Directions: Follow the instructions to go through the simulation. Respond to the questions and prompts in the orange boxes.
Vocabulary: controlled experiment, harmonic motion, oscillation, pendulum, period, spring, spring constant
Prior Knowledge Questions (Do these BEFORE using the Gizmo.)
- A bungee jumper launches herself off a bridge. How would you describe her motion?
| She will continue to bounce, like the pendulum we examined in our lab - although up and down. |
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- A child goes to the playground and gets on a swing. How would you describe his motion?
| The child’s actions will reflect the pendulum but they’ll have a continuous force (the child swinging). |
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- What do the motions of the bungee jumper and swinger have in common?
| They’ll both reflect pendulum-like motions. |
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Gizmo Warm-up
Harmonic motion is repeating back-and-forth or up-and-down movement. The Simple Harmonic Motion Gizmo allows you to compare the harmonic motions of a spring and a pendulum.
To begin, open the TOOLS tab on the bottom right and drag one arrow to the bottom of the spring so that the weight just touches the tip of the arrow, as shown. Drag a second arrow so that the pendulum just touches the tip of the arrow when it swings to the left.
- When the spring touches the arrow, click the green button on the stopwatch. Count the movements, or oscillations. Click the green button again after the tenth oscillation.
| What is the time for 10 oscillations of the spring? | 4.5 seconds |
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| Divide this time by 10 to find the period of the spring: | 0.45 seconds |
- Click the red button to reset the stopwatch. Use the same procedure on the pendulum. What is the
| period of the pendulum? | 1.245 seconds |
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| Activity A: Period of a spring | Get the Gizmo ready: Click the red button to reset the stopwatch. | |
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Introduction: The spring constant is a measure of the stiffness of a spring. The greater the spring constant, the harder it is to stretch or compress the spring. The Simple Harmonic Motion Gizmo allows you to manipulate the mass on the end of the spring (m), the spring constant (k), and the gravitational acceleration (g).
Question: Which factors affect the period of a spring?
- Predict: Which factors do you think will increase or decrease the period of a spring? Which do you think will have no effect on the period of a spring?
| The gravity, mass, and the force (k). |
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- Gather data: In a controlled experiment, only one factor changes at a time. Controlled experiments are a fair way to compare the effects of each variable on a system.
For each of the combinations given in the table below, measure the time for 10 oscillations of the spring and calculate the period. If necessary, reduce the Sim. speed to improve your accuracy.
| m (kg) | k (N/m) | g (m/s2) | Time for 10 oscillations (s) | Period (s) |
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| 0.5 kg | 100 N/m | 9.8 m/s2 | 4.5 | 10 |
| 2.0 kg | 100 N/m | 9.8 m/s2 | 4.3 | 10 |
| m (kg) | k (N/m) | g (m/s2) | Time for 10 oscillations (s) | Period (s) |
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| 1.0 kg | 50 N/m | 9.8 m/s2 | 2.23 | 10 |
| 1.0 kg | 200 N/m | 9.8 m/s2 | 9 | 10 |
| m (kg) | k (N/m) | g (m/s2) | Time for 10 oscillations (s) | Period (s) |
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| 1.0 kg | 100 N/m | 5.0 m/s2 | 9.2 | 10 |
| 1.0 kg | 100 N/m | 20.0 m/s2 | 7.3 | 10 |
- Analyze: Examine the results of each experiment.
| Which factors affected the period of the spring? | String length and force. |
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| Which factor had no effect on the period? | Mass |
| What is the effect of quadrupling the mass? | Nothing. |
| What is the effect of quadrupling the spring constant? | The time for one movement increased. |
| Activity B: Period of a pendulum | Get the Gizmo ready: Reset the stopwatch. Set L to 1.0 m and g to 9.8 m/s2. | |
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Introduction: The Simple Harmonic Motion Gizmo allows you to manipulate three variables for the pendulum: its mass (m), its length (L), and the gravitational acceleration (g).
Question: Which factors affect the period of a pendulum?
- Predict: Which factors do you think will increase or decrease the period of a pendulum? Which do you think will have no effect at all?
