Materials Included In Kit

Additional Materials Required

Safety Precautions

The binder clips may slip off the rubber band ends if too much weight is placed at the end. This could cause the rubber band to snap back toward the support clamp. Remind students not to stand directly over the stretched rubber band or place anything under the hanging mass (including hands and feet). Students should wear safety glasses when performing this experiment. Also, instruct students to use caution when hanging the masses on the binder clips. Students should make sure the binder clips have a secure grip on the rubber band before releasing the mass.

Disposal

All materials may be saved and stored for future use.

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.
• Hooked masses are recommended owing to their ease of use and premarked mass (Flinn Catalog No. OB2066). Slotted masses and slotted-mass hangers may also be used. In fact, any kind of mass can be used as the load for this experiment, as long as it is not too heavy for the rubber band or spring. If the mass is not marked, make sure students measure the mass of the object using a balance and record this in the data table.

Teacher Tips

• The guided-inquiry design and procedure section was developed to subtly guide students through the experimental design process. You may decide to provide less/more information than is included. So long as the students are challenged to think critically.
• Rubber bands follow a nonlinear force–stretch relationship, different from the linear relationship Hooke’s law predicts for springs. This is the result of the many long-chain polymers that must uncoil in the elastic material, and this typically is not uniform. The metallic bonds in a spring do not actually stretch or uncoil, but the shape of the material changes, which causes the spring to increase in length, and then springs back due to the rigidity of the metal structure and metallic bonds. Elastic materials, such as rubber, are also affected more by temperature than metal springs.
• The period of the oscillations is how long it takes for one complete oscillation. The period (T) is therefore the inverse of the frequency (T = 1/f). So, the period of an oscillating spring in simple harmonic motion is equal to:
  {13789_Tips_Equation_3}

  Equation 3 shows that reducing the mass of the hanging weight will cause the period of the oscillations to decrease as well. The shorter period of oscillations will result in a greater frequency (the spring will oscillate very quickly).

Further Extensions

Opportunities for Inquiry

Following the determination of the spring constant via simple harmonic motion, use the hanging-mass method to determine the masses of conventional items (e.g., car keys, mittens). “Double-up” the rubber band or springs and determine the spring constant for a linear combination of rubber bands and/or springs: how does the combination spring constant compare to the original spring constants? Measure the spring constant of an uncut rubber band. Design a bungee-jumping experiment where knowledge of k is necessary to design an apparatus that will not result in harm to the bungee jumper. Finally, students can qualitatively rank springs or rubber bands and then compare qualitative rankings to rankings generated from empirical investigations.

Alignment to the Curriculum Framework for AP^® Physics 1

Enduring Understandings and Essential Knowledge
Classically, the acceleration of an object interacting with other objects can be predicted by using a = ΣF/m. (3B)
3B2: Free-body diagrams are useful tools for visualizing forces being exerted on a single object and writing the equations that represent a physical situation.
3B3: Restoring forces can result in oscillatory motion. When a linear restoring force is exerted on an object displaced from an equilibrium position, the object will undergo a special type of motion called simple harmonic motion. Examples should include gravitational force exerted by the Earth on a simple pendulum, mass-spring oscillator.

Learning Objectives
3B2.1: The student is able to create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively.
3B3.1: The student is able to predict which properties determine the motion of a simple harmonic oscillator and what the dependence of the motion is on those properties.
3B3.2: The student is able to design a plan and collect data in order to ascertain the characteristics of the motion of a system undergoing oscillatory motion caused by a restoring force.
3B3.3: The student can analyze data to identify qualitative or quantitative relationships between given values and variables (i.e., force, displacement, acceleration, velocity, period of motion, frequency, spring constant, string length, mass) associated with objects in oscillatory motion to use that data to determine the value of an unknown.
3B3.4: The student is able to construct a qualitative and/or a quantitative explanation of oscillatory behavior given evidence of a restoring force.

Science Practices
1.4 The student can use representations and models to analyze situations or solve problems qualitatively and quantitatively.
3.1 The student can pose scientific questions.
3.2 The student can refine scientific questions.
3.3 The student can evaluate scientific questions.
4.1 The student can justify the selection of the kind of data needed to answer a particular scientific question.
4.2 The student can design a plan for collecting data to answer a particular scientific question.
4.3 The student can collect data to answer a particular scientific question.
4.4 The student can evaluate sources of data to answer a particular scientific question.
5.1 The student can analyze data to identify patterns or relationships.
5.2 The student can refine observations and measurements based on data analysis.
5.3 The student can evaluate the evidence provided by data sets in relation to a particular scientific question.
6.1 The student can justify claims with evidence.
6.2 The student can construct explanations of phenomena based on evidence produced through scientific practices.
6.4 The student can make claims and predictions about natural phenomena based on scientific theories and models.

