CSEC Physics Investigation: This experiment explores whether the amplitude (initial angular displacement) of a pendulum affects its period of oscillation. While the simple pendulum formula T = 2π√(l/g) assumes small angles, real pendulums may show amplitude dependence. This investigation helps students understand the limitations of physical models and the importance of experimental verification.

Amplitude vs Period Relationship
🔍 Key Question:
📐
Small angles (<15°): Period approximately constant
📏
Large angles (>30°): Period increases with amplitude
📊
Expected graph: Period vs Amplitude shows upward curve for large angles

Experimental Aim

1 What We’re Investigating

Aim: To investigate how the amplitude (initial angular displacement) of a simple pendulum affects its period of oscillation.

Hypothesis: We predict that for small angles (less than 15°), the period will be approximately constant, but for larger angles, the period will increase slightly with increasing amplitude.

Variables: What We Change, Measure, and Control

📐 Independent Variable

What: Amplitude/initial angular displacement (θ)

How varied: Change release angle (e.g., 5°, 10°, 15°, 30°, 45°, 60°, 75°)

Measurement: Large protractor ±1°

⏱️ Dependent Variable

What: Period of oscillation (T)

How measured: Time for 10 oscillations, then calculate T = time/10

Measurement: Stopwatch ±0.1 s

⚖️ Controlled Variables (Keep Constant)

Length of pendulum
Same string length throughout
Mass of bob
Use same bob
Size/shape of bob
Don’t change bob
Location
Same g value (same place)
Type of string
Same string material
Point of suspension
Same fixed support

Apparatus Required

🔧 Equipment List
Retort stand & clamp
Stable support
Split cork
To hold string
String/thread
≈2.0 m, light
Heavy metal bob
Dense, ≈2-3 cm
Stopwatch
Digital (±0.01 s)
Large protractor
Measure angles (±1°)
Meter rule
Measure constant length
Plumb line
Determine vertical

Note: Use a long pendulum (≈2m) to make period changes more noticeable and to reduce percentage timing errors.

Method: Step-by-Step Procedure

📋 Experimental Procedure
Step 1: Setup Long Pendulum
Set up retort stand with clamp and split cork. Use long string (≈2.0 m). Attach heavy bob. Measure and record exact length from pivot to center of bob.
Step 2: Set Up Protractor Reference
Set up plumb line behind pendulum to mark vertical. Place large protractor so its center is at pivot point and 0° aligns with vertical.
Step 3: Measure Smallest Amplitude First
Displace bob to smallest angle (e.g., 5°). Use protractor to measure precisely. Mark reference point on floor/bench for timing.
Step 4: Time Oscillations for Small Amplitude
Release bob without pushing. Time 10 complete oscillations (from reference point back to same point, same direction). Record time. Repeat twice more.
Step 5: Calculate Period for Small Amplitude
Calculate average time for 10 oscillations. Divide by 10 to get period T. Record T.
Step 6: Increase Amplitude Gradually
Repeat steps 3-5 for increasing amplitudes: 10°, 15°, 30°, 45°, 60°, 75°. Always use same release method.
Step 7: Ensure Consistency
After each large amplitude measurement, check that pendulum length hasn’t changed (string slipping). Allow pendulum to come to rest completely between measurements.

Why Use a Long Pendulum? A long pendulum (≈2m) has a longer period (≈2.8s). This makes timing errors smaller as a percentage of the period. Also, amplitude effects are more noticeable with longer pendulums, making changes in period easier to detect.

Data Collection Table

📊 Sample Data Table (Length = 2.000 m constant)
Amplitude, θ (°) Time for 10 oscillations (s) Mean time for 10 oscillations (s) Period, T (s)
T = mean time ÷ 10		 % Change from T at 5°
5 28.4, 28.5, 28.3 28.40 2.840 0.0% (reference)
10 28.4, 28.5, 28.4 28.43 2.843 +0.11%
15 28.5, 28.6, 28.5 28.53 2.853 +0.46%
30 28.7, 28.8, 28.7 28.73 2.873 +1.16%
45 29.0, 29.1, 29.0 29.03 2.903 +2.22%
60 29.4, 29.5, 29.4 29.43 2.943 +3.63%
75 30.0, 30.1, 30.0 30.03 3.003 +5.74%

Note: In this sample data, period increases with amplitude, especially for angles >30°. The increase is small (5.7% at 75°) but measurable.