- Gather data: For each of the combinations given in the table below, measure the time for 10 oscillations of the pendulum and calculate the period. If necessary, change the Sim. speed to improve the accuracy of your measurements.
| m (kg) | L (m) | g (m/s2) | Time for 10 oscillations (s) | Period (s) |
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| 0.5 kg | 1.0 m | 9.8 m/s2 | 4.3 | 10 |
| 2.0 kg | 1.0 m | 9.8 m/s2 | 4.5 | 10 |
| m (kg) | L (m) | g (m/s2) | Time for 10 oscillations (s) | Period (s) |
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| 1.0 kg | 0.5 m | 9.8 m/s2 | 2.2 | 10 |
| 1.0 kg | 2.0 m | 9.8 m/s2 | 5.3 | 10 |
| m (kg) | L (m) | g (m/s2) | Time for 10 oscillations (s) | Period (s) |
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| 1.0 kg | 1.0 m | 5.0 m/s2 | 3.2 | 10 |
| 1.0 kg | 1.0 m | 20.0 m/s2 | 4.3 | 10 |
- Analyze: Examine the results of each experiment.
| Which factors affected the period of the pendulum? | String length and force. |
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| Which factor had no effect on the period? | Mass |
| What is the effect of quadrupling the length? | An increase in speed. |
| What is the effect of quadrupling the gravitational acceleration? | An increase in speed. |
| Activity C: Calculating periods | Get the Gizmo ready: Reset the stopwatch. Set g to 9.8 m/s2. | |
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Introduction: In activity A, you found that the period of a spring depends only on its mass and the spring constant. In activity B, you found that the period of a pendulum depends on its length and gravitational acceleration. Now, you will determine formulas for the period of each object.
Question: How are the periods of pendulums and springs calculated?
- Summarize: Look at your results from prior activities.
| How did quadrupling m affect the period of the spring? | Nothing. |
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| How did quadrupling k affect the period of the spring? | A large amount of time was taken for one swing. |
| How did quadrupling L affect the period of the pendulum? | An larger amount of time was taken for one speed. |
| How did quadrupling g affect the period of the pendulum? | A shorter amount of time was taken for one swing. |
- Gather data: Use the Gizmo to find the period of the spring for three sets of m and k values. Then find the period of the pendulum for three sets of L and g values.
Spring
| m (kg) | k (N/m) | Time for 10 oscillations (s) | Period (s) |  |  | Period |
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| 10 | 2 | 4.3 | 10 | 5 | 2.3… | 0.23… |
| 2 | 3 | 2.1 | 10 | 0.666… | 0.2434… | 0.02434… |
| 6 | 2 | 6 | 10 | 3 | 1.23… | 0.0123… |
Pendulum
| L (m) | g (m/s2) | Time for 10 oscillations (s) | Period (s) |  |  | Period |
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| 10 | 2 | 3.2 | 10 | 5 | 2.3… | 0.23… |
| 2 | 3 | 1.2 | 10 | 0.666… | 0.2434… | 0.02434… |
| 6 | 2 | 5.4 | 10 | 3 | 1.23… | 0.0123… |
Calculate: Calculate the ratios listed in the last three columns of each table.
Analyze: Examine your data tables.
| No matter what values of m and k you used for the spring, what is the ratio of the period to ? | 0.1 |
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| No matter what values of L and g you used for the pendulum, what is the ratio of the period to ? | 0.1 |
- Challenge: Put together what you learned in this activity to come up with formulas for the period (T) of a spring based on its mass (m) and spring constant (k) and for the period of a pendulum based on its length (L) and gravitational acceleration (g).
| TSpring = | (x)1.2 |
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| TPendulum = | (x)1.3 |
Have your teacher check your formulas when they are complete.
- Apply: How long would a pendulum on Earth (where g = 9.8 m/s2) have to be to have the same period as a spring with a mass of 10.0 kg and a spring constant of 25 N/m?
- Think and discuss: Which is more likely to keep accurate time on the Moon, a spring-driven watch or a pendulum clock? Explain your answer.
| Spring watch, a pendulum requires gravity. |
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