CSEC Physics Practical: This experiment investigates the relationship between the length of a simple pendulum and its period of oscillation. By systematically varying length and measuring time, you’ll verify Galileo’s discovery that T² ∝ l (the square of the period is proportional to length). This is a classic CSEC Physics investigation that demonstrates experimental design, data collection, and graphical analysis skills.
The Experiment: Length vs. Period Relationship
🔍 What You’ll Discover:
📏
As length increases → Period increases
📈
T vs l graph: Curve upward
📊
T² vs l graph: Straight line through origin
🧮
Mathematical relationship: T² ∝ l or T = 2π√(l/g)
Experimental Aim
1 What We’re Investigating
Aim: To investigate how the length of a simple pendulum affects its period of oscillation and to determine the mathematical relationship between them.
Hypothesis: We predict that as the length of the pendulum increases, the period will also increase. Based on Galileo’s work, we expect to find that T² is directly proportional to l (T² ∝ l).
Variables: What We Change, Measure, and Control
📐 Independent Variable
What: Length of pendulum (l)
How varied: Change suspension length (e.g., 0.20 m to 1.00 m in 0.10 m increments)
Measurement: Meter rule ±0.001 m (to nearest mm)
⏱️ Dependent Variable
What: Period of oscillation (T)
How measured: Time for 10 oscillations, then calculate T = time/10
Measurement: Stopwatch ±0.1 s (for total time)
⚖️ Controlled Variables (Keep Constant)
Mass of bob
Use same bob throughout
Amplitude (angle)
Keep constant (e.g., 10°)
Size/shape of bob
Don’t change bob
Location
Same g value (same place)
Apparatus Required
🔧 Equipment List
Retort stand & clamp
Stable support
Split cork
To hold string
String/thread
≈1.5 m, light
Metal bob
≈2 cm diameter
Stopwatch
Digital (±0.01 s)
Meter rule
±0.001 m
Protractor
Measure angle
Marker/tape
Reference point
Method: Step-by-Step Procedure
📋 Experimental Procedure
Step 1: Setup
Set up retort stand with clamp. Attach split cork in clamp. Thread string through cork. Attach bob to other end.
Set up retort stand with clamp. Attach split cork in clamp. Thread string through cork. Attach bob to other end.
Step 2: Measure Length
Adjust string so length from pivot to center of bob is 1.000 m (first length). Measure with meter rule. Mark this length on string for consistency.
Adjust string so length from pivot to center of bob is 1.000 m (first length). Measure with meter rule. Mark this length on string for consistency.
Step 3: Set Amplitude
Displace bob sideways about 10° (use protractor). Mark reference point on bench below equilibrium position.
Displace bob sideways about 10° (use protractor). Mark reference point on bench below equilibrium position.
Step 4: Time Oscillations
Release bob (don’t push!). Time 10 complete oscillations (from reference point, back to same point, same direction). Record time.
Release bob (don’t push!). Time 10 complete oscillations (from reference point, back to same point, same direction). Record time.
Step 5: Repeat Timing
Repeat timing for same length 2 more times (3 readings total). Calculate average time for 10 oscillations.
Repeat timing for same length 2 more times (3 readings total). Calculate average time for 10 oscillations.
Step 6: Change Length
Shorten pendulum to next length (e.g., 0.900 m). Repeat steps 3-5.
Shorten pendulum to next length (e.g., 0.900 m). Repeat steps 3-5.
Step 7: Continue
Continue with lengths: 0.800 m, 0.700 m, 0.600 m, 0.500 m, 0.400 m, 0.300 m (or as many as time allows).
Continue with lengths: 0.800 m, 0.700 m, 0.600 m, 0.500 m, 0.400 m, 0.300 m (or as many as time allows).
Data Collection Table
📊 Sample Data Table
| Length, l (m) | Time for 10 oscillations, t₁₀ (s) | Mean t₁₀ (s) | Period, T (s) T = mean t₁₀ ÷ 10 |
T² (s²) |
|---|---|---|---|---|
| 1.000 | 20.1, 20.0, 20.2 | 20.1 | 2.01 | 4.04 |
| 0.900 | 19.0, 19.1, 19.1 | 19.07 | 1.907 | 3.64 |
| 0.800 | 18.0, 17.9, 18.1 | 18.00 | 1.800 | 3.24 |
| 0.700 | 16.8, 16.9, 16.7 | 16.80 | 1.680 | 2.82 |
| 0.600 | 15.5, 15.6, 15.5 | 15.53 | 1.553 | 2.41 |
| 0.500 | 14.2, 14.1, 14.3 | 14.20 | 1.420 | 2.02 |
| 0.400 | 12.7, 12.6, 12.8 | 12.70 | 1.270 | 1.61 |
| 0.300 | 11.0, 11.1, 10.9 | 11.00 | 1.100 | 1.21 |
Note: These are sample values. Your actual measurements will vary based on your setup and location (g value).
