Teacher Tips

• Students may be surprised that a smaller radius circle requires a much faster rotation of the stopper when all other variables remain constant. This provides the opportunity to differentiate between angular speed (rpm or rad/s) and tangential (or linear) speed.

Further Extensions

Opportunities 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.

Alignment to the Curriculum Framework for AP^® Physics 1 

Enduring Understandings and Essential Knowledge
All forces share certain common characteristics when considered by observers in inertial reference frames. (3A)
3A2: Forces are described by vectors.
3A3: A force exerted on an object is always due to the interaction of that object with another object.

Classically, the acceleration of an object interacting with other objects can be predicted by using a = ΣF/m. (3B)
3B1: If an object of interest interacts with several other objects, the net force is the vector sum of the individual forces.

Learning Objectives
3A2.1 The student is able to represent forces in diagrams or mathematically using appropriately labeled vectors with magnitude, direction, and units during the analysis of a situation.
3A3.1 The student is able to analyze a scenario and make claims (develop arguments, justify assertions) about the forces exerted on an object by other objects for different types of forces or components of forces.
3A3.3 The student is able to describe a force as an interaction between two objects and identify both objects for any force.
3B1.2 The student is able to design a plan to collect and analyze data for motion (static, constant, or accelerating) from force measurements and carry out an analysis to determine the relationship between the net force and the vector sum of the individual forces.

Science Practices
1.1 The student can create representations and models of natural or man-made phenomena and systems in the domain.
1.4 The student can use representations and models to analyze situations or solve problems qualitatively and quantitatively.
4.2 The student can design a plan for collecting data to answer a particular scientific question.
5.1 The student can analyze data to identify patterns or relationships.
6.4 The student can make claims and predictions about natural phenomena based on scientific theories and models.
7.2 The student can connect concepts in and across domain(s) to generalize or extrapolate in and/or across enduring understandings and/or big ideas.

Answers to Prelab Questions

1. Consider an object moving in a circle at a constant speed.
   1. Are all the forces acting on the object balanced?

      No, if the object is moving in a circle, it is constantly changing direction. An unbalanced force must be causing this change.

   2. Explain why or why not in terms of Newton’s laws.

      According to Newton’s first law, an object will remain in uniform motion (same speed and direction) unless acted upon by an unbalanced force. An object changing direction is accelerating. According to Newton’s second law, an object will experience acceleration when an unbalanced force acts on it.

2. Figure 2 shows an overhead view of a ball traveling in a circle in a clockwise direction.
   1. At point A, draw an arrow indicating the direction of the tangential velocity of the ball.
   2. At point B, draw the direction of the centripetal force on the ball.
   3. At point C, draw the direction of the centripetal acceleration of the ball.
   4. If the string were to break as the ball reached point D, draw an arrow from point D to indicate the path the ball would take.
      {13785_PreLabAnswers_Figure_2}
3. Read the entire Background section and Procedure for the Introductory Activity. In Figures 1 and 3, the hand holding the string causes tension in the string, which exerts a force on the rubber stopper. What causes the tension in the string in Figure 5?

   The tension in the string is caused by the weight of the objects attached to the string: the washers, alligator clip, and the paper clip. The combined weight is the total mass of the objects multiplied by the acceleration due to gravity (F = mg).

Sample Data

Introductory Activity

Mass of Stopper (m₁): 0.014 kg
Radius: 0.75 m

Graph A Graph B 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 (m₁): 0.014 kg
Mass of Washers + Clips (m₂): 0.111 kg Changing Mass Data

Radius: 0.5 m
Mass of Washers + Clips (m₂): 0.111 kg 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 0.5-m 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 0.25-m radius, so an analysis for this experiment is shown. 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/s² Stopper mass 0.014 kg: a_c = (39.9 m/s)²/0.5 m = 79.8 m/s² Stopper mass 0.026 kg: a_c = (21.9 m/s)²/0.5 m = 43.8 m/s²

Answers to Questions

Guided-Inquiry Discussion Questions

1. Review the centripetal force procedure. Calculate the percent error between the theoretical tension in the string, F_T, to the experimental centripetal force, F_c for each set of washers.
   {13785_Answers_Equation_1}

   For 12 washers: 17%
   For 18 washers: 3.5%

2. Identify random sources of error in the experiment.
   1. Which measurement has the greatest potential for error?

      Random sources of error in the measured quantities include the length of the radius, the mass of the stopper, the mass of the washers and clips, and the time for 20 revolutions. Based on the variations in the recorded data, it is evident that the measurement of time has the greatest potential for error.

   2. What are some ways this error might be reduced?

      This error could be reduced by having more than one person time the revolutions and by conducting more trials. With more trials, outliers might be more apparent and could be disregarded, with only the closer times contributing to the mean.