Data Analysis & Graphing

Graphical Analysis: Period vs Amplitude
Expected Result
Two Regions:
1. θ < 15°: Nearly horizontal
2. θ > 30°: Curves upward
Period increases with amplitude for large angles
Key Feature
Small-Angle Approximation
For θ < 15°:
sinθ ≈ θ (in radians)
Error < 1%
Period nearly constant
📈 What Your Graph Should Show:
Horizontal section: For θ < 15°, points cluster around constant T
Upward curve: For θ > 30°, T increases with θ
No linear relationship: Not straight line – shows T increases faster at larger θ
Theoretical prediction: T = T₀[1 + (1/16)θ² + …] where T₀ = 2π√(l/g)

Theoretical Background

🧮 Exact vs Approximate Period Formula
Small-Angle Approximation (θ in radians):
sinθ ≈ θ (for θ < 0.26 rad ≈ 15°)
This gives simple formula: T ≈ 2π√(l/g)
Period independent of amplitude (to first approximation)
Exact Formula (Large Angles):
T = 2π√(l/g) × [1 + (1/4)sin²(θ/2) + (9/64)sin⁴(θ/2) + …]
For θ = 30°: T ≈ T₀ × 1.017 (1.7% increase)
For θ = 60°: T ≈ T₀ × 1.037 (3.7% increase)
For θ = 90°: T ≈ T₀ × 1.073 (7.3% increase)
Conclusion: Period increases with amplitude for large angles

Why Amplitude Affects Period for Large Angles

⚠️ Physical Explanation

For large amplitudes, the restoring force is no longer proportional to displacement:

  • Small angles: Restoring force F ≈ -mgθ (linear, leads to SHM)
  • Large angles: Restoring force F = -mg sinθ (non-linear)
  • sinθ < θ for θ > 0, so restoring force is weaker than linear approximation
  • Weaker restoring force means slower acceleration back to center
  • Slower acceleration means longer period
  • Analogy: Pushing a child on a swing – larger arcs take slightly longer to complete

CSEC Insight: The simple pendulum formula T = 2π√(l/g) is an approximation valid only for small angles (<15°). Your experiment verifies this limitation.

Experimental Challenges & Solutions

🎯 Measurement Challenges

  • Amplitude decay: Due to air resistance, amplitude decreases during timing
  • Solution: Time first few oscillations only; use heavy bob to reduce decay rate
  • Angle measurement: Hard to measure large angles accurately
  • Solution: Use large protractor with plumb line reference; measure from side view
  • Consistent release: Releasing at exact angle without pushing
  • Solution: Use marker/stop; release gently without imparting lateral motion

📝 Data Quality Issues

  • Timing variation: Period changes as amplitude decays
  • Solution: Time small number of oscillations (5-10); note amplitude at start/end
  • Elliptical motion: Bob may swing in ellipse not arc
  • Solution: Release carefully; check from side view; use guide if needed
  • Air currents: Affect large-amplitude swings more
  • Solution: Conduct in still air; away from windows/fans

Calculations & Analysis of Results

📊 Analyzing Your Data

1. Calculate percentage change from smallest amplitude:

For θ = 30°: % change = [(2.873 – 2.840)/2.840] × 100% = 1.16%
For θ = 60°: % change = [(2.943 – 2.840)/2.840] × 100% = 3.63%
For θ = 75°: % change = [(3.003 – 2.840)/2.840] × 100% = 5.74%

2. Compare to theoretical predictions:

Theoretical T(30°) = T₀ × 1.017 = 2.840 × 1.017 = 2.888 s
Actual T(30°) = 2.873 s (99.5% of theoretical)
Theoretical T(60°) = T₀ × 1.037 = 2.840 × 1.037 = 2.945 s
Actual T(60°) = 2.943 s (99.9% of theoretical)

3. Assess significance of changes:

Timing error for 10 oscillations ≈ ±0.3s (reaction time)
Error in T ≈ ±0.03s (±1% of T ≈ 2.84s)
Changes at 30° (0.033s) and above exceed measurement error
Changes at 10° (0.003s) are within measurement error

Conclusion from sample data: Period is effectively constant for θ < 15° (changes within experimental error) but increases significantly for θ > 30° (changes exceed experimental error).