Answers to Prelab Questions

1. Consider a puck (mass = 0.40 kg) that moves over a frictionless surface in one-dimensional motion along an x-axis. A force of 3.3 N, directed along the x-axis, is exerted on the puck. What is the acceleration of the puck?

   By Newton’s second law, F = ma.
   3.3 N = 0.40 kg x a
   a = 3.3 N/0.40 kg = 8.25 N/kg = 8.25 m/s².

2. Consider the same puck hanging vertically from a rubber bungee cord 1.24 m above the ground. What is the magnitude of the force exerted on the puck in the downward direction?

   In this scenario, F = ma becomes F = mg. F = 0.40 kg x 9.8 m/s² = 3.92 N x m.

3. The puck is pulled from its stationary state and the bungee cord stretches approximately 4 cm along the y-axis. The puck is then released and the bungee cord/puck system exhibits simple harmonic motion. In a 20-second period, the system moves up and down a total of 18 times. Calculate the spring constant, k, of the bungee cord.

   k = 4π²f²m

   {13789_PreLab_Equation_1}
4. An Olympic biathlete shoots several rounds of ammunition at a target and, though all rounds miss the bull’s-eye they strike the target in very close proximity to each other. Apply the ideas of accuracy and precision to this example.

   Accuracy is determined by how close a measurement comes to the true value. In this case, an accurate shot is one that hits near the bull’s-eye. Precision is determined by looking at how repeatable the measurements are. A precise Olympic biathelete would be able to hit the target in a small grouping. The Olympic biathelete’s shots were in a small group but not near the bull’s-eye. The athlete is precise, but inaccurate.

Sample Data

Introductory Activity

• Which material more strictly obeyed Hooke’s law? Justify your answer.

  A graph of F versus x for the rubber band is noticeably nonlinear. The spring constant for the rubber band changes as its stretched distance changes. At small stretch distances, the spring constant is large. As the rubber band stretches, the spring constant decreases. In contrast, the graph of F versus x for the spring is linear with a constant slope. The slope represents the proportionality between F and x, which is the spring constant, k.

• It is not possible to compare the experimental spring constants to known reference values because no such values exist. Explain how Newton’s second law can be used to assess accuracy and precision of the spring constants given the mass was stationary at the end of the material.

  When the mass was not moving at the end of the spring and rubber band, there was no net acceleration. The force of gravity was acting down on the mass, while the force of the spring was acting up on the mass. These forces were equal and opposite, resulting in a net force equal to zero. By setting the equations for the force of the spring and the force of gravity equal to each other, the value of g, the acceleration due to gravity, can be calculated. The accepted value of g = 9.81 m/s².

  F_net = 0 = F_spring – F_gravity
  F_spring = F_gravity
  –kx = mg

  For Spring Trial 1:

  –6.1 N/m x 0.032 m = 0.020 kg x g
  g = 9.76 m/s²

  The spring constant values calculated for the rubber band are fairly accurate, but not precise. The average value of g calculated from the spring constants is 9.78 m/s², which is pretty close to the accepted value, 9.81 m/s², indicating accuracy. However, the measurements were not precise. The three values of g covered a large range (9.70 to 9.84), showing the rubber band is not a reliable material to determine the value of g.
  
  The spring constant values calculated for the spring are fairly accurate and highly precise. The average value of g calculated from the spring constants is 9.78 m/s², which is close to the accepted value. The three values of g were also close in value, indicating the spring method allowed for repeatable measurements.

Answers to Questions

Answers to Guided-Inquiry Discussion Questions

1. Was it easier to induce simple harmonic motion in the rubber band or the spring? Does one material seem more difficult to stretch than the other the rubber band or the spring? Qualitatively assess the spring constant of each material. How does your qualitative assessment compare to your quantitative determination of k via simple harmonic motion?

   Student answers to these questions will vary. This first set of questions is designed to promote a level of critical thinking in students. They should manipulate the rubber bands and springs and qualitatively associate stiffness with spring constants. Some groups may not find it difficult to induce simple harmonic motion in either the rubber band or spring, whereas some groups may note minor trouble promoting simple harmonic motion in the rubber band. We noted minor difficulty getting the rubber band to exhibit simple harmonic motion in our trials.

