The Simple Pendulum Experiment

In short

The simple pendulum experiment is the textbook CBSE class-11 method for measuring g, the acceleration due to gravity, in a school laboratory. Hang a heavy bob from a light string of length L, release it from a small angle, and time the oscillations with a stopwatch. From the simple pendulum theory the time period is

T = 2\pi\sqrt{\frac{L}{g}} \quad \Longleftrightarrow \quad T^2 = \frac{4\pi^2}{g}\,L.

The square-of-period is linear in length, with slope = 4\pi^2/g and intercept = 0. So:

For each of five or six different lengths L, time a large number N of oscillations (typically N = 20) and divide by N to get the mean period T.
Plot T^2 (vertical) against L (horizontal). The points should lie on a straight line through the origin.
Measure the slope m of the best-fit line. Then

\boxed{\; g \;=\; \frac{4\pi^2}{m}. \;}

The trick that makes the experiment work is timing N oscillations rather than one. A stopwatch started and stopped by hand has a reaction-time error of about \pm 0.2 s. Divide that over 20 oscillations and the error in T drops to \pm 0.01 s — a factor-of-20 improvement for free. Done carefully with a 1 m pendulum and N = 20, this experiment gives g = 9.8 m/s² to about \pm 0.1 m/s² — a 1% measurement of a fundamental physical constant, done in one lab period with a ruler, a bob, a stopwatch, and a piece of string.

There is a scene that has played out in every class-11 physics lab in India for the last sixty years. A rickety ring-stand clamped to the corner of a wooden bench. A length of cotton thread tied to the stand, passing through a small metal ring fixed at the top. At the bottom of the thread, a heavy brass bob. A metre scale leaning against the wall. A stopwatch, clicked on and off by a student whose hand is shaking slightly from the afternoon's chai. On the blackboard, a teacher is writing T = 2\pi\sqrt{L/g} in slightly sloping chalk, and next to it, g = 9.8\ \text{m/s}^2. The instruction has been: "measure g with this apparatus."

The remarkable thing — genuinely remarkable, and the thing that makes this experiment a rite of passage in the Indian physics curriculum — is that when you do it carefully, with twenty oscillations per measurement and five or six different lengths, and plot T^2 against L, the points really do fall on a straight line through the origin, and the slope really does give you g to within about 1%. A piece of string, a weight, a watch, and an afternoon: you have just measured the acceleration at which every dropped object on Earth accelerates, without dropping anything. Galileo could only dream of apparatus this good.

This article is the companion to the theoretical Simple Pendulum article. That one derives T = 2\pi\sqrt{L/g} from the small-angle approximation. This one is the practical lab counterpart: the apparatus, the procedure, the plot, the error sources, and the number you should believe at the end. If you have a lab session scheduled tomorrow, read this to understand exactly what you are going to do and why each step matters.

The apparatus

A "simple pendulum" in the lab is a close approximation to the idealised point bob on a massless string. Typical apparatus:

Bob: a heavy, dense, symmetric object with a small hook. Brass is classic (density ~8.4 g/cm³); steel also works. A spherical shape matters because the string should attach to the centre-of-mass axis. Mass 30–80 g is typical.
String: light, inextensible, non-stretching cotton or silk thread. Length about 1.5 m (giving you room to vary L down to 40 cm or so).
Rigid support: a clamp-and-stand, or (better) a wall-mounted hook. The support must be truly rigid — if it sways by even a millimetre as the pendulum swings, the effective pivot is moving and the theory of a fixed pivot no longer applies.
Split cork at the pivot: a two-piece cork held together with the thread sandwiched between. This defines a sharp pivot point. Otherwise the thread rubs on the support, friction varies with angle, and the effective pivot point wanders.
Metre scale with a least count of 1 mm.
Stopwatch: digital, least count 0.01 s, or analog stopwatch with 0.1 s divisions.
Vernier callipers: to measure the radius of the bob so you can add it to the string length (see below).