3. Identify any sources of systematic error in this experiment.

   The stopper does not actually rotate in a horizontal plane that is parallel to the ground. It approaches this plane at faster speeds, but still rotates at a small angle. A vertical component of the tension in the string must exist to counteract the force of gravity acting upon the stopper. Therefore the length of the string is not exactly the same as the radius of the circle. In addition, the string may also stretch slightly as the stopper is being whirled around. Some amount of friction exists between the string and the glass tube. Keeping the washers at exactly the same level throughout the experiment was challenging.

4. The independent variable in an experiment is the variable that is changed by the experimenter, while the dependent variable responds to or depends on the changes in the independent variable.
   1. Name the independent and dependent variables in Part B of the Introductory Activity.

      The independent variable was the weight of the washers that created the tension in the string. The dependent variable was the velocity of the stopper.

   2. What other variables could be tested?

      The mass of the stopper or the radius of the circle could be changed to test its effect on the velocity of the stopper for a given amount of tension in the string. Note: You may decide to have half of the groups test the effect of the radius and the other half test the effect of the mass of the stopper.

Answers to the Review Questions for AP^® Physics 1

1. A rider on a bicycle with a total mass of 80 kg rounds a curve with a radius of 20 meters at a speed of 10 km/hr.
   1. What is the centripetal acceleration of the cyclist? (Hint: Convert km/hr to m/s.)
      {13785_Answers_Equation_3}

      a = (2.8 m /s)²/ 20 m = 0.39 m/s²

   2. What is the amount of centripetal force acting on the bicycle?

      F_c = ma = 80 kg x 0.39 m/s² = 31.2 N

   3. What is the source of the centripetal force on the bicycle?

      The centripetal force is provided by the force of friction from the interaction of the bicycle tires and the ground.

2. A 0.6-kg ball is attached to a cord and spun in a horizontal circle with a radius of 1.2 m. The maximum tension the cord can withstand is 60 N.
   1. What is the maximum speed the ball can attain before the cord breaks?

      The tension in the cord is causing the centripetal force, so F_T = mv²/r. Velocity can be expressed as

      {13785_Answers_Equation_4}
      {13785_Answers_Equation_5}
   2. If you wanted to maintain the speed of the ball from part a, would it be better to increase or decrease the radius to ensure the cord would not break?

      A larger radius would reduce the acceleration of the ball and the centripetal force required to keep it moving in a circle would be less. Therefore the tension on the string would be reduced.

3. The Earth’s orbit around the Sun is nearly circular with an average radius of 1.5 x 10¹¹ m. Assume the Earth is in uniform circular motion as it travels around the Sun.
   1. What is the source of the centripetal force acting on the Earth?

      The gravity of the Sun provides the centripetal force acting on the Earth.

   2. What is the centripetal acceleration of the Earth?

      First determine the velocity of the Earth, 2πr/T.
      T = 365 days x 24 hr/day x 60 min/hr x 60 s/min = 3.15 x 10⁷ s
      v = 9.4 x 10¹¹ m/3.15 x 10⁷ s = 2.98 x 10⁴ m/s

      {13785_Answers_Equation_6}

References

AP^® Physics 1: Algebra-Based and Physics 2: Algebra-Based Curriculum Framework; The College Board: New York, NY, 2014.

Introduction

Can 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

• Centripetal force

• Centripetal acceleration
• Velocity
• Newton’s laws of motion

Background

Centripetal 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.

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. where

F is the force (N)
m is the mass (kg)
a is the acceleration (m/s²)

The formula for centripetal acceleration (Equation 2) can be derived from vector analysis of the forces acting on an object as it travels in a circle. where

a is the centripetal acceleration
v_t is the tangent velocity of the object
r is the radius of the circular path of the object

According to Equation 2 and Figure 1, an object whirled around should accelerate toward the center of the circle, because the centripetal force is pulling it inward. But why does it not get pulled right into the center of the circle? The reason for this is the object’s tangent velocity, which prevents it from being pulled into the center.

If we substitute Equation 2 into Equation 1, centripetal force (F_c) can be expressed as 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. 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. where

v_t is the tangent velocity (m/s)
r is the radius of the circular path (m)
T is the period—time for one revolution (s)

Experiment Overview

The 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 guided-inquiry design of experiments to determine what factors affect the centripetal acceleration of an object in circular motion. Sources of experimental error will be identified.

