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

The materials should be saved and stored for future use.

Lab Hints

• Enough materials are provided in this kit for 24 students working in groups of three or for 8 groups of students. This laboratory can reasonably be completed in one 50-minute class period. All materials are reusable.
• Commercial hooked masses are recommended due to their ease of use and premarked mass. Slotted masses and slotted-mass hangers may also be used. Actually, 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.
• Students can “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? Students can also measure the spring constant of an uncut rubber band.

Teacher Tips

• 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 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 (P) is therefore the inverse of the frequency (P = 1/f). So, the period of an oscillating spring in simple harmonic motion is equal to:
  {12583_Tips_Equation_6}
  Equation 6 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).
• The Bungee-Jumping Egg—Student Laboratory Kit (Catalog No. AP6391) is an excellent supplemental activity to teach Hooke’s law and the conservation of energy principle, and is available through Flinn Scientific, Inc.

Sample Data

Observations

Hooke’s law

The rubber band and spring forces increase as they stretch.

Harmonic motion

As the rubber band oscillates the height of the up-and-down motion (amplitude) decreases. The rubber band is hardly oscillating after 15 seconds. However, the period of each oscillation appears to remain constant. The spring oscillates smoothly and constantly for a long time. It only gradually loses height in the up-and-down motion (amplitude) over time.

Force Data Table Oscillation Data Table

Answers to Questions

1. Calculate the stretch distance for each trial by subtracting the final length by the initial length. Record the results in the Force Data Table.

   Sample calculation: 16.7 cm – 15.5 cm = 1.2 cm

2. Graph the stretch distance versus the hanging mass (force) for both the rubber band and the spring. Which stretchable material appears to follow the linear, straight-line relationship expressed by Hooke’s law? Support your answer.
   {12583_Answers_Figure_4}
   {12583_Answers_Figure_5}
   The spring appears to follow Hooke’s law much better than the rubber band. The graph shows a straight line for the spring, but a curved line for the rubber band.
3. Calculate the magnitude of the spring constant for both the rubber band and spring for each hanging mass. Note: Convert grams to kilograms, centimeters to meters, and use 9.8 m/s² for the acceleration due to gravity constant, g. How “constant” are the spring constants for each material?

   Sample calculation
   (200 g × 1 kg/1000 g × 9.8 m/s²)/(3.7 cm × 1 m/100 cm) = 52.97 N/m
   The “spring constants” for the rubber band are not constant, and actually decreased with increasing mass (force). The spring’s “spring constant” remained constant for each added mass.

4. How did the oscillations of the rubber band compare to the oscillations of the spring?

   The rubber band did not oscillate for very long, and the oscillation heights (the amplitude) decreased (was dampened) very quickly. The spring oscillated uniformly for a long period of time and the oscillation heights gradually decreased.

5. Calculate the average number of oscillations in 10 seconds for both the rubber band and the spring, and then calculate the average frequency of the oscillations by dividing the average number of oscillations by the time.

   Sample calculations
   (16 + 16 + 17)/3 = 16.33 → 16 oscillations (rubber band)
   (17 + 17 + 17)/3 = 17 oscillations (spring)
   16/10 s = 1.6 s^–1 (hz) (rubber band)
   17/10 s = 1.7 s^–1 (hz) (spring)

6. Use the average frequencies calculated in Question 5, and Equation 5 to calculate the spring constant for both the rubber band and the spring. Note: Convert grams to kilograms.

   Sample calculations
   k = 4π²f²m
   k = 4π²(1.6 s^–1)2(0.3 kg) = 30.3 kg/s² → 30 N/m (rubber band)
   k = 4π²(1.7 s^–1)2(0.05 kg) = 5.7 kg/s² → 5.7 N/m (spring)

7. How do “spring constants” of the rubber band and spring compare for both experiments? Explain any discrepancies.

   The rubber band’s “spring constant” calculated using the oscillation experiment was much different from the spring constants calculated in the force experiment. However, it was shown in the first experiment that the rubber band does not follow Hooke’s law, so the discrepancy is the result of the rubber band not being modeled well by Hooke’s law. The second experiment also revealed that Hooke’s law does not represent the elasticity of a rubber band very well. The “spring constants” for the spring were relatively close for both experiments. The spring is modeled by Hooke’s law very well.

8. True or False
   1. The force exerted by any stretchable material remains constant as the material is stretched.

      (False; The force exerted by a stretchable material increases as the material is stretched.)

   2. The spring constant unit, N/m, can also be represented as kg/s².

      (True; kg•m/s²•m = N/m)

9. Use Equation 4 from the Background section to predict how increasing the hanging mass would effect the frequency of an oscillating spring. Write your prediction as an “If/then” statement.

   If the hanging mass at the end of a spring increases, then the oscillating frequency will decrease.

Advanced Postlab Question

10. Explain why one stretchable material would show linear stretching properties, following Hooke’s law, better than another.

    The spring shows better linear stretching properties that follow Hooke’s law better than the rubber band because the stretchiness of the spring is based on the properties of the coils, and not entirely on the composition of the metal. Stiffer and softer metals would affect the stretchiness of the spring, but the properties would be more uniform throughout the spring. The elasticity of the rubber band depends on the many long-chain polymers that need to uncoil in the rubber. These properties most likely will not be uniform throughout the rubber band, or be uniform for different stretch lengths.

Introduction

How do the stretch properties of a rubber band compare to that of a spring? Does one material stretch more evenly than another? Explore the stretchiness of a rubber band and a spring, and compare them to the model developed by the physicist Robert Hooke (1635–1703).