The simple pendulum apparatus. The effective length $L$ runs from the centre of the bob (the bob's CM, because the point-mass idealisation is most accurate there) to the bottom edge of the split cork (where the pivot pinches the thread). Measure the string length $\ell$ with a metre scale, then add the bob radius $r$ measured with vernier callipers: $L = \ell + r$. The initial release angle $\theta_0$ should be small — no more than 5°, or 1/20 of a radian — so the small-angle approximation holds.

Measuring the effective length L

This is the most common source of systematic error in the experiment, and the most common student mistake. The theory derives T = 2\pi\sqrt{L/g} with L being the length from the pivot point to the centre of mass of the bob — not to the top of the bob, and not to the bottom. If you measure only the string length \ell (pivot to top of bob) and forget the bob's radius r, your L is too small by r \sim 1.5 cm for a typical 3 cm-diameter brass bob. That sounds tiny, but for a string of 50 cm it is a 3% error, directly translated into a 3% error in g — bigger than any other error in a careful experiment.

So the recipe is:

Measure the string length \ell from the bottom of the pivot (split cork's tip) to the top of the bob — use a metre scale, record to the nearest millimetre.
Measure the bob diameter d with vernier callipers to the nearest 0.01 cm. Then r = d/2.
The effective length is L = \ell + r.

Record L to three significant figures. A 1 metre pendulum measured this way has an uncertainty of about \pm 1\ \text{mm} — a 0.1% fractional uncertainty. That is comfortably the best-measured quantity in the whole experiment.

The procedure

The canonical CBSE procedure, the one expected in a class-11 practical exam, runs like this.

Step 1: set up and check. Clamp the split cork firmly to a rigid support. Thread the string through the split cork so the pivot is sharp. Tie the bob to the lower end. Verify by eye that the bob hangs vertically with the string taut when undisturbed.

Step 2: choose a length. Start with a convenient length — 100 cm is traditional. Adjust the string so that \ell + r = L = 100.0 cm. Record L.

Step 3: release at a small angle. Pull the bob sideways by at most 5° (about 9 cm of horizontal displacement for a 1 m string) and release it gently so it starts swinging in a single vertical plane — no orbiting, no wobbling. If it wobbles, stop and restart.

Why: the derivation T = 2\pi\sqrt{L/g} uses \sin\theta \approx \theta. At \theta = 5° (~0.087 rad), \sin 5° = 0.0872 — the error of approximation is 0.1%. At \theta = 30° the error rises to 4%, and your answer for g would be wrong by 4%. Keep the angle small.

Step 4: time many oscillations. Let the pendulum make a few oscillations to stabilise its motion. Then start the stopwatch at the instant the bob passes the lowest point moving in a chosen direction (say, left-to-right). Count N = 20 complete oscillations (one oscillation = back to the same position moving in the same direction). Stop the stopwatch. Record the total elapsed time t_1.

Why: timing the bob as it passes the lowest point — where it is moving fastest — is less error-prone than timing it at the turnaround points. At the highest point the bob is momentarily at rest and the transition is slow, making the stopwatch click ambiguous by several centiseconds. At the lowest point the bob flashes by, and the timing decision is sharp.

Step 5: repeat for precision. Do Step 4 twice more, getting times t_2 and t_3 for 20 oscillations each. Compute the mean:

t_{\text{mean}} = \frac{t_1 + t_2 + t_3}{3}, \qquad T = \frac{t_{\text{mean}}}{20}

Why: three independent measurements of a time-of-20 oscillations reduce the random (reaction-time) error by \sqrt{3} \approx 1.73. This, combined with the 20× reduction from timing multiple oscillations, brings the fractional error in T to about 0.1 percent — the smallest error in the experiment.

Step 6: change length, repeat. Shorten the string to L = 80 cm and repeat Steps 3–5. Then L = 60, 50, 40 cm. You should have at least five data points of (L, T).

Step 7: tabulate L, T, and T^2.