Expected Results & Conclusion

✅ What Your Experiment Should Show

  • Small angles (<15°): Period constant within experimental error (±1-2%)
  • Large angles (>30°): Period increases with amplitude
  • Graph shape: Nearly horizontal for small θ, upward curving for large θ
  • Magnitude of effect: ~1-2% increase at 30°, ~3-4% at 60°, ~5-7% at 75°
  • Theory confirmation: Small-angle approximation valid for θ < 15°
  • Practical implication: For accurate timekeeping, pendulum clocks use small amplitudes

Writing Your Conclusion (CSEC Format):

  • State the relationship: “The results show that for small amplitudes (<15°), the period is approximately constant, but for larger amplitudes, the period increases with increasing amplitude.”
  • Reference your data: “The period increased by only 0.46% from 5° to 15°, but by 5.74% from 5° to 75°.”
  • Compare to theory: “This supports the theoretical prediction that the small-angle approximation (T = 2π√(l/g)) is valid only for θ < 15°, and that the exact period formula includes amplitude-dependent terms.”
  • Discuss significance: “The small-angle approximation is accurate to within 1% for θ < 15°, which explains why pendulum clocks use small amplitudes for accurate timekeeping.”
  • Acknowledge limitations: “Amplitude decay during measurements and angle measurement errors may have affected the results, especially at large amplitudes where air resistance is more significant.”
  • Suggest improvements: “To obtain more accurate results, the experiment could be conducted in a vacuum to eliminate air resistance, or with automated timing and angle measurement systems.”

CSEC Exam Practice: Amplitude and Pendulum Period

CSEC Exam Practice: Amplitude Effects on Pendulum Motion
Question 1: Why is the formula T = 2π√(l/g) only valid for small amplitudes (θ < 15°)?
Answer: The formula T = 2π√(l/g) is derived using the small-angle approximation sinθ ≈ θ, which is only valid for small angles.

Detailed explanation:
  • The restoring force for a pendulum is F = -mg sinθ
  • For simple harmonic motion, we need F ∝ -θ (linear restoring force)
  • Only when sinθ ≈ θ (for small θ in radians) do we get F ≈ -mgθ (linear)
  • When sinθ ≈ θ is valid, the motion is approximately SHM with T = 2π√(l/g)
  • For large θ, sinθ < θ, so the restoring force is weaker than linear
  • Weaker restoring force means slower acceleration → longer period

Numerical check: sin(15°) = 0.2588, 15° in radians = 0.2618, error = 1.1% (acceptable). sin(30°) = 0.5, 30° in radians = 0.5236, error = 4.5% (significant).

Question 2: A pendulum of length 1.00 m has a period of 2.00 s at small amplitude. If the amplitude is increased to 60°, approximately what would you expect the period to become?
Answer: Approximately 2.07-2.08 s

Calculation:
For θ = 60°, period increases by about 3.7% from theoretical formula
Increase factor = 1 + (1/4)sin²(θ/2) + … ≈ 1.037 for θ = 60°
New period T ≈ 2.00 s × 1.037 = 2.074 s
Rounded to 3 significant figures: 2.07 s

Alternative estimation: From sample data, 60° gave 3.63% increase, so T ≈ 2.00 × 1.0363 = 2.073 s
Question 3: In this experiment, why is it recommended to use a long pendulum (≈2 m) rather than a short one (≈0.5 m)?
Answer: A long pendulum has several advantages for this investigation:

1. Longer period: 2m pendulum has T ≈ 2.8s vs 0.5m pendulum T ≈ 1.4s. Timing errors (reaction time ±0.3s) become smaller as percentage of period.

2. More noticeable changes: Absolute changes in period are larger with longer pendulums, making them easier to measure accurately.