2. Are the materials’ spring constants truly constant or do the materials become more difficult/less difficult to stretch as their lengths increase? Manipulate the rubber band and spring and note any variability in their spring constants.

   Students may note that the rubber band becomes decreasingly difficult to stretch as the stretch length increases. If students do not qualitatively observe this disobedience to Hooke’s law, they will note it quantitatively as they progress through the guided-inquiry process. In contrast, students will likely note the spring does not become less difficult/more difficult to stretch as the stretch length increases. There is a notable smoothness in stretching the spring. Again, part of the idea here is to encourage students to thoughtfully manipulate the materials.

3. Quantitative confirmation of a material’s obedience or disobedience to Hooke’s law occurs when F = –kx for the material. Although it is possible to derive a material’s spring constant from a single experiment’s data, or one pair of F/x values, why is this approach not prudent?

   The information provided in steps 2 and 3 is meant to intimate that any determination as to whether a material obeys Hooke’s Law must rely on data gathered over a wide range of stretched distances because some materials have variable spring constants (i.e., they do not obey Hooke’s law). If the stretching forces, and consequent stretched distances, are not adequately varied, the variability in the spring constant of a material (one that does not obey Hooke’s law) may not present. That is, if x is varied multiple times over a narrow range, the spring constant may appear constant.

4. Newton’s second law permits measurable application of a stretching force(s). Application of a stretching force results in a difference between a material’s nonstretched (relaxed) and stretched lengths. The difference is given by x. Explain how to measure F and x.

   Newton’s second law is given by the mathematical equation: F = ma. Because the only force acting on a spring occurs in the y direction, F = –kx becomes ma = –kx, or mg = –kx. The mass of the hanging mass can be systematically varied and the x measured simply with a ruler. Because “g” is a constant value, the spring constant k can be determined with these two values.

5. When determining the spring constant of a material, is it best to use a wide range of masses (and stretching forces) or a narrow range? For example, would it be better to use 5 masses ranging from 50 to 60 g, separated by 2-g increments, or 5 masses ranging from 50 to 250 kg, separated by 50-g increments? Justify your answer. Revisit your answers to the questions posed in steps 2 and 3 for guidance.

   Drawing on information given in steps 2 and 3, students should opt to use a wide range of masses, and thus x values, to assess obedience or disobedience to Hooke’s law.

6. How can you present coupled data, in which manipulation of one variable causes a change in another, in a succinct and illustrative form?

   A graph, or (x,y) scatter plot, is a great way to describe the relationship, mathematically expressed as direct, between F and x.

7. Write detailed, step-by-step procedures for quantitatively determining the spring constants of the rubber band and spring. Will the materials require different ranges of stretching forces?

   Affix masses of variable value over a wide range (e.g., 50-g increments from 50 to 250 g) to the hanging spring/rubber band so that a large range of x values is observed. Newton’s second law allows for calculation of the stretching force and x can be measured with a ruler and taken as the difference between the materials relaxed and stretched lengths. Students should take care to measure the stretched distance from the same well-defined beginning and ending points on the rubber band and spring so as to minimize experimental error.

Answers to the Review Questions for AP^® Physics 1

1. How might you determine the mass of a common object, such as a set of car keys or a wool cap, using the ideas presented herein?

   First determine the spring constant for the spring via the hanging-mass method. Once you have determined the spring constant and verified its invariability, attach an item (e.g., a set of car keys) to the hanging end. Measure x, the difference between the stretched and nonstretched spring lengths. F = –kx may be rewritten as ma = –kx. Because a is a constant, k has been determined experimentally, and x is easily measurable, the equation may be rearranged to solve for m: m = kx/a.

2. 50-g and 300-g masses were used to induce simple harmonic motion in the spring and rubber band, respectively. How would increasing these masses impact the oscillating frequencies of the rubber band and spring?

   Based on the equation, k = 4π²f²m, m and f are inversely proportional and so as m increases, the oscillating frequency will decrease.

3. Identify two sources of error that may arise in the experimental determination of spring constants and explain how the sources of error may affect results.

   Most students neglect to consider the mass of the rubber band or spring. In addition, it is not easy to count the number of oscillations as this is done visually and the oscillating frequency can be quite high. As the frequency increases, the difficulty associated with counting oscillations would also increase.