L (cm)	t_1 (s)	t_2 (s)	t_3 (s)	T (s)	T^2 (s²)
100.0	40.20	40.15	40.22	2.010	4.040
80.0	35.90	35.95	35.88	1.796	3.226
60.0	31.10	31.08	31.12	1.555	2.418
50.0	28.35	28.40	28.38	1.419	2.014
40.0	25.41	25.38	25.43	1.270	1.613

These are representative numbers you might see in a CBSE class-11 lab. The first column is the effective length in cm, columns 2–4 are three separate measurements of the time for 20 oscillations, column 5 is the mean period (column 2–4 average, divided by 20), and column 6 is T^2 for plotting.

The T² vs L plot — why the straight line matters

Rearrange the theoretical relation:

T = 2\pi\sqrt{\frac{L}{g}} \quad\Longrightarrow\quad T^2 = \frac{4\pi^2}{g}\,L \tag{1}

This is a straight-line relationship between T^2 and L, passing through the origin. The slope is 4\pi^2/g; the intercept is zero.

Why square both sides? Because the square relation is linear in L. A plot of T against L alone is a square-root curve — eye-fitting a slope to a curve is hard and unreliable. A plot of T against \sqrt{L} is also linear, but people are bad at computing \sqrt{L} mentally. The cleanest move is to square the period. The plot then has the form "y = mx + 0" — a straight line through the origin with slope m = 4\pi^2/g.

Interactive $T^2$ vs $L$ plot. Five data points (dark dots) from a typical class-11 experiment, and a straight line through the origin whose slope is $4\pi^2/g$. Drag the red point to change the value of $g$ used for the line; only $g \approx 9.81$ m/s² makes the line pass through the data. The value you extract from the slope of the best-fit line is the value of $g$ in the lab.

Extracting g from the slope

From equation (1), the slope of the best-fit T^2-vs-L line is m = 4\pi^2/g. Rearrange to solve for g:

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

Using the tabulated data from Step 7 (converted to SI: L in metres, T^2 in s²), the five points (L, T^2) are approximately (0.40, 1.613), (0.50, 2.014), (0.60, 2.418), (0.80, 3.226), (1.00, 4.040). A best-fit line through the origin (using m = \sum L_i T_i^2 / \sum L_i^2 or simply eye-fitting on graph paper) gives m \approx 4.02 s²/m. Then

g = \frac{4\pi^2}{4.02} = \frac{39.48}{4.02} = 9.82\ \text{m/s}^2

which agrees with the standard value 9.81 m/s² to within about 0.1% — better than the experiment's intrinsic errors would suggest, because the errors happen to cancel in this particular dataset.

Worked examples

Example 1: Computing g from a single length measurement

In a standard class-11 lab in Lucknow, a student measures a pendulum of effective length L = 0.800 m and times 20 oscillations at 35.92 s. Compute the time period T, and use the formula T = 2\pi\sqrt{L/g} to find g.

A pendulum of 0.8 m length timed over 20 oscillations. From the elapsed time, extract the period $T$ and then compute $g$.

Step 1. Compute the time period.

T = \frac{t}{N} = \frac{35.92\ \text{s}}{20} = 1.796\ \text{s}

Why: 20 oscillations took 35.92 seconds, so one oscillation takes 35.92/20 seconds. This is why we time many oscillations — dividing by 20 also divides the stopwatch error by 20, as the error-analysis section will show.

Step 2. Square the relation T = 2\pi\sqrt{L/g}.

T^2 = 4\pi^2 \cdot \frac{L}{g}

Why: squaring makes the formula easier to solve for g — the square root is gone and the remaining algebra is one rearrangement.

Step 3. Solve for g.

g = \frac{4\pi^2 L}{T^2}

Step 4. Substitute.

g = \frac{4 \times (3.14159)^2 \times 0.800}{(1.796)^2} = \frac{4 \times 9.8696 \times 0.800}{3.2256} = \frac{31.583}{3.2256} = 9.79\ \text{m/s}^2

Why: plug in \pi^2 = 9.8696, L = 0.800 m, T^2 = 1.796^2 = 3.2256 s². The resulting 9.79 m/s² is the experimental value of g from this one measurement.

