Experiment 1: The Pendulum & Gravity

A pendulum is the simplest clock in physics: a mass on a string, swinging under gravity alone. Galileo reportedly timed one with his pulse; three centuries of clockmakers trusted the same physics. In this experiment you will measure how the period — the time for one complete back-and-forth swing — depends on string length, and you will turn your stopwatch readings into a measurement of g, the acceleration due to gravity, using nothing but a ruler, a phone timer and a graph.

The physics behind the practical

For small swings (release angles below about 15°), a simple pendulum obeys T = 2π√(L/g), where T is the period in seconds, L is the length in metres measured from the pivot to the centre of the bob, and g ≈ 9.81 m/s² on Earth. Read the formula carefully: the bob's mass does not appear in it, and neither does the swing size. Only length and gravity set the tempo, because a heavier bob is pulled harder by gravity but is also harder to accelerate — the two effects cancel exactly.

Squaring both sides gives T² = (4π²/g) × L. That rearrangement is the experimental key: T² is directly proportional to L, so a graph of T² against L is a straight line through the origin whose gradient is 4π²/g. Measure the gradient and you have measured gravity itself.

Variables and controls

Timing 10 swings instead of 1 is not laziness — it is error control. Your start/stop reaction error is roughly ±0.1 s however many swings you time, so spreading it over 10 swings cuts the error in each period to about ±0.01 s.

In the virtual lab

The simulation panel below mirrors the bench experiment. Set the string length to 1.00 m and gravity to 9.8 m/s² and read the Period and Frequency readouts (expect T ≈ 2.01 s and f ≈ 0.50 Hz, since f = 1/T). Drag the length down to 0.50 m: the period falls to about 1.42 s — not half, because T grows with √L. Press Try on the Moon to rerun the same pendulum at g = 1.6 m/s²: the swing slows dramatically, and pressing Back to Earth restores it. Slide the length slowly from 0.2 m to 2.0 m and watch the orange dot travel along the Period-vs-length curve in the Data Visualization panel: the curve steepens quickly at first, then flattens — the square-root shape made visible. Use the Observation Log to record at least three (length, gravity, period) combinations before you touch the real string.

Hands-on: timing the real pendulum

Tie the bob firmly, tape the string to a table edge so it hangs clear of everything, and measure L from the pivot to the centre of the bob. Pull the bob aside about 15°, release it without pushing, and start the stopwatch as it passes the bottom. Count complete swings — one swing means back to the same point moving the same way — and stop after the 10th. Repeat three times per length, then change the length and repeat. Keep people and breakables out of the swing plane, and make sure the knot cannot slip: a flying bob is the one real hazard here.

A worked investigation

One class group recorded these times for 10 swings:

Step 1 — Periods. Divide each mean 10-swing time by 10: at 1.00 m, T = 20.1 ÷ 10 = 2.01 s. Squaring gives T² = 4.04 s², and likewise down the table.

Step 2 — Gradient. Plot T² (vertical) against L (horizontal). The points lie on a straight line through the origin, confirming T² ∝ L. Gradient = (4.04 − 1.01) ÷ (1.00 − 0.25) = 3.03 ÷ 0.75 = 4.04 s²/m.

Step 3 — Extract g. Since gradient = 4π²/g, g = 4π² ÷ gradient = 39.48 ÷ 4.04 = 9.8 m/s² (9.77 before rounding) — within 0.4% of the accepted 9.81 m/s², from string and a stopwatch.

Step 4 — Check a prediction. The formula predicts the Moon period for a 1.00 m pendulum: T = 2π√(1.00/1.6) = 4.97 s. The sim's Moon setting gives just under 5 s — model and measurement agree.

Conclusion and evaluation

The data support the theory on every point: T² is proportional to L, the line passes through the origin, and the extracted g is within half a percent of the accepted value. The main uncertainty is timing: ±0.1 s on a 20.1 s total is ±0.5%, which becomes about ±1% in T² and so roughly ±0.15 m/s² in g. Length measurement adds a little more (the bob's centre is easy to misjudge by ±0.5 cm, which is ±0.5% at 1.00 m but ±2% at 0.25 m — one reason the shortest length is the least reliable point). Improvements: time 20 swings instead of 10, measure length with the bob hanging under tension, and use a fiducial mark at the bottom of the swing for sharper start/stop judgement. A systematic trap to avoid: at release angles above ~20° the true period lengthens slightly, so a group that swings wide will consistently over-measure T.

