# Centripetal Force

## Student Activity Kit

### Materials Included In Kit

Handle tube
Paper clips, 2
Rubber stopper, two-hole
String, 1.5 m
Washers, 20

(for each lab group)
Balance, 0.1-g precision
Graph paper
Meter stick
Stopwatch or clock with second hand

### Safety Precautions

This lab is best conducted outdoors, in an open gymnasium or other large open area. All students should wear safety glasses whenever anyone is conducting the experiment. You might check each experimental setup before the twirling begins. Have students spread out as much as possible. Please follow all laboratory safety guidelines.

### Disposal

All materials in the kit may be reused many times.

### Teacher Tips

• Be very alert about safety during this lab. If students spin the apparatus too violently, projectiles can fly around unexpectedly. Try to find an open area where students can spread out and thus decrease the chance of flying objects hitting anyone or anything. Be sure that students spin the rubber stoppers and not the paper clips or washers!
• The amount of theory to cover relative to centripetal force will be dictated by your students and course goals. Most physics texts describe a derivation of the formula provided in the student background. The derivation involves vector solutions. Use your judgment about how much of the derivation should be discussed. Please consult a physics textbook for a more thorough explanation.

### Science & Engineering Practices

Developing and using models
Analyzing and interpreting data
Planning and carrying out investigations
Using mathematics and computational thinking

### Disciplinary Core Ideas

MS-PS2.A: Forces and Motion
MS-PS2.B: Types of Interactions
MS-PS3.B: Conservation of Energy and Energy Transfer
MS-PS3.C: Relationship between Energy and Forces
HS-PS2.A: Forces and Motion
HS-PS3.A: Definitions of Energy

### Crosscutting Concepts

Patterns
Cause and effect
Systems and system models
Scale, proportion, and quantity
Energy and matter

### Performance Expectations

MS-PS2-1: Apply Newton’s Third Law to design a solution to a problem involving the motion of two colliding objects.
MS-PS2-2: Plan an investigation to provide evidence that the change in an object’s motion depends on the sum of the forces on the object and the mass of the object
MS-PS2-4: Construct and present arguments using evidence to support the claim that gravitational interactions are attractive and depend on the masses of interacting objects
MS-PS3-1: Construct and interpret graphical displays of data to describe the relationships of kinetic energy to the mass of an object and to the speed of an object.
HS-PS2-1: 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.
HS-PS2-4: Use mathematical representations of Newton’s Law of Gravitation and Coulomb’s Law to describe and predict the gravitational and electrostatic forces between objects.

### Sample Data

Part II. Measuring the Force

{12267_Data_Table_1}

Part I. Feeling the Force

1. What happened to the force on the string as the speed of the revolving stopper increased?

The force increased as the speed increased.

2. Predict what would happen if you were to let go of the string. (Do not actually do this.)

The centripetal force would be removed and the stopper would travel away from the tube in a straight line following the path of the tangent velocity.

Part II. Measuring the Force
1. In this activity, the relationship between centripetal force and velocity is being tested. What variables must be held constant in order to do this? Hint: See Equation 3 in the Background section.

The radius of the object’s circular path and the object’s mass must be held constant.

2. Calculate the velocity, v, of the rubber stopper using Equation 5 from the Background section. Record your answers in the the data table.
3. Calculate the centripetal force, F, for each trial using Equation 3 in the Background section. Show key calculations, include units of kg•m/sec2 or newtons (N). Record your answers in the data table.
4. State the relationship between the velocity of the rubber stopper and the centripetal force.

As the centripetal force increases, the velocity increases. This is because when centripetal force increases, the rubber stopper is being pulled to the center with a greater amount of force. In order to prevent it from being pulled in by this force, a faster tangential velocity is needed to keep the radius at one meter. If the speed was not increased, the rubber stopper would be pulled toward the center decreasing the radius of its circular path.

5. On a piece of graph paper, plot the calculated centripetal force versus the number of washers for trials 1–3.
6. The washers are actually creating the centripetal force needed to make the stopper move in a circle. Using the information from the graph, state the relationship between the number of washers used and the newtons of centripetal force applied to the spinning stopper.

As the number of washers was increased, the force in newtons increased.

# Centripetal Force

### Introduction

Have you ever been on a human centrifuge at an amusement park? You can certainly feel the tremendous force being imposed upon your body as you travel in a circular fashion—so much force that when the floor drops away, you are pinned against the wall!

### 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 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.

{12267_Background_Figure_1}
The mathematical expression for centripetal force is the same as for any other force, based on Newton’s second law of motion:
{12267_Background_Equation_1}
where

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

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. The formula for centripetal acceleration can be derived from vector analysis of the forces on a circle:
{12267_Background_Equation_2}
where

a is the centripetal acceleration
v is the velocity of the object (tangent velocity)
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 due to the tangent velocity of the object. The tangent velocity of the object actually prevents it from being pulled into the center. For more information on the relationship between tangent velocity and centripetal acceleration, consult a physics textbook.

Now if we substitute Equation 2 into Equation 1, centripetal force can be expressed as Equation 3.
{12267_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 (v) of an object can be used to determine the velocity.
{12267_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.
{12267_Background_Equation_5}
where

v is the 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 lab activity is to determine the relationship between the velocity and centripetal force of an object moving in a circle.

### Materials

Balance, 0.1-g precision
Graph paper
Handle tube
Meter stick
Paper clips, 2
Rubber stopper, two-hole
Stopwatch or clock with second hand
String, 1.5 m
Washers, 20

### 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 your group or anyone else is conducting the experiment in the area. Please follow all laboratory safety guidelines.

### Procedure

Part I. Feeling the Force

1. Use a balance to measure the mass of the rubber stopper. Record the mass in kilograms on the data table in Part II of the Centripetal Force Worksheet. Note: 1 g = 0.001 kg
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. Leave about one meter of string between the handle and the rubber stopper. See Figure 2 for the basic setup.
{12267_Procedure_Figure_2}
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 2). Caution: Be sure you are in an open area clear of people and any breakable items.
5. Twirl the rubber stopper slowly in a horizontal circle over your head and gradually increase the speed of the rubber stopper. Feel the pull on the string as the speed of the stopper increases. Be sure to hold on tight to the bottom string.
6. Answer the questions for Part I on the Centripetal Force Worksheet.
Part II. Measuring the Force
1. Return the string to its original one-meter length between the stopper and the tube. Slip a paper clip over the string just below the handle tube (see Figure 3). This is the marker paper clip.
{12267_Procedure_Figure_3_Basic setup for centripetal force measurements}
2. Tie a loop in the free end of the string about 12 inches below the handle tube.
3. Slip the loop through the center of six washers and hold the washers in place by inserting a bent paper clip through the loop as shown in Figure 4.
{12267_Procedure_Figure_4}
4. Holding the tube in one hand, slowly begin to twirl the rubber stopper overhead. Increase the speed of rotation until the marker paper 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 one meter.
5. Twirl the stopper evenly so that the marker paper clip remains stationary just below the bottom of the handle tube, but not touching the handle tube.
6. With a partner, count the number of revolutions the rubber stopper makes in a 20-second period. Record the data in the table for Part II on the Centripetal Force Worksheet.
7. Increase the number of washers to 12 and repeat steps 11 and 12. Make sure the marker paper clip is still in the proper location. It should be just under the handle tube when the radius is one meter in length.
8. Repeat steps 11 and 12 using 18 washers.
9. For each trial, calculate the time needed to complete one revolution. Record the result in the data table under Period.
10. Answer the questions in Part II on the Centripetal Force Worksheet.

### Student Worksheet PDF

12267_Student1.pdf

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