Concepts

• Hooke’s law
• Forces
• Simple harmonic motion

Background

The force produced by a stretched spring is directly proportional to the distance the spring is stretched compared to its unstretched state, expressed mathematically in Equation 1. In graphical terms, the relationship between force and stretch distance will show a straight line. This connection between spring force and stretch length was discovered by Robert Hooke, and therefore this principle is better known as Hooke’s law. The negative sign in Equation 1 signifies that the force produced by a spring is a restoring force. In other words, the force wants to bring the spring back to its unstretched, or equilibrium state.

F = force produced by a spring
k = spring constant
x = stretch distance (the difference between the stretched and unstretched length of the spring)

The spring constant for a spring can be calculated by rearranging Equation 1, and using only the magnitude (absolute value) of the value of F. The spring constant, k, is a constant physical property of a spring that is based on properties such as the stiffness of the material used to make the spring, the number of coils, and the length of the spring. The units of the spring constant are newtons per meter (N/m).

By hanging a mass with a known value from the end of the spring and measuring the total length of the stretched spring, the spring constant of the spring can be calculated (Equation 3). Where m_u is equal to the mass value, g is the acceleration of gravity constant (9.8 m/s²), and x_u is the stretch distance of the spring as a result of the hanging mass, m_u. Remember that the stretch distance of the spring is the stretched length minus the unstretched length.

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 4.

f = oscillating frequency
k = spring constant
m = mass of hanging weight

The spring constant, k, can be determined from the frequency by rearranging Equation 4. 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

Determine the “spring constant” of a rubber band and a spring using a force experiment and an oscillation experiment. Compare the spring constants to the Hooke’s law model to determine which material has more consistent stretching properties.

Materials

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

Hooke’s Law

1. Obtain a thick rubber band.
2. Use scissors to cut the rubber band so that it becomes one long rubber strand.
3. Stretch the rubber band by the ends to feel the force exerted by the rubber band. Does the force feel the same throughout the stretch? Record observations in the Hooke’s Law Worksheet.
4. Fold a small portion (about 1 cm) of the rubber band’s end over itself (see Figure 1).
   {12583_Procedure_Figure_1}
5. Attach a binder clip to the “doubled-up” section of the rubber band (see Figure 2).
   {12583_Procedure_Figure_2}
6. Obtain a second binder clip and repeat steps 4 and 5 on the other end of the rubber band.
7. Set up a support stand, clamp and S-hook as shown in Figure 3.
   {12583_Procedure_Figure_3}
8. Hang one of the binder clips attached to the rubber band from the end of the S-hook. The rubber band should hang straight down.
9. Use a ruler to measure the initial (hanging) length of the rubber band. Measure the distance between the pinching edges of the two binder clips. It may be necessary to pull the rubber band taut, without stretching it, to measure the unstretched length accurately. Record the initial rubber band length in the Force Data Table.
10. Carefully hang a 100-g mass from the hanging binder clip. Allow the rubber band to slowly extend until it reaches a constant (equilibrium) length. Note: Do not allow the mass to hang on the rubber band for more than 30 seconds, it may weaken the rubber band. Be prepared to measure the length of the rubber band before hanging the mass.
11. Use a ruler to measure the extended length of the rubber band. Measure the distance between the pinching edges of the two binder clips. Record the hanging mass and the stretched rubber band length in the Force Data Table.
12. Remove the 100-g mass.
13. Repeat steps 10–12 twice more using a 200-g mass and a 300-g mass. Record the hanging mass and corresponding stretched rubber band length measurement in the Force Data Table.
14. Obtain the spring and hang the spring on the end of the S-hook by one of the loops at the end of the spring.
15. Measure the length of the unstretched spring with a ruler. Note: Measure from the top of the support loop to the bottom of the end loop of the spring. Record the length in the Force Data Table.
16. Carefully hang a 20-g mass from the bottom loop of the spring. Allow the spring to slowly extend until it reaches a constant length. Note: Do not allow the mass to hang on the spring for more than 30 seconds, it may weaken the spring. Be prepared to measure the length of the spring before hanging the mass.
17. Use a ruler to measure the extended length of the spring. Measure from the top of the support loop to the bottom of the end loop of the spring. Record the hanging mass and the stretched spring length in the Force Data Table.
18. Remove the 20-g mass.
19. Repeat steps 16–18 twice more using a 50-g mass and a 100-g mass. Record the mass and corresponding stretched spring length measurement in the Force Data Table.
20. Answer the Calculations and Post-Lab Questions 1–4 on the Hooke’s Law Worksheet.

Simple Harmonic Motion

21. Obtain the rubber band, two binder clips and 300-g mass and set up the hanging rubber band as described in steps 4–8.
22. Carefully hang a 300-g mass from the hanging binder clip. Allow the rubber band to slowly extend until it reaches a constant length.
23. With a timer with a second hand ready, pull the hanging mass straight down 2–3 centimeters 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.
24. 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.
25. Record the number of oscillations in the Oscillations Data Table. What happens to the rubber band oscillations during the time measurement? Record your observations in the worksheet.
26. Repeat steps 23–25 two more times. Record the number of oscillations in the Oscillations Data Table.
27. Obtain the spring and a 50-g mass and set up the hanging spring as described in step 14.
28. 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.
29. With a timer with a second hand ready, pull the hanging mass straight down 2–3 centimeters 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.
30. 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.
31. Record the number of oscillations in the Oscillations Data Table. What happens to the spring oscillations during the time measurement? Record your observations in the worksheet.
32. Repeat steps 29–31 two more times. Record the number of oscillations in the Oscillations Data Table.
33. Consult your instructor for appropriate storage procedures.

Student Worksheet PDF

12583_Student1.pdf