4. Given your answers to Questions 2 and 3, how might you mitigate the error associated with counting the number of oscillations exhibited by the spring or rubber band.

   Use a heavier mass to slow down the oscillations, thereby making them easier to count.

5. Why is there a discrepancy between the rubber band spring constants derived via the simple harmonic motion and the hanging-mass methods?

   The discrepancy arises owing to the fact that the spring constant for the rubber band is not actually a constant. Rather, the spring constant varies as the stretched length increases. This question is meant to induce such a realization—which has experimental-design implications—more than it is intended to probe the concept of Hooke’s law.

6. Estimate the spring constant for the spring in a car’s shock absorber.

   The spring component in an automobile shock absorber must be very rigid (compared to the materials investigated in this laboratory) owing to an automobile’s significant mass. The highly rigid spring would have a very high spring constant.

References

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

Introduction

How do the stretch properties of a rubber band compare to that of a spring? Does one material stretch more evenly than another? Simple stretching and compression are enough to approximate the relative rigidities, or spring constants, of rubber bands and springs. Spring constants can also be determined by setting a mass at the end of an elastic material in a repetitive, up-and-down motion. Investigate the spring constants of rubber bands and springs by studying the concepts of simple harmonic motion and Hooke’s law.

Concepts

• Hooke’s law
• Forces
• Simple harmonic motion
• Accuracy and precision
• Newton’s second law

Background

A material that returns to its original dimensions following removal of an applied force is considered elastic. Elastic materials are very common. Automobile shock absorbers, mechanical pens and pencils, clothing and sporting equipment often rely on springs or rubber band–type materials to perform properly. Though qualitative differentiation among elastic materials can be straightforward (the difference between the force required to stretch a small rubber band and a more rigid bungee cord is easily measured) quantitative differentiation is often necessary for applications-based uses. For example, bungee cord manufacturers must derive empirical values to describe cords’ rigidities to provide jumpers thrilling yet safe jumps (i.e., ones that provide significant stretch and recoil distances but avoid contact with the ground).

The spring constant is a quantitative description of an elastic material’s rigidity. Materials with low spring constants are easy to deform, or stretch, whereas materials with high spring constants are more rigid in accordance with Hooke’s law:

F is the spring force, or force needed to stretch or compress a spring or elastic material.
k is the proportionality, or spring constant, reported in newtons per meter (N/m).
x is the amount of deformation, or difference between a material’s stretched and nonstretched lengths.

The spring constant also influences the oscillating properties of a spring. A vertically hanging spring that is pulled and then released will exhibit a periodic, or repetitive, up-and-down motion. Periodic motion is better known as simple harmonic motion. The frequency of the oscillating spring exhibiting simple harmonic motion is based on the spring constant of the spring, and the hanging mass at the loose end of the spring, and is represented by Equation 2,

where f is the oscillating frequency, k is the spring constant, and m is the mass of hanging weight. Frequency is a measure of how many oscillations (up-and-down movements) occur per second. The units for frequency are hertz (Hz), and are equal to 1/s, or s^–1.

Experiment Overview

The purpose of this advanced inquiry investigation is to quantify the relationship between the force needed to stretch a spring and the stretch distance—the difference between the spring’s stretched and non-stretched (relaxed) lengths. An introductory activity is performed to derive the spring constants for a rubber band and spring using simple harmonic motion. An alternative experiment to determine the spring constants and confirm adherence to Hooke’s law will then be designed. Finally, ways to creatively assess the success of these guided-inquiry experiments will be considered.

Materials

Prelab Questions

1. Consider a puck (mass = 0.40 kg) that moves over a frictionless surface in one-dimensional motion along an x-axis. A force of 3.3 N, directed along the x-axis, is exerted on the puck. What is the acceleration of the puck?
2. Consider the same puck hanging vertically from a rubber bungee cord 1.24 m above the ground. What is the magnitude of the force exerted on the puck in the downward direction?
3. The puck is pulled from its stationary state and the bungee cord stretches approximately 4 cm along the y-axis. The puck is then released and the bungee cord/puck system exhibits simple harmonic motion. In a 20-second period, the system moves up and down a total of 18 times. Calculate the spring constant, k, of the bungee cord.
4. An Olympic biathlete shoots several rounds of ammunition at a target and, though all rounds miss the bull’s-eye they strike the target in very close proximity to each other. Apply the ideas of accuracy and precision to this example.