Result: g = 9.79\ \text{m/s}^2, which compares to the standard value of 9.81\ \text{m/s}^2 within 0.2% — a superb result for a one-length measurement, much better than you should trust for a single trial.

What this shows: A single length, timed carefully, gives g to about 1%. The plot method using multiple lengths gives even better precision because it averages out random errors. The key measurement is the time period T, because g \propto 1/T^2 and T's error gets squared into the error of g.

Example 2: Why timing 20 oscillations is twenty times better than one

A student's reaction time when clicking the stopwatch is about \sigma_{\text{rxn}} = \pm 0.2 s (on starting) and \pm 0.2 s (on stopping). Using a pendulum with T \approx 2.0 s, compute the fractional error in T if the student times (a) 1 oscillation, (b) 20 oscillations, (c) 100 oscillations.

Step 1. Model the measurement error.

For any time interval measured by hand, the stopwatch error is independent at the start and the stop. By the rule for independent errors adding in quadrature:

\sigma_t = \sqrt{\sigma_{\text{rxn, start}}^2 + \sigma_{\text{rxn, stop}}^2} = \sqrt{0.2^2 + 0.2^2} = 0.283\ \text{s}

Why: reaction-time errors on pressing the start and stop buttons are independent (different instants, independent attention lapses). Independent errors add in quadrature, not linearly — the total uncertainty is the square root of the sum of squares.

Step 2. For N oscillations, the period is T = t/N where t is the measured total time. The error propagates as:

\sigma_T = \frac{\sigma_t}{N}

Why: when you divide a measured quantity by an exact integer, the uncertainty divides by the same integer. No squaring, no quadrature — just plain linear scaling.

Step 3. Compute for the three cases.

N	\sigma_T	Fractional error \sigma_T/T
1	0.283 s	14%
20	0.0142 s	0.71%
100	0.00283 s	0.14%

Why: at N = 1, the stopwatch error is the dominant uncertainty and the experiment is useless — 14% error in T gives about 28% error in g (because g \propto 1/T^2). At N = 20 the error drops by a factor of 20, and the experiment is usable. At N = 100 the error drops by another factor of 5, but air resistance and timing fatigue start to matter — there is no free lunch beyond about N = 30.

Step 4. Propagate the T-error to a g-error. Since g = 4\pi^2 L/T^2, the fractional error in g is:

\frac{\sigma_g}{g} = \sqrt{\left(\frac{\sigma_L}{L}\right)^2 + \left(2\frac{\sigma_T}{T}\right)^2}

Why: for a quantity g = \text{constant} \times L \cdot T^{-2}, the fractional uncertainty is the sum-in-quadrature of the fractional uncertainties of L and of T^{-2}. The power of 2 on T doubles the T-fraction.

For \sigma_L/L = 0.1\% (a 1 mm error on a 1 m pendulum) and \sigma_T/T = 0.71\% (from N = 20):

\frac{\sigma_g}{g} = \sqrt{(0.001)^2 + (2 \times 0.0071)^2} = \sqrt{10^{-6} + 2.02 \times 10^{-4}} \approx 1.4\%

So the final g is good to about 1.4%, or \pm 0.14 m/s² — consistent with a measurement of g = 9.8 \pm 0.1 m/s². The dominant error is the stopwatch reaction time; length measurement is barely a whisper in comparison.

Percentage error in the time period $T$ as a function of the number of oscillations $N$ timed. The 1/N decay is dramatic at small $N$: going from 1 to 20 cuts the error by a factor of 20. Beyond about $N = 30$ the stopwatch is no longer the dominant error, and further increase buys little.

Result: Timing 20 oscillations gives an error in T of 0.7% instead of 14% — a factor-of-20 improvement that costs nothing. Timing 100 oscillations improves further but hits diminishing returns because other errors (air resistance, angular decay, human timing fatigue) start to matter. The sweet spot for a class-11 lab is N = 20 to 30.

What this shows: The most important trick of the simple-pendulum experiment is the 1/N error reduction. This is why every physics lab manual in India — whether NCERT, CBSE, or the Tata Institute's school-outreach manuals — says "time 20 oscillations". It is not a tradition. It is the single intervention that turns a useless measurement into an accurate one.

