Simple Pendulum Experiment: Time Period Formula, Procedure, Observation Table, and Complete CBSE Guide

Of all the experiments in the CBSE and ICSE Class 11 physics practical syllabus, the simple pendulum experiment is the most iconic. It is one of the oldest and most beautifully simple demonstrations in classical physics — a brass bob on a thread, swinging back and forth under the influence of gravity — yet it contains within it the precise numerical value of one of the most fundamental constants in physics: the acceleration due to gravity, g = 9.8 m/s².

The simple pendulum experiment teaches students three critical laboratory skills simultaneously: (1) how to use a metre scale and vernier calipers for length measurement, (2) how to use a stop clock for precise time measurement, (3) how to apply the concept of mean and error analysis to repeated timing measurements. It also introduces students to the concept of plotting a graph (T² vs. L) and determining a physical constant from the graph’s slope — a skill that is used in nearly every subsequent physics experiment at the graduate level.

This guide covers the simple pendulum experiment completely: the apparatus required, working principle, the time period formula and its derivation, the step-by-step experimental procedure, the observation table format, how to calculate g from the readings, the L vs. T² graph method, precautions, and sources of error. All apparatus described is manufactured and supplied by AJKANT Overseas from Ambala, India.

1. Aim of the Experiment

The aim of the simple pendulum experiment (CBSE Class 11 Physics Practical) is:

1. To measure the time period (T) of a simple pendulum for different effective lengths (L) of the pendulum.
2. To plot a graph of L vs. T and L vs. T².
3. To determine the value of acceleration due to gravity (g) at the location of the experiment from the slope of the L vs. T² graph.

2. Apparatus Required

3. Theory — What is a Simple Pendulum?

An ideal simple pendulum consists of a point mass (the bob) suspended from a fixed, frictionless pivot by an inextensible, massless string of length L. When displaced from its equilibrium (vertical) position by a small angle θ and released, it oscillates back and forth under the restoring force of gravity in a motion called Simple Harmonic Motion (SHM).

The key physical insight is that the restoring force on the bob at displacement angle θ is the component of gravity tangential to the arc of swing: F = −mg sinθ. For small angles (typically θ < 15°, where sinθ ≈ θ in radians), this force is approximately proportional to the displacement — which is the defining condition for SHM. This approximation is why it is critical to keep the amplitude of swing small in the experiment.

In a real laboratory pendulum (as opposed to the ideal), the “effective length” L is measured from the pivot point (centre of the split cork) to the centre of mass of the bob (which is at the geometric centre of the spherical bob). This is why the bob’s radius must be measured with a vernier caliper and added to the thread length.

4. Time Period Formula and Derivation

The time period T of a simple pendulum (for small oscillations) is given by:

Key Observations from the Formula

• T depends on L: Longer pendulums have longer time periods. A 1-metre pendulum has T ≈ 2.007 s; a 0.25-metre pendulum has T ≈ 1.004 s.
• T is independent of mass: The time period does NOT depend on the mass of the bob. A 50 g brass bob and a 200 g brass bob of the same radius swing with identical time periods (for the same L). This is Galileo’s famous discovery.
• T is independent of amplitude (for small angles): As long as the amplitude remains small (<15°), the time period does not change when the amplitude changes. This is called isochronism of the pendulum.
• T depends on g: A pendulum on the Moon (g ≈ 1.6 m/s²) swings about 2.5 times slower than the same pendulum on Earth (g ≈ 9.8 m/s²) for the same length.