Data Analysis & Graphing
Graphical Analysis: Finding the Relationship
Graph 1: T vs l
• Plot T (y-axis) vs l (x-axis)
• Result: Curved line
• Shows T increases with l
• But exact relationship unclear
Graph 2: T² vs l
• Plot T² (y-axis) vs l (x-axis)
• Result: Straight line
• Through origin (0,0)
• Proves T² ∝ l
📈 What the T² vs l Graph Tells Us:
Straight line through origin means: T² ∝ l
Equation: T² = k × l, where k is the gradient (slope)
From theory: k = 4π²/g, so g = 4π²/k
Calculations & Results
🧮 Sample Calculations
1. Calculating Period (T) from timed oscillations:
For l = 1.000 m, mean time for 10 oscillations = 20.1 s
T = 20.1 ÷ 10 = 2.01 s
2. Calculating T²:
T = 2.01 s
T² = (2.01)² = 4.0401 ≈ 4.04 s² (3 significant figures)
3. Finding gradient of T² vs l graph:
Choose two points on best-fit line: (0.3, 1.2) and (0.9, 3.6)
Gradient = (3.6 – 1.2) ÷ (0.9 – 0.3) = 2.4 ÷ 0.6 = 4.0 s²/m
So T² = 4.0 × l (approximately)
🌍 Calculating g from Your Results
From T² = (4π²/g) × l, the gradient k = 4π²/g
If gradient k = 4.0 s²/m (from example)
Then g = 4π²/k = 4π²/4.0
4π² ≈ 39.478
g = 39.478/4.0 = 9.87 m/s²
Compare to accepted value (9.81 m/s²). Calculate percentage error:
% error = |(9.87 – 9.81)/9.81| × 100% = 0.61%
Precautions & Sources of Error
⚠️ Common Experimental Errors and How to Minimize Them
| Error/Source | Effect on Results | How to Minimize |
|---|---|---|
| Reaction time (starting/stopping stopwatch) |
Random error in T measurements | Time multiple oscillations (10-20); use same person timing |
| Parallax error (reading meter rule) |
Systematic error in length measurements | View scale at eye level perpendicular to rule |
| Amplitude change (friction/air resistance) |
Period may change during timing | Use small amplitude (10°); start timing after 2-3 swings |
| String stretch/slip | Length changes during experiment | Use non-stretch string; secure tightly in clamp |
| Non-vertical swing (elliptical motion) |
Affects period formula validity | Release carefully without pushing; check from side view |
| Air currents/drafts | Irregular swinging | Conduct experiment away from windows/fans |
CSEC Pro Tip: In your SBA (School-Based Assessment), always mention precautions you actually took and suggest improvements. For example: “To reduce reaction time error, we timed 20 oscillations instead of 10 for the shorter pendulums where period was less than 1 second.”
Expected Results & Conclusion
✅ What Your Experiment Should Show
- T vs l graph: Curved line showing T increases with l
- T² vs l graph: Straight line through origin (or very close)
- Relationship: T² ∝ l (verified by straight line through origin)
- Gradient: k ≈ 4.0 s²/m (varies with location’s g value)
- Calculated g: Around 9.8 m/s² (typically 9.6-10.0 m/s² is acceptable)
Writing Your Conclusion (CSEC Format):
- State the relationship: “The results show that T² is directly proportional to l, confirming that T² ∝ l.”
- Reference your graph: “This is evidenced by the straight line through the origin on the T² vs l graph.”
- Mention calculated value: “From the gradient of the graph, g was calculated to be 9.87 m/s².”
- Compare to accepted value: “This is close to the accepted value of 9.81 m/s², with a 0.6% error.”
- Acknowledge limitations: “Sources of error include reaction time and small air currents, but these were minimized by timing multiple oscillations and conducting the experiment in a still area.”
CSEC Exam Practice: Pendulum Experiment
CSEC Exam Practice: Pendulum Length Experiment
Question 1: In this experiment, why is it important to time multiple oscillations (e.g., 10) rather than timing just one oscillation?
Answer: Timing multiple oscillations reduces the percentage error due to reaction time.
Detailed explanation:
Detailed explanation:
- Human reaction time for starting/stopping a stopwatch is typically ±0.2-0.3 seconds
- If you time 1 oscillation of period ~2s, this error is about 10-15%
- If you time 10 oscillations (~20s total), the same reaction error becomes only 1-1.5%
- The error is “spread” over more oscillations, reducing its effect on the calculated period
- Also helps average out any irregularities in individual swings
Question 2: A student obtains these results: Length = 0.400 m, time for 10 oscillations = 12.7 s. Calculate the period T and T².