Common confusions

"Use the longest pendulum for best accuracy." Only to a point. Longer pendulums have longer periods, so timing errors take a smaller fraction of each period. But longer pendulums also swing at lower frequencies, meaning you collect fewer oscillations in a given time — and air resistance accumulates more per oscillation, so the late oscillations have smaller amplitudes than the early ones. A length of 0.5–1.5 m is the standard compromise in a school lab.
"The amplitude doesn't affect the period, so it doesn't matter how far you pull the bob." True for small angles (under about 10°), false for large. The period grows as T = T_0 (1 + \theta_0^2/16 + \dots) for amplitude \theta_0 in radians. At \theta_0 = 30° (~0.52 rad), the correction is 0.52^2/16 = 1.7\% — bigger than your stopwatch error. Keep the release angle below 5°. See Simple Pendulum for the finite-amplitude derivation.
"Measure the length from pivot to the top of the bob." No — measure to the centre of mass of the bob. A 3 cm diameter brass bob has a centre 1.5 cm below the top of the bob, so forgetting the bob radius introduces a 1.5 cm systematic error in L (about 1.5% on a 1 m pendulum, directly giving a 1.5% error in g). Always measure \ell (string length) and add r (bob radius) for the effective length L.
"If the string isn't perfectly vertical at rest, the experiment is ruined." Not really. A slight off-vertical at rest means your zero crossings are still at the fastest-moving point; the period is unchanged. What does ruin the experiment is if the bob orbits (swings in a circle or an ellipse) instead of staying in a single plane — that is a different dynamical system. Restart the swing carefully if you see any elliptical motion.
"Air resistance slows the bob, so g comes out less than 9.8." Air resistance reduces the amplitude over time (the bob eventually stops), but for a dense brass bob in still air over 20 oscillations, it barely affects the period — under 0.1%. This is because viscous damping doesn't change the natural frequency of a lightly-damped oscillator; it only reduces the amplitude. See Damped Oscillations for the exact formula.
"The stopwatch is the most accurate instrument in the apparatus, so timing error is negligible." Exactly backwards. The stopwatch itself is precise (0.01 s readout), but the human pressing it has reaction-time error of 0.2 s at each end, which is 20 times worse. That is why we time 20 oscillations — to bring the human-induced timing error down to the level of the other measurements.

If you came here to do the class-11 experiment and write it up for your practical file, you have what you need. What follows is the linear-regression formula for the best-fit slope (since eye-fitting introduces its own bias), a proper derivation of how amplitude correction modifies the period, and a note on one of the more elegant pendulum measurements in Indian science history.

The least-squares slope for a line through the origin

Eye-fitting a line on graph paper works but introduces a subjective bias. For a more rigorous slope, use linear regression through the origin — the line y = mx + 0 that minimises the sum of squared vertical deviations \sum_i (y_i - m x_i)^2 where (x_i, y_i) = (L_i, T_i^2) for each data row.

Differentiate with respect to m, set to zero:

\frac{d}{dm}\sum_i (y_i - m x_i)^2 = -2 \sum_i x_i(y_i - m x_i) = 0

\sum_i x_i y_i = m \sum_i x_i^2

m = \frac{\sum_i x_i y_i}{\sum_i x_i^2} = \frac{\sum_i L_i \cdot T_i^2}{\sum_i L_i^2}

For the five rows of Step 7 (L in metres, T^2 in s²):

L_i	T_i^2	L_i T_i^2	L_i^2
1.000	4.040	4.040	1.000
0.800	3.226	2.581	0.640
0.600	2.418	1.451	0.360
0.500	2.014	1.007	0.250
0.400	1.613	0.645	0.160
Σ		9.724	2.410

m = 9.724 / 2.410 = 4.035 s²/m, and g = 4\pi^2/m = 39.478/4.035 = 9.784 m/s² — within 0.3% of the accepted value 9.81 m/s². The linear-regression approach removes eye-fitting bias and is the method used in university physics labs.

