6.3 – Determining Acceleration Due to Gravity Using a Simple Pendulum Revision Notes for Leaving Cert Physics

6.3 – Determining Acceleration Due to Gravity Using a Simple Pendulum

Aim

This experiment allows us to find the value of acceleration due to gravity (g) by studying how a simple pendulum behaves. We want to prove that the mathematical relationship $T = 2\pi\sqrt{\frac{l}{g}}$ accurately describes the connection between a pendulum's period and its length.

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This experiment demonstrates one of the most elegant ways to measure a fundamental physical constant using simple equipment. The pendulum's predictable behaviour has been studied for centuries and forms the basis for understanding oscillatory motion.

Apparatus needed

You'll need the following equipment to carry out this experiment:

Retort stand with boss head and clamp
String (approximately 1 metre long)
Small metal bob (the weight that swings)
Ruler for measuring length
Stopwatch for timing
Protractor to check swing angles

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Equipment Tips: The choice of bob material matters - use a dense material like metal rather than plastic to minimise air resistance effects. Ensure your string is non-elastic (cotton or fishing line works well) to maintain consistent pendulum length throughout the experiment.

Theory behind the experiment

When a pendulum swings through small angles (less than 10°), it demonstrates simple harmonic motion. The key relationship we use is:

$T = 2\pi\sqrt{\frac{l}{g}}$

Where:

T = period (time for one complete oscillation)
l = length of the pendulum
g = acceleration due to gravity

By rearranging this equation and squaring both sides, we get:

$T^2 = \frac{4\pi^2}{g} \times l$

This shows us that when we plot $T^2$ against length (l), we should get a straight line passing through the origin. The slope of this line equals $\frac{4\pi^2}{g}$ , which means we can calculate g using:

$g = \frac{4\pi^2}{\text{slope}}$

chatImportant

The linear relationship $T^2 \propto l$ is only valid for small amplitude oscillations. If the pendulum swings through large angles, the motion becomes more complex and this simple relationship breaks down.

Step-by-step procedure

Follow these steps carefully to get accurate results:

Set up your pendulum - Attach the metal bob to the string and clamp the other end to the retort stand so the bob hangs freely
Measure the length - Use your ruler to measure from the fixed point down to the centre of the bob
Create small oscillations - Displace the bob slightly (keeping the angle less than 10°) and release it gently without pushing
Time multiple oscillations - Use your stopwatch to measure the time for 20 complete swings, then divide by 20 to find the period T
Repeat with different lengths - Try at least 5 different lengths ranging from about 0.3m to 1.0m
Calculate and plot - Work out $T^2$ for each measurement and create a graph of $T^2$ (y-axis) versus length (x-axis)

infoNote

Why time 20 oscillations? Timing multiple oscillations and then dividing by the number of swings significantly reduces the impact of human reaction time on your measurements. This technique, called "timing multiple events," is a standard method for improving precision in physics experiments.

Sample results

Here's what typical experimental data might look like:

Length (m)	Time for 20 Oscillations (s)	Period T (s)	T² (s²)
0.30	21.8	1.09	1.19
0.40	25.3	1.27	1.61
0.50	28.3	1.42	2.02
0.60	31.1	1.56	2.43
0.80	35.7	1.79	3.20

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Notice how the period increases as the length increases, and how $T^2$ shows an even clearer relationship with length. This data should produce a straight-line graph when $T^2$ is plotted against length.

Calculations

Once you've plotted your graph of $T^2$ versus length, you can find the slope (m) of the straight line. Then calculate g using:

$g = \frac{4\pi^2}{m}$

where m is the slope of your graph.

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Worked Example: Calculating g from the slope

Step 1: Determine the slope from your graph If your graph shows $\text{slope} = 4.0 \, \text{s}^2/\text{m}$

Step 2: Apply the formula $g = \frac{4\pi^2}{m} = \frac{4\pi^2}{4.0} = \frac{39.48}{4.0} = 9.87 \, \text{m/s}^2$

Step 3: Compare with the accepted value The accepted value is $9.8 \, \text{m/s}^2$ , so this result shows excellent agreement with theory.

Expected results

If your experiment works well, you should find that:

The graph of $T^2$ versus length gives a straight line through the origin
Your calculated value for g should be close to 9.8 m/s²

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Small deviations from the expected value are normal and can result from experimental uncertainties, air resistance, or slight measurement errors. Values within 5-10% of the accepted value indicate a successful experiment.

Important precautions

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Critical Points for Accuracy:

To get the best results, make sure you:

Keep swing angles small - Always less than 10° for simple harmonic motion
Measure length accurately - From the suspension point to the centre of the bob
Time multiple oscillations - This reduces timing errors and gives more precise periods
Use appropriate string - Make sure it's light, doesn't stretch, and won't be affected by air currents
Avoid parallax errors - When measuring length, ensure your eye is level with the measurement point
Check for consistent oscillations - Ensure the pendulum swings in a single plane without wobbling

Conclusion

This experiment demonstrates that T² is directly proportional to length for a simple pendulum. The mathematical relationship $T = 2\pi\sqrt{\frac{l}{g}}$ accurately models how pendulums behave and allows us to determine the acceleration due to gravity near Earth's surface.

The success of this experiment validates our understanding of simple harmonic motion and shows how fundamental physical constants can be measured using basic laboratory equipment.

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Key Points to Remember:

A simple pendulum consists of a mass hanging from a string that can swing freely
The period $T$ is the time for one complete oscillation (swing and back)
Small angles (less than 10°) are essential for simple harmonic motion
The relationship $T^2 \propto l$ gives us a straight-line graph we can use to find g
Timing multiple oscillations and dividing gives more accurate period measurements
This method provides an elegant way to measure one of nature's fundamental constants

6.3 – Determining Acceleration Due to Gravity Using a Simple Pendulum (Leaving Cert Physics): Revision Notes