Safety Precautions

The binder clips may slip off the rubber band ends if too much weight is placed at the end. This could cause the rubber band to snap back toward the support clamp. Do not stand directly over the stretched rubber band, or place anything under the hanging mass (including hands and feet). Wear safety glasses when performing this experiment. Use caution when hanging the masses on the binder clips. Make sure the binder clips have a secure grip on the rubber band before releasing.

Procedure

Introductory Activity

Simple Harmonic Motion

1. Obtain a rubber band.
2. Use scissors to cut the rubber band so that it becomes one long rubber strand.
3. Fold a small portion (about 1 cm) of the rubber band’s end over itself (see Figure 1).
   {13789_Procedure_Figure_1}
4. Attach a binder clip to the “doubled-up” section of the rubber band (see Figure 2).
   {13789_Procedure_Figure_2}
5. Repeat steps 3 and 4 on the other end of the rubber band with a second binder clip.
6. Set up a support stand, clamp and S-hook as shown in Figure 3a.
   {13789_Procedure_Figure_3}
7. Hang one of the binder clips from the end of the S-hook. The rubber band should hang straight down.
8. Carefully hang a 300-g mass from the other binder clip. Allow the rubber band to slowly extend until it reaches a constant length.
9. With a timer ready, pull the hanging mass straight down 3 cm and then release the mass. Allow the oscillations to equilibrate for 1–2 seconds before starting the timer. Note: The hanging mass should oscillate smoothly up and down, with very little side-to-side motion.
10. Count the number of up-and-down oscillations that occur in 10 seconds. Note: Down and up (or up and down) is equal to one oscillation.
11. Record the number of oscillations.
12. Repeat steps 9–11 two more times. Record the number of oscillations.
13. Using the spring and a 50-g mass, set up the hanging spring as shown in Figure 3b.
14. Carefully hang the 50-g mass from the bottom loop of the spring. Allow the spring to slowly extend until it reaches a constant length.
15. With a timer ready, pull the hanging mass straight down 3 cm and then release the mass. Allow the oscillations to equilibrate for a few seconds. Note: The hanging mass should oscillate smoothly up and down, with very little side-to-side motion.
16. Count the number of up-and-down oscillations that occur in 10 seconds. Note: Down and up (or up and down) is equal to one oscillation.
17. Record the number of oscillations.
18. Repeat steps 15–16 two more times. Record the number of oscillations.

Analyze the Results

Determine the frequency of oscillations in each trial. Calculate the spring constants for the rubber band and spring based on your data.

Guided-Inquiry Design and Procedure

1. Was it easier to induce simple harmonic motion in the rubber band or the spring? Does one material seem more difficult to stretch than the other? Qualitatively assess the spring constant of each material. How does your qualitative assessment compare to your quantitative determination of k via simple harmonic motion?
2. Are the materials’ spring constants truly constant or do the materials become more difficult/less difficult to stretch as their lengths increase? Manipulate the rubber band and spring and note any variability in their spring constants.
3. Quantitative confirmation of a material’s obedience or disobedience to Hooke’s law occurs when F = –kx for the material. Although it is possible to derive a material’s spring constant from a single experiment’s data, or one pair of F/x values, why is this approach not prudent?
4. Newton’s second law permits measurable application of a stretching force(s). Application of a stretching force results in a difference between a material’s nonstretched (relaxed) and stretched lengths. The difference is given by x. Explain how to measure F and x.
5. When determining the spring constant of a material, is it best to use a wide range of masses (and stretching forces) or a narrow range? For example, would it be better to use 5 masses ranging from 50 to 60 g, separated by 2-g increments, or 5 masses ranging from 50 to 250 kg, separated by 50-g increments? Justify your answer. Revisit your answers to the questions posed in steps 2 and 3 for guidance.
6. How can you present coupled data, in which manipulation of one variable causes a change in another, in a succinct and illustrative form?
7. Write detailed, step-by-step procedures for quantitatively determining the spring constants of the rubber band and spring. Will the materials require different ranges of stretching forces?

Analyze the Results

• Calculate the spring constant for the rubber band and spring for each trial.
• Prepare a data table to clearly and concisely present the collected and calculated data.
• Prepare graphs comparing the stretch distance versus the hanging mass for both the rubber band and spring.
• Which material more strictly obeyed Hooke’s law? Justify your answer.
• It is not possible to compare the experimental spring constants to known reference values because no such values exist. Explain how Newton’s second law can be used to assess accuracy and precision of the spring constants given the mass was stationary at the end of the material.

Student Worksheet PDF

13789_Student1.pdf