Safety

Tie and check the knot before every set of trials, swing the pendulum in a clear space away from faces and glass, and never use a bob so heavy that a slipped string could injure someone — a bunch of keys or a metal nut is plenty.

Common mistakes

• Measuring L to the top of the bob instead of its centre — this shortens every length by the bob's radius and bends your graph.
• Counting the release as swing 1, so you actually time 9 swings and every period comes out 10% small.
• Pushing the bob at release, which adds energy and can make the swing elliptical.
• Using a big release angle "to see it better" — beyond about 15–20° the small-angle formula no longer holds.
• Stopping the watch when the bob reaches the side rather than completing the swing at the same point and direction.

Exam Experience

Question 1 — calculation [4 marks]. A student times 20 complete swings of a pendulum of length 0.64 m as 32.1 s. Calculate the period, and use it to find a value for g.

Scoring guide:

• Period T = 32.1 ÷ 20 = 1.605 s [1]
• T² = 2.58 s² [1]
• g = 4π²L/T² = 39.48 × 0.64 ÷ 2.58 [1]
• g = 9.8 m/s², matching the accepted value [1]

Question 2 — evaluation and design [4 marks]. A student claims that a heavier pendulum bob swings more slowly "because gravity has more to pull". Design a fair test of whether bob mass affects the period, and state how you would decide from the data whether any effect is real.

Scoring guide:

• Independent variable: bob mass (e.g. 50, 100, 200 g); dependent variable: period [1]
• Controls held fixed: string length (measured to bob centre), release angle, same string/pivot [1]
• Times many swings (e.g. 20) and repeats each mass at least three times, averaging [1]
• Decision rule: an effect is real only if the difference between mass means exceeds the spread (range) within repeats at one mass [1]

Multiple choice 1. The string length of a pendulum is doubled. Its period:

• A. Doubles
• B. Increases by a factor of √2 (about 1.41)
• C. Increases by a factor of 4
• D. Stays the same

Answer: B. T ∝ √L, so doubling L multiplies T by √2 ≈ 1.41 — exactly what the sim's curve shows between 0.50 m and 1.00 m.

Multiple choice 2. Timing 10 swings and dividing by 10 gives a better period than timing 1 swing because:

• A. The pendulum speeds up over the first 10 swings
• B. The stopwatch only works accurately above 10 s
• C. The fixed reaction-time error is shared across 10 periods, so the error per period is ten times smaller
• D. Averaging removes the effect of gravity

Answer: C. Your ±0.1 s reaction error is roughly constant; over 10 swings it contributes only ±0.01 s to each period.

Going deeper: the small-angle secret

The formula T = 2π√(L/g) is an *approximation* — it assumes the swing angle stays small, so that the restoring force stays proportional to displacement. Open the swing wide and the true period lengthens: to a good approximation T ≈ T₀ × (1 + θ₀²/16), with θ₀ the release angle in radians. At θ₀ = 30° = 0.524 rad, the correction is 1 + 0.274/16 = 1.017 — a 1.7% longer period. On a 2.01 s period that is 0.03 s per swing, or 0.3 s across ten swings: measurable with your stopwatch if you are careless about release angle, invisible if you keep every release under about 10°. This is why the controls table insists on a fixed, small release angle — it is not pedantry, it is a 2% effect wearing a disguise.

Two more refinements separate careful work from lucky work. Time the swings as they pass the bottom of the arc, not the extremes: at the bottom the bob moves fastest, so the instant of crossing a marker is sharpest and your timing judgement error smallest — the same reason race officials time at a line, not at a turnaround. And use a fiducial marker (a card with a line) behind the lowest point so "crossing" means the same thing on swing 1 and swing 10. With those two habits, ten-swing timing error drops towards ±0.05 s total, and your g estimate tightens to well under 1% — research-grade care from a piece of string.