3. Reduced percentage errors: With T ≈ 2.8s, a 1% change = 0.028s, which is more measurable than 0.014s for T ≈ 1.4s.

4. Slower amplitude decay: Longer pendulums swing more slowly, so air resistance has less effect per swing, reducing amplitude decay during timing.

5. Practical: Easier to measure large angles accurately with long string.
Question 4: During the experiment, the amplitude decreases slightly with each swing due to air resistance. How does this affect the measured period, and how can you minimize this effect?
Answer:

Effect on measured period: As amplitude decreases during timing, the period also decreases slightly (since period decreases with decreasing amplitude). This means the measured period for a given initial amplitude will be slightly lower than the true period for that constant amplitude.

How to minimize:
  1. Time fewer oscillations: Time only 5 or 10 oscillations instead of 20 to reduce amplitude decay during measurement
  2. Use heavy, dense bob: Reduces effect of air resistance relative to weight
  3. Start timing after 1-2 swings: Let amplitude stabilize slightly after release
  4. Use still air environment: Conduct away from drafts, windows, fans
  5. Note amplitude at start and end: Record how much amplitude decays during timing and consider this in analysis
  6. Use average: For large amplitudes where decay is significant, take period as average of first few oscillations
Question 5: A student’s results show no change in period from 5° to 75° amplitude. Give TWO possible reasons for this unexpected result.
Answer:

1. Timing errors masking the effect: If the pendulum is short (e.g., 0.5m) with period ~1.4s, a 5% increase at 75° is only 0.07s. With reaction time errors of ±0.3s for timing 10 oscillations (±0.03s per period), this change might be within experimental uncertainty and not detectable.

2. Incorrect angle measurement: The student might not be measuring angles accurately. For example, what they think is 75° might actually be much less, or they might not be measuring from the true vertical.

Other possibilities:
  • Using wrong formula: Calculating period incorrectly from timing data
  • Inconsistent timing: Not using same reference point for all amplitudes
  • Very heavy damping: Extreme air resistance or friction causing irregular motion
  • Data recording errors: Misreading stopwatch or recording wrong values

How to check: Use longer pendulum (2m) to amplify the effect, verify angle measurements with protractor and plumb line, take multiple readings, calculate percentage changes carefully.

Question 6: Explain why pendulum clocks are designed to operate with small amplitudes (typically 5-10°).
Answer: Pendulum clocks use small amplitudes for several important reasons:

1. Constant period: For small amplitudes (<15°), period is essentially constant (independent of amplitude). This ensures accurate timekeeping even if the amplitude varies slightly due to mechanical imperfections or changes in driving force.

2. Reduced air resistance effects: At small amplitudes, the bob moves more slowly, so air resistance is smaller and more predictable. This reduces energy loss and amplitude decay.

3. Less stress on mechanism: Small amplitudes mean smaller forces on the pivot and escapement mechanism, reducing wear and tear.

4. More consistent operation: The small-angle approximation (T = 2π√(l/g)) is accurate to better than 0.5% for θ < 10°, ensuring consistent period.

5. Historical practice: Early clockmakers like Christiaan Huygens discovered that small amplitudes gave more accurate timekeeping and designed their clocks accordingly.

CSEC Connection: This practical application demonstrates why understanding the limitations of physical formulas (like the small-angle approximation) is important in engineering and technology.

🎯 Amplitude and Pendulum Period: Key Points for CSEC

  • Small-angle approximation: Valid for θ < 15°, gives T = 2π√(l/g)
  • Large angles: Period increases with amplitude (T > T₀)
  • Magnitude: ~1-2% increase at 30°, ~3-4% at 60°, ~5-7% at 75°
  • Reason: sinθ < θ for θ > 0 → weaker restoring force → longer period
  • Experimental design: Use long pendulum (2m) to amplify effect and reduce timing errors
  • Graph: Period vs Amplitude: horizontal for small θ, upward curve for large θ
  • Practical application: Pendulum clocks use small amplitudes (5-10°) for accuracy
  • CSEC SBA: Excellent investigation for showing understanding of model limitations
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