Materials Included In Kit

Additional Materials Required

Safety Precautions

The plumb bobs contain lead. Wash hands thoroughly with soap and water before leaving the laboratory. Please follow normal laboratory safety guidelines.

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. The Introductory Activity may be completed in one 50-minute class period. It is important to allow ample 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.
• To protect against the hazards of lead, the plumb bobs can be dipped into melted wax, or coated with a clear coat of paint. The additional mass will not affect the results of the experiment.
• Working in pairs, one student should measure the angle of the pendulum with a protractor and then release the plumb bob, while the other student times and counts the oscillations.
• Advise students to release the pendulum so that it swings back and forth in one plane and does not rotate as it oscillates. Lightly holding the plumb bob from the bottom with only a fingertip as it is pulled to the appropriate release angle and then lowering the finger to release the bob will provide a smooth release and even swing.
• If a computer spreadsheet program is unavailable, students can draw the data table and perform graphical analysis with graph paper and pencil.
• Students should choose string lengths of 75 cm or shorter. The pendulum should not exceed the length of the support stand rod.
• Use PSworks™ equipment from Flinn Scientific to obtain more precise measurements. Students can use the PSworks photogate timer to digitally display the pendulum’s period measurements. Visit www.flinnsci.com for more information.

Further Extensions

Opportunities for Inquiry

Challenge Lab
Make it a challenge! Design a pendulum that will have a specific period for time-keeping purposes.

Measuring the Energy of a Pendulum
Consider using a toy car to measure the energy of a swinging pendulum. Set up either the large or small pendulum on the benchtop with a support stand ring clamp. Obtain a toy car and place it directly underneath the hanging plumb bob so that the bob rests on the car. Pull the plumb bob to an angle ≤ 15° and release the bob so that it strikes the toy car. Using the meter stick, measure the distance the toy car travels. What happens to the pendulum’s potential energy when the plumb bob is released? Test the displacement of the car from various angles to explore the interconversion of potential and kinetic energy.

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 for 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.

Science Practices
1.4 The student can use representations and models to analyze situations or solve problems qualitatively and quantitatively.
2.2 The student can apply mathematical routines to quantities that describe natural phenomena.
3.1 The student can pose scientific questions.
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. Galileo first studied swinging pendulums when he noticed the swinging motion of a chandelier. Provide three other real-world examples of a swinging pendulum.
   Examples may include a trapeze artist, child on a swing, and grandfather clock time-keeping swing.
2. Draw arrows and label the forces mg, T, mg cosθ and mg sinθ acting on the plumb bob at its maximum displacement in the diagram of a swinging pendulum (see Figure 2).
   
   {13790_PreLab_Figure_2}
3. Define the period of a pendulum's motion.
   The period is the time in seconds that a swinging pendulum takes for each complete oscillation or each back and forth motion.
4. Describe the characteristics of simple harmonic motion and explain why a pendulum’s motion fits these requirements when the angle of displacement is relatively small (< 15°).
   A pendulum’s motion is simple harmonic because the restoring force is proportional to the displacement of the bob. If we consider a simple pendulum, the only force acting on the bob in the direction of the motion is gravity. Gravity always pushes or pulls the bob towards equilibrium.

Sample Data

Introductory Activity

Guided-Inquiry Procedure

Table 1. Table 2.

*% error = (measured–predicted)/predicted x 100%

Answers to Questions

Answers to Guided-Inquiry Discussion Questions

1. In the Introductory Activity, the number of complete oscillations was counted for a specified time. Identify additional features or variables in the design and construction of the pendulum that may be tested to determine how they affect the motion of a pendulum.

   The release angle, mass of the plumb bob, and length of the string should be tested to determine how they affect the motion of the pendulum.

2. How reproducible was the measurement in the Introductory Activity? In order to explore how different variables influence pendulum motion, how many trials would you recommend for any value of each variable to be tested in future experiments?

   All three trials in the Introductory Activity showed consistent results. Each variable should be tested at least three times.

3. A pendulum’s swing is generally assumed to be simple harmonic for small angles of displacements, ≤ 15°. How many release angles should be tested for each value of any variable in this activity?

   Two or more release angles that are ≤ 15° should be tested.

4. Construct additional pendulums to test the different variables. Plan and carry out survey experiments with the pendulums to determine the effect of each variable.

   Construct a pendulum with the small plumb bob and an additional piece of 75-cm string. Test each pendulum at different release angles following the procedure from the Introductory Activity. See sample data table 1.

5. After survey experiments are completed, determine any variable(s) that affect the period of a pendulum’s swing. Select additional values of this variable to determine and confirm the mathematical relationship between this variable and the period.

   The release angle and size of the plumb bob does not affect the number of oscillations and thus the period of a swinging pendulum. The other variable to test is the length of the pendulum. At least four lengths should be tested.

6. Calculate the average experimental value of the period (T) for each different pendulum length you studied.

   See Sample Data Table 2.