Finite-amplitude correction — the second-order period

The small-angle derivation gives T_0 = 2\pi\sqrt{L/g}. For a finite release angle \theta_0 (in radians), the exact period is given by a complete elliptic integral:

T = \frac{2}{\pi} T_0 \, K\!\left(\sin\tfrac{\theta_0}{2}\right)

where K(k) is the complete elliptic integral of the first kind. For small \theta_0, a series expansion gives:

T = T_0 \left(1 + \frac{\theta_0^2}{16} + \frac{11\,\theta_0^4}{3072} + \dots\right)

At \theta_0 = 5° (~0.087 rad), the correction is (0.087)^2/16 = 4.7 \times 10^{-4}, i.e. 0.05% — well below the stopwatch error. At \theta_0 = 30° (~0.524 rad), the correction is (0.524)^2/16 = 1.7\% — larger than any measurement error, making the experiment biased toward a larger T and therefore a smaller apparent g. The rule "\theta_0 < 5°" is not fussiness — it is the working limit beyond which the small-angle approximation fails by more than the experimental precision.

Historical note: the Survey of India pendulum expeditions

Between 1819 and 1828, the Great Trigonometrical Survey of India carried out a series of pendulum measurements at stations across the subcontinent — from Thiruvananthapuram to Kashmir — to map how the acceleration due to gravity varied with latitude. The idea was simple: if the Earth is a perfect sphere, g is the same everywhere. If the Earth is oblate (flattened at the poles), g is larger at high latitude and smaller at low latitude, because you are closer to or farther from the centre of the spheroidal Earth. By timing identical pendulums at each station, the surveyors measured this variation and used it to pin down the Earth's polar flattening. The survey contributed to the geodetic understanding of the Earth's shape — on the same scale and era as the French geodetic expedition to Peru.

Vikram Sarabhai, before he founded ISRO, used similar precision pendulum measurements during his doctoral work to study cosmic-ray intensity variations with altitude and latitude at research stations in Kodaikanal and Gulmarg. The ordinary-looking school-lab pendulum, when instrumented carefully, is accurate enough to probe the non-uniformity of the Earth's gravitational field itself — a reminder that this experiment is not a toy, but a scaled-down version of real geodetic science.

What to write in your practical file

The CBSE class-11 practical examination typically asks for:

Aim: to determine the value of g at the place of the experiment using a simple pendulum.
Apparatus: list (bob, thread, split cork, clamp stand, metre scale, stopwatch, vernier callipers).
Theory: state T = 2\pi\sqrt{L/g} and its square T^2 = (4\pi^2/g)\,L with a derivation reference.
Procedure: the seven steps above.
Observation table: six columns as in Step 7, with at least 5 rows.
Graph: plotted on millimetre graph paper, T^2 on y-axis, L on x-axis, best-fit straight line through origin with slope calculated.
Calculation: slope from graph, g = 4\pi^2/m.
Result: g = (\text{value}) \pm (\text{error}) m/s².
Sources of error: list stopwatch reaction time, length measurement, finite amplitude, air resistance.
Precautions: small amplitude, rigid support, string mass negligible, bob hung from centre of mass.

A neat presentation of the above, with clean numbers and a straight-looking graph, is what earns full marks. Most examiners will not penalise a 2% disagreement with 9.81 m/s²; they will penalise a missing error analysis or a graph that does not pass through the origin when it should.

Where this leads next

Simple Pendulum — the full derivation of T = 2\pi\sqrt{L/g} from Newton's second law and the small-angle approximation, plus the finite-amplitude correction.
Errors in Measurement — the general theory of measurement uncertainty, error propagation, and significant figures that underlies the error analysis here.
Vernier Callipers and Screw Gauge — Experiments — the measuring instruments you use to measure the bob radius and, more generally, any small dimension in the lab.
Equation of SHM and Phase — the simple-harmonic-motion framework that applies to the pendulum in the small-angle limit.
Damped Oscillations — what happens when you do not neglect air resistance, and why it reduces amplitude but barely touches the period.