Spooling Around with Physics

Introduction

Demonstrate the unusual motion of a spool pulled by a string. Will the spool roll away? Will it roll towards you? Maybe it will not roll at all. A great demonstration to discuss torque, free-body diagrams, and net forces.

Concepts

• Torque
• Free-body diagrams
• Frictional force
• Discrepant events

Materials

Safety Precautions

Although this activity is considered nonhazardous, please follow normal laboratory safety guidelines.

Disposal

The spools and string may be saved for future use.

Prelab Preparation

1. Cut two 50-cm lengths of string.
2. Tie an end of one string to the center of the axle of one spool. Tie an end of the second string to the center of the axle of the second spool (see Figure 1).
   {13971_Preparation_Figure_1}
3. Place a small piece of transparent tape on the tied ends of the strings to secure the ends to the axles and prevent them from slipping.
4. Tightly wrap the string around the center of each spool until there is approximately 15–20 cm of unwrapped string remaining.
5. After winding the string, pull on the loose end to make sure the string does not slip or unravel. If the string slips, unwrap it, place another piece of tape on the tied end of the string to secure it to the axle and prevent it from slipping. Then, rewind the string around the axle.

Procedure

Simple Discrepent Event

1. Place the large spool on the tabletop so that the free end of the string extends from the bottom of the axle (see Figure 2).
   {13971_Procedure_Figure_2}
2. Ask students to predict the direction the spool will roll when the string is pulled.
3. If students predict the spool will roll away from your hand, slowly pull on the string with a small angle with respect to the tabletop. (Watch the spool roll towards your hand.)
4. If students predict that the spool with roll towards your hand, slowly pull the string with a larger angle. (The spool will roll away from your hand.) With practice, the difference between the “large” and “small” angle will be small and hardly noticeable to the students.
5. Discuss the results with the students.

Physics of a Pulled Spool (Optional)

6. Use a ruler or calipers to measure the outside diameters of the axle and flange of both spools.
7. From the measured diameters, use Equation 3 on page 3 to calculate the predicted angle at which the spools will slide without rolling.
8. Carefully and slowly pull on the string and adjust the angle of the string until the spool begins to slide without rolling.
9. Have a student use a protractor to measure the angle the string makes with respect to the horizontal for the case in which the spool slides but does not roll (see Figure 3).
   {13971_Procedure_Figure_3}
10. How close does the measured angle compare to the predicted value?
11. Repeat steps 8 and 9 for the second spool.
12. Obtain the spool with the larger-diameter flanges.
13. Use transparent tape to secure two, three or four nuts to the outside of each flange. The nuts need to be equidistant so the weight is balanced (see Figure 4).
    {13971_Procedure_Figure_4}
14. Repeat steps 8 and 9.
15. Did the “critical angle” change?
16. Discuss the results with the students.

Teacher Tips

• Make sure the string is tightly wrapped around the middle of the axle. The loose end of the string should extend directly from the middle of the axle so the spool does not twist as it is pulled. Pull on the string slowly to prevent the spool from turning.
• Using Equation 3 from the Discussion, the critical angle for the small spool is equal to:
  {13971_Tips_Equation_1}
  The critical angle for the large spool is equal to:
  {13971_Tips_Equation_2}
• The measured critical angles for the large and small spools were 71° and 50°, respectively. With the added mass, the critical angle for the large spool was still 72°.

Discussion

When the spool is on the verge of slipping without rolling, there are four forces acting on it—the force of gravity (mg), the normal force (N), the tension in the string (T) and the force of friction (μN), where μ is the coefficient of static friction between the spool’s flanges and the tabletop (see Figure 5).

At the moment just before the spool begins to slide without rolling, the net horizontal force is zero and therefore the frictional force and the horizontal component of the string tension are balanced (Equation 1). The tension in the string and the frictional force both must produce equal and opposite torques about the center of the spool if the spool is to slip and not rotate. Torque is equivalent to a rotational force and is equal to the applied force multiplied by the distance away from the point of rotation (the center of the spool in this activity). The torque produced by the tension in the string is therefore equal to R₁T. The torque produced by the frictional force is equal to R₂μN. To determine the equilibrium angle at which the torque is balanced and no rotation occurs, set the two torques equal to each other (Equation 2). Rearranging Equation 2 and substituting μN/ T = R₁/R₂ into Equation 1 yields: Equation 3 shows that the critical angle at which the spool slides without rolling depends only on the ratio of the spool’s axle radius to the flange radius. The tension in the string, the amount of friction, and the mass of the spool do not affect the critical angle, and therefore do not affect the direction the spool will roll based on the angle at which the string is pulled. If the string is pulled with an angle smaller than the critical angle, the spool will roll in the direction of the pull. If the string is pulled at an angle greater than the critical angle, the spool will roll away from the pulling force.