Analyze the Results

• Consider the following possible mathematical formulas for the relationship between the period of a pendulum and its length. Which formula has the correct units to calculate T in seconds?
  {13790_Answers_Equation_1}

  The correct equation must have T measured in seconds. Units of the variables: g = m/s², L = m, and T = s. The correct equation is a because it gives the correct units of T (seconds). Choice b is incorrect because it gives units of √s²/m and choice c is incorrect because it gives units of s². The period of the pendulum is proportional to the square root of the pendulum length.

• Calculate the theoretical value of the period, using the appropriate formula, for each pendulum length tested and determine the experimental error between the observed and calculated values.

  See Sample Data table 2.

• Create three separate graphs: period vs. pendulum length, period vs. square root of pendulum length, and period vs. square of pendulum length. Plot the data and draw a trendline starting from (0,0). Is the trendline a straight line in each of the three graphs? If so, what does that say about the relationship between the period and the length of the pendulum?
  {13790_Answers_Figure_4}

  The second graph, Period vs. Square Root of Pendulum Length, has a linear correlation coefficient of 0.994 for the best-fit line and confirms the trends observed during data collection. Also, the graph correlates with the correct equation to calculate the period of a swinging pendulum, where it was determined that the period of the pendulum is proportional to the square root of the pendulum length. Rearranging the equation shows the relationship:

  {13790_Answers_Equation_4}

Review Questions for AP^® Physics 1

1. A pendulum of 3.00 m in length completes 20 back-and-forth swings in 60 seconds. What is the acceleration of gravity?
   {13790_Answers_Equation_6}

   g = 13.2 m/s²

2. The time-keeping swing of a grandfather clock has a period of 2 seconds. How tall is the grandfather clock?
   
   
   {13790_Answers_Equation_8}

   L = 0.968 m

3. Consider three pendulums with the same length, 4.500 m. Two of the pendulums are located on Earth: one in Fairbanks, Alaska, and the other in Jakarta, Indonesia. The third is located on the moon. Look up the acceleration due to gravity for each location and determine the period of each pendulum.

   Fairbanks, Alaska, g = 9.829 m/s²
   Jakarta, Indonesia, g = 9.777 m/s²
   Moon, g = 1.624 m/s²
   North Pole, T = 4.251 s
   Jakarta, Indonesia, T = 4.263 s
   Moon, T = 10.459 s

   {13790_Answers_Equation_9}
4. Consider a pendulum that swings through maximum displacements on both sides of equilibrium. It swings through a total displacement of 26°. Determine the amplitude.
   1. What is the length of the string?

      L = h + L x cosθ
      L – L x cosθ = h
      L x (1 – cosθ) = h

      {13790_Answers_Equation_11}

      L = 5.05 m

      {13790_Answers_Figure_1}
   2. What is the period of the pendulum? Assume the pendulum is on the surface of the Earth.
      {13790_Answers_Equation_12}

      T = 4.51 s

References

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

Cooper, J. H., Smith, A. W. The Elements of Physics, 6th ed.; McGraw-Hill: York, PA, 1957; pp 162–165.

Faughn, J. S., Serway, R. A. Holt Physics; Holt, Rinehart and Winston: Austin, Texas, 1999; pp 438–451.

Tipler, Paul A. Physics for Scientists and Engineers, 3rd ed., Vol. 1; Worth: New York, 1990; pp 382–385.

Introduction

Swinging pendulums are common experiences and observations—from swinging on a playground to the time-keeping swing of a grandfather clock. A pendulum will oscillate back and forth along an arc, following the same path and always reaching the same maximum displacement away from its equilibrium position. When each complete oscillation takes the same amount of time, the motion of a pendulum is known as simple harmonic motion. Model these real-world examples to investigate and design simple pendulums in the lab and explore simple harmonic motion.

Concepts

• Pendulums
• Simple harmonic motion
• Period of oscillation
• Gravity

Background

It is said that the “father of modern physics,” Galileo Galilei (1564–1632), first observed or measured the motion of a swinging pendulum during a church service. While lighting the candles at the service, someone bumped into a chandelier and it began swinging back and forth. This sparked the famous physicist’s curiosity and he measured the motion of the chandelier by timing its swings to his pulse. The swinging chandelier is a real-life example of a simple pendulum, which is composed of string tied to a rigid object at one end, the anchoring point, with a freely hanging mass (m), also known as a plumb bob, tied to the other end (see Figure 1). When the pendulum is at rest, the plumb bob will hang directly below the anchoring point, and the string will be vertical. The only external forces acting on the plumb bob are the pull of gravity (mg) and the tension in the string (T) (see Figure 1). When the pendulum is vertical, these forces are balanced.