5. Procedure — Step-by-Step

1. Set Up the Pendulum Support

   Clamp the retort stand firmly to the lab bench or against the wall. Attach the boss head and horizontal arm at the top. Insert the split cork into the horizontal clamp with the split face pointing downward. Ensure the entire support is absolutely rigid and vertical.
2. Measure the Bob’s Diameter with a Vernier Caliper

   Using a vernier caliper (LC = 0.02 mm), measure the diameter of the brass bob at three positions (0°, 90°, 45°). Calculate the mean diameter D. The radius r = D/2. This radius will be added to the thread length to get the effective pendulum length L.
3. Set the First Pendulum Length (L = 60 cm)

   Tie the thread to the bob’s hook. Thread the upper end through the split cork and tighten the cork in the clamp. Measure the length from the bottom of the split cork (pivot point) to the top of the bob using a metre scale. Adjust until this length equals (L − r), so the effective length from pivot to bob’s centre is exactly L = 60 cm. Note: L = thread length + radius of bob.
4. Displace and Release for Small Oscillations

   Pull the bob to one side by not more than 5–10 cm (corresponding to an angle less than 15° for a 60 cm pendulum) and release it gently without pushing. The bob should swing in a single vertical plane — no elliptical or conical motion. Discard the first 2–3 oscillations if the motion is not clean and planar.
5. Time 20 Complete Oscillations

   Start the stop clock when the bob passes through the mean (equilibrium) position moving in one direction. Count oscillations as it passes the mean position in the SAME direction. Count up to 20 complete oscillations and stop the clock when the bob passes the mean position for the 20th time. Record the time (t₁). Repeat two more times (t₂, t₃) for the same length.
6. Calculate Time Period for This Length

   Calculate mean time: t̄ = (t₁ + t₂ + t₃) / 3.
   Time period: T = t̄ / 20 seconds.
   Calculate T². Record L, t₁, t₂, t₃, t̄, T, and T² in the observation table.
7. Repeat for 5–6 Different Lengths

   Increase the effective length by 10 cm each time: L = 60, 70, 80, 90, 100, 110 cm. For each length, repeat steps 3–6, recording three timing measurements and calculating T and T². Use at least 5–6 different lengths for a reliable graph.
8. Plot the L vs. T² Graph and Calculate g

   Plot L (in metres, on Y-axis) vs. T² (in s², on X-axis) on graph paper. Draw the best-fit straight line through the origin. The slope of this line = L/T² = g/(4π²). Therefore: g = 4π² × slope of (L vs. T²) graph.

6. Observation Table

Diameter of bob (D) = ______ cm  |  Radius of bob (r = D/2) = ______ cm  |  Least count of stop clock = 0.1 s

7. Calculations — Finding the Value of g

Method 1: Direct Calculation from Each Reading

For each set of L and T values, use the formula directly:

Calculate g for each of the 5–6 readings. Take the mean of all calculated values of g. The mean value is the experimental determination of g.

8. Graph Method — T² vs. L Plot

9. Precautions and Sources of Error

Precautions to Follow

• ✅
  Keep amplitude small: Displace the bob by no more than 5–10 cm (angle < 15°). Larger amplitudes violate the small-angle approximation sinθ ≈ θ, making T slightly larger than the formula predicts.
• ✅
  Ensure planar motion only: The bob must swing in a single vertical plane. Elliptical or conical motion causes the observed time period to deviate from the simple pendulum formula.
• ✅
  Measure effective length carefully: The effective length must be from the pivot (centre of split cork) to the centre of the bob. Measure the thread length to the top of the bob and add the bob’s radius measured by vernier caliper.
• ✅
  Time from the mean position, not the extreme: Always start and stop timing as the bob crosses the central equilibrium position, not at the extreme of the swing. The bob moves fastest at the centre (easiest to observe precisely) and slowest at the extreme (hardest to identify exactly).
• ✅
  Count 20 oscillations (not 10): Timing 20 oscillations and dividing by 20 reduces the percentage error in T. If LC of stop clock is 0.1 s, the error in 20 oscillations is 0.1/20T ≈ 0.26% (for T ≈ 1.9 s) vs. 0.52% for 10 oscillations.
• ✅
  Rigid support is essential: Any wobble or vibration of the retort stand or suspension point adds oscillatory energy to the pendulum and distorts the time period measurement.

Common Sources of Error

10. Frequently Asked Questions (FAQ)