Answer: T = 1.27 s, T² = 1.61 s²
Calculations:
Significant figures note: 12.7 has 3 sig figs, 10 is exact count, so T has 3 sig figs (1.27), and T² has 3 sig figs (1.61).
Calculations:
Period T = total time ÷ number of oscillations
T = 12.7 s ÷ 10 = 1.27 s
T² = (1.27)² = 1.6129 s²
Rounding to 3 significant figures: T² = 1.61 s²
Significant figures note: 12.7 has 3 sig figs, 10 is exact count, so T has 3 sig figs (1.27), and T² has 3 sig figs (1.61).
Question 3: On a graph of T² against l, what would the gradient (slope) of the line represent?
Answer: The gradient represents 4π²/g, where g is the acceleration due to gravity.
Explanation:
Example: If gradient = 4.0 s²/m, then g = 4π²/4.0 = 39.48/4.0 = 9.87 m/s²
Explanation:
From T = 2π√(l/g), we square both sides: T² = 4π²l/g
This is in the form y = mx + c, where y = T², x = l, m = 4π²/g, c = 0
So gradient m = 4π²/g
Therefore, g = 4π²/m (we can calculate g from the gradient)
Example: If gradient = 4.0 s²/m, then g = 4π²/4.0 = 39.48/4.0 = 9.87 m/s²
Question 4: List THREE variables that must be kept constant during this experiment, and explain why each is important to control.
Answer:
1. Mass of the bob: Must be kept constant because we’re investigating the effect of length only. Though theory says period is independent of mass for an ideal simple pendulum, in practice changing mass might affect air resistance or the pendulum’s behavior.
2. Amplitude (angle of swing): Must be kept constant because for large amplitudes (>15°), period depends on amplitude. By keeping amplitude small and constant, we ensure any changes in period are due to length changes only.
3. Shape/size of bob: Must be kept constant because a larger bob experiences more air resistance, which could affect the period. Using the same bob ensures consistent air resistance effects.
Additional: Location (same g value), type of string, method of release, etc.
1. Mass of the bob: Must be kept constant because we’re investigating the effect of length only. Though theory says period is independent of mass for an ideal simple pendulum, in practice changing mass might affect air resistance or the pendulum’s behavior.
2. Amplitude (angle of swing): Must be kept constant because for large amplitudes (>15°), period depends on amplitude. By keeping amplitude small and constant, we ensure any changes in period are due to length changes only.
3. Shape/size of bob: Must be kept constant because a larger bob experiences more air resistance, which could affect the period. Using the same bob ensures consistent air resistance effects.
Additional: Location (same g value), type of string, method of release, etc.
Question 5: If the T² vs l graph does not pass through the origin (0,0), what might this indicate about the experiment?
Answer: A non-zero intercept suggests systematic error in length measurements.
Possible reasons:
How to check: Re-examine how length was measured. For a true simple pendulum, T² should be 0 when l = 0 (no length means no pendulum). A positive intercept on T² axis suggests you’re effectively adding an extra constant length to all measurements.
Possible reasons:
- Zero error in length measurement: Not measuring from the actual pivot point to the center of the bob
- Incorrect length: Measuring to top of bob instead of center of mass
- String stretch: Length changes during swinging
- Consistent timing error: Systematic error in starting/stopping stopwatch
How to check: Re-examine how length was measured. For a true simple pendulum, T² should be 0 when l = 0 (no length means no pendulum). A positive intercept on T² axis suggests you’re effectively adding an extra constant length to all measurements.
Question 6: A student’s T² vs l graph has a gradient of 4.03 s²/m. Calculate the value of g from this gradient.
Answer: g = 9.79 m/s²
Calculation:
Interpretation: This is very close to the accepted value of 9.81 m/s², suggesting accurate measurements.
Calculation:
Gradient m = 4π²/g = 4.03 s²/m
So g = 4π²/4.03
4π² = 4 × (3.1416)² = 4 × 9.8696 = 39.4784
g = 39.4784 ÷ 4.03 = 9.795 m/s²
To 3 significant figures: g = 9.80 m/s²
Interpretation: This is very close to the accepted value of 9.81 m/s², suggesting accurate measurements.
🎯 Pendulum Length Experiment: Key Points for CSEC
- Relationship: T² ∝ l (verified by straight line through origin on T² vs l graph)
- Variables: Independent = length (l), Dependent = period (T), Controlled = mass, amplitude, bob size
- Timing: Time multiple oscillations (10-20) to reduce reaction time error
- Graphs: Plot both T vs l (curve) and T² vs l (straight line)
- Gradient: of T² vs l graph = 4π²/g, so g = 4π² ÷ gradient
- Typical gradient: ~4.0 s²/m (giving g ≈ 9.87 m/s²)
- Precautions: Small constant amplitude, secure string, still air, proper length measurement
- CSEC SBA: This experiment is excellent for School-Based Assessment – shows clear variables, good data, clear relationship