When the plumb bob of the pendulum is moved away from its equilibrium position along the arc of the pendulum swing and then released, gravity and the tension in the string are still the only forces acting on the plumb bob. However, now these forces are no longer balanced. The unbalanced forces result in a restoring force (mg sinθ) that moves the plumb bob back toward the equilibrium position along the arc of the swing. Because of momentum, the plumb bob will continue to swing past the equilibrium position. Once it passes equilibrium, the plumb bob will swing up along the pendulum’s arc and a restoring force will again act on the plumb bob to slow it down until it momentarily stops, and then falls back down towards its equilibrium position. The cycle will repeat itself and the pendulum will continue to oscillate back and forth this way indefinitely if no other forces, such as friction, act on it.

For small displacements, the restoring force acting on the plumb bob is directly proportional to the displacement away from the equilibrium position. Thus, the farther away from equilibrium, the larger the restoring force. As the plumb bob swings closer to equilibrium, the restoring force decreases evenly. When the restoring force is directly proportional to the displacement, the oscillations are said to exhibit simple harmonic motion. In simple harmonic motion, the pendulum will oscillate back and forth along an arc and reach the same displacement away from equilibrium each time. The time it takes for each complete oscillation will be constant. The displacement away from equilibrium is also called the amplitude of the oscillation, and the time in seconds for one complete oscillation is the period of the oscillation. As long as the amplitude is relatively small, the oscillations will exhibit simple harmonic motion and the period will not depend on the amplitude.

Experiment Overview

The purpose of this advanced inquiry activity is to investigate the variable(s) that affect the period of a pendulum’s swing. The lab begins with an introductory activity where a simple pendulum is constructed and tested. The guided-inquiry activity leads to further exploration of variables that may affect the period of the pendulum’s swing. The data will be interpreted using graphical analysis to confirm the tested relationships.

Materials

Prelab Questions

1. Galileo first studied swinging pendulums when he noticed the swinging motion of a chandelier. Provide three other real-world examples of a swinging pendulum.
2. Draw arrows and label the forces mg, T, mg cosθ and mg sinθ acting on the plumb bob at its maximum displacement in the diagram of a swinging pendulum (see Figure 2).
   {13790_PreLab_Figure_2}
3. Define the period of a pendulum’s motion.
4. Describe the characteristics of simple harmonic motion and explain why a pendulum’s motion fits these requirements when the angle of displacement is relatively small (< 15°).

Safety Precautions

The plumb bobs contain lead. Wash hands thoroughly with soap and water before leaving the laboratory. Please follow normal laboratory safety guidelines.

Procedure

Introductory Activity

Construct a Simple Pendulum

1. Obtain a support stand and ring clamp.
2. Cut a length of string 75 cm long.
3. Tie the large plumb bob to the end of the string. Cut off any extra string from the knot to keep excess string to a minimum.
4. Clamp the string to the side of the ring with the clothespin.
5. Position the support stand so that the plumb bob can dangle over the edge of the table and swing freely.
6. Using a meter stick, adjust the length of the pendulum so that it is 30 cm from the bottom of the plumb bob.
7. Use a protractor to measure the angle of the string. Pull the plumb bob along its swing arc to an angle, θ, of 5 degrees from equilibrium.
8. Release the plumb bob and immediately begin timing. Count the number of times the plumb bob swings through one complete oscillation within a total time of 30 seconds.
9. Repeat the measurement at least twice to determine the reproducibility.

Guided-Inquiry and Procedure

1. In the Introductory Activity, the number of complete oscillations was counted for a specified time. Identify additional features or variables in the design and construction of the pendulum that may be tested to determine how they affect the motion of a pendulum.
2. How reproducible was the measurement in the Introductory Activity? In order to explore how different variables influence pendulum motion, how many trials would you recommend for any value of any variable to be tested in future experiments?
3. A pendulum’s swing is generally assumed to be simple harmonic for small angles of displacements, ≤ 15°. How many release angles should be tested for each value of any variable in this activity?
4. Construct additional pendulums to test the different variables. Plan and carry out survey experiments with the pendulums to determine the effect of each variable.
5. After survey experiments are completed, determine any variable(s) that affect the period of a pendulum’s swing. Select additional values of this variable to determine and confirm the mathematical relationship between this variable and the period.
6. Calculate the average experimental value of the period (T) for each different pendulum length you studied.

Analyze the Results

• Consider the following possible mathematical formulas for the relationship between the period of a pendulum and its length. Which formula has the correct units to calculate T in seconds?
  {13790_Procedure_Equation_1}
• Calculate the theoretical value of the period, using the appropriate formula, for each pendulum length tested and determine the experimental error between the observed and calculated values.
• Create three separate graphs: period vs. pendulum length, period vs. square root of pendulum length, and period vs. square of pendulum length. Plot the data and draw a trendline starting from (0,0). Is the trendline a straight line in each of the three graphs? If so, what does that say about the relationship between the period and the length of the pendulum?

Student Worksheet PDF

13790_Student1.pdf