Materials

Prelab Questions

1. Consider an object moving in a circle at a constant speed.
   1. Are all the forces acting on the object balanced?
   2. Explain why or why not in terms of Newton’s laws.
2. Figure 2 shows an overhead view of a ball traveling in a circle in a clockwise direction.
   1. At point A, draw an arrow indicating the direction of the tangent velocity of the ball.
   2. At point B, draw the direction of the centripetal force on the ball.
   3. At point C, draw the direction of the centripetal acceleration of the ball.
   4. If the string were to break as the ball reached point D, draw an arrow from point D to indicate the path the ball would take.
      {13785_PreLab_Figure_2}
3. Read the entire Background and Procedure for the Introductory Activity. In Figures 1 and 3, the hand holding the string causes tension in the string, which exerts a force on the rubber stopper. What causes the tension in the string in Figure 5?

Safety Precautions

The 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.

Procedure

Introductory Activity

Read the entire Procedure before beginning. Construct an appropriate data table to record measurements and the results of calculations.

Part A. Feeling the Force

1. Use a balance to measure the mass of the rubber stopper (m₁). Record the mass in kilograms on the data table.
2. Thread the string through one hole in the rubber stopper and then back through the other hole. Tie the stopper securely to the end of the string. Tie a few knots to make sure the stopper is secure.
3. Thread the free end of the string through the handle tube. Measure 0.75 m of string from the center of the rubber stopper to the handle tube. See Figure 3 for the basic setup.
   {13785_Procedure_Figure_3}
4. Hold the bottom of the free end of the string firmly in one hand and the handle tube in the other hand (see Figure 3). Caution: Be sure you are in an open area clear of people and any breakable items. Everyone in the area should be wearing safety glasses.
5. Twirl the rubber stopper slowly in a horizontal circle over your head and gradually increase the speed of the rubber stopper. Note what happens to the pull on the string as the speed of the stopper increases. Be sure to hold on tight to the bottom string.

Part B. Measuring the Force

6. Tie a loop in the free end of the string about 30 cm below the handle tube.
7. Measure and record the mass of six washers plus the paper clip and alligator clip.
8. Return the string to its original 0.75 m length between the center of the stopper and the tube. Attach an alligator clip onto the string just below the handle tube (see Figure 4). This is the marker to ensure a constant radius.
   {13785_Procedure_Figure_4}
9. Slip the loop through the center of the six washers and hold the washers in place by inserting a bent paper clip through the loop as shown in Figure 5.
   {13785_Procedure_Figure_5}
10. Holding the tube in one hand, slowly begin to twirl the rubber stopper horizontally overhead. Increase the speed of rotation until the marker alligator clip is just below the bottom of the handle tube but not touching the handle. This is to ensure that the radius of the stopper’s circular path will be 0.75 m.
11. Twirl the stopper evenly so that the alligator clip remains stationary just below the bottom of the handle tube, but not touching the handle tube.
12. With a partner, time how long it takes for the stopper to complete 20 revolutions.
13. Repeat steps 11–13 two more times.
14. Increase the number of washers to 12 and repeat steps 8–13.
15. Repeat steps 8–13 with 18 washers.

Analyze the Results

• Calculate the period, T, for each set of washers using the average time to complete 20 revolutions.
• Calculate the tangent velocity, v_t, of the stopper for each set of washers.
• Calculate the centripetal force on the stopper for each set of washers.
• The washers combined with the alligator clip and paper clip pull down on the string with their combined mass (m₂). Therefore the tension in the string must be equal to the weight of m₂. This can be written as 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 (x-axis) versus the theoretical centripetal force in newtons (y-axis). Explain the shape of each graph.

Guided-Inquiry Design and Procedure

Form a working group with other students and discuss the following questions.

1. Review the centripetal force procedure. Calculate the percent error between the theoretical tension in the string, F_T, and the experimental centripetal force, F_c for each set of washers.
   {13785_Procedure_Equation_7}
2. Identify random sources of error in the experiment.
   1. Which measurement has the greatest potential for error?
   2. What are some ways this error might be reduced?
3. Identify any sources of systematic error in this experiment.
4. The independent variable in an experiment is the variable that is changed by the experimenter, while the dependent variable responds to or depends on the changes in the independent variable.
   1. Name the independent and dependent variables in Part B of the Introductory Activity.
   2. What other variables could be tested?
5. Plan, discuss, execute and evaluate an experiment to test one of the independent variables identified above.
   1. Write the question the experiment is designed to explore and predict the outcome of the experiment.
   2. Discuss and design a controlled procedure to test the prediction. Which variables will remain constant? What will provide the tension in the string?
   3. List any safety concerns and the precautions that will be implemented to keep yourself, your classmates, and your instructor safe during the experimental phase of this laboratory.
   4. Determine what data you will collect and how it will be recorded.
   5. How will you analyze the data?

Student Worksheet PDF

13785_Student1.pdf