Acceleration due to gravity by free fall lab report

1. Title and aim

Uniformly accelerated motion and determination of acceleration due to gravity by free fall timing. Determine the acceleration due to terrestrial gravity g by measuring transit elapsed times of a dense metallic plummet released from rest across varying vertical drop heights h using precision spark chronometry, and verify the classical quadratic kinematic relation h = 1/2 g t² through ordinary least-squares linear regression.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Acceleration due to gravity of 19.6153 m s⁻², differing by 0.010% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Precision Vertical Drop Column and Millimeter Scale: Provides a rigid, vibration-isolated vertical structural guide 1.5 m in height, equipped with a graduated millimeter rule and dual-axis leveling tripod to maintain exact plumb alignment.
• Electromagnetic Plummet Release Solenoid: Suspends the ferromagnetic plummet prior to release and rapidly demagnetizes via an inductive quench diode upon circuit interruption to ensure zero initial velocity without mechanical disturbance.
• Precision Digital Microsecond Chronometer: Measures the transit time interval from the instant of electrical circuit interruption to mechanical strike-plate contact with a timebase resolution of 0.1 ms (0.0001 s).
• Aerodynamic Cylindrical Steel Plummet and Cushion Trap: A dense, precision-machined steel cylinder with a conical nose designed to minimize air drag and provide a repeatable electrical/mechanical impact surface, landing in an energy-absorbing foam pad.
• Height scale calibrated with precision vertical millimeter rule; digital spark chronometer resolution 0.1 ms (0.0001 s).
• Electromagnetic bob release solenoid fitted with fast-demagnetization transient suppressor to eliminate release latency.
• Reference gravitational acceleration g = 9.80665 m s⁻².

4. Theory and graph choice

The study of bodies falling freely under the influence of Earth's gravity represents one of the foundational cornerstones of modern empirical physics. Prior to the seventeenth century, Aristotelian natural philosophy asserted that heavy objects fall inherently faster than light ones, with speed proportional to their mass. This misconception persisted for nearly two millennia until Galileo Galilei dismantled it through rigorous experimentation and deductive thought experiments. Galileo established that, in the absence of aerodynamic medium resistance, all physical bodies near the Earth's surface accelerate downwards at the exact same constant rate, regardless of their material composition, shape, or weight. Under uniform gravitational acceleration g, the velocity acquired by a falling body increases linearly with elapsed time according to v(t) = v₀ + gt, while the total vertical displacement from rest satisfies the quadratic relation h(t) = 1/2 gt². Measuring this motion with precision requires overcoming the physiological limitations of human reflex timing: human visual-motor reaction time averages approximately 200 milliseconds, introducing unacceptably large percentage errors when observing drop times of less than half a second. To achieve laboratory accuracy, experimental physicists developed automated electronic drop towers, spark gap recording timers, and optical photogate chronometers capable of microsecond time resolution. By releasing a streamlined steel plummet from an electromagnetic holder and recording the elapsed fall time across a calibrated series of heights, students directly capture the parabolic trajectory of free fall. Plotting the drop height h as a function of the square of the fall time t² linearizes the theoretical curve, yielding a straight line whose slope is precisely one-half the gravitational acceleration. Linear regression over multiple independent heights filters out random measurement scatter and provides a robust, unbiased determination of local g.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units m s⁻².

5. Method

1. Inspect the vertical column using a dual-axis spirit level positioned against the guide rail. Adjust the leveling foot screws at the tripod base until the spirit level bubbles are perfectly centered along two orthogonal axes. Ensure the column is clamped securely to prevent wobble during plummet impact.
2. Energize the release solenoid with the designated DC holding current. Place the ferromagnetic apex of the steel plummet against the pole piece. Verify that the plummet hangs freely and centralizes precisely over the impact strike pad. Test the quick-release switch to confirm that residual core magnetization does not cause mechanical sticking upon opening.
3. Lower the plummet manually until its bottom edge touches the top surface of the mechanical strike pad. Check the height reading on the millimeter vernier carriage. Record any index zero-offset correction so that all subsequent height settings reflect the true transit distance from the bottom of the bob to the strike plate.
4. Position the release carriage at the initial test height of 0.20 m (200 mm) using the vernier index mark. Lock the clamping screw firmly. Reset the digital microsecond chronometer display to 0.0000 s. Suspend the steel plummet from the electromagnet and confirm the readiness status indicator.
5. Depress the electronic release trigger to de-energize the electromagnet simultaneously with the starting of the digital timer. The plummet falls freely under gravity until its base strikes the landing pad, instantly opening the timer gate circuit. Record the elapsed fall time t from the digital counter to 0.0001 s. Repeat for three successive trials to confirm repeatability.
6. Repeat the release and timing sequence at heights of 0.40 m, 0.60 m, 0.80 m, and 1.00 m. Calculate the square of the mean fall time t² for each height station. Construct a graph of h versus t², evaluate the slope S via ordinary least squares, and compute the experimental acceleration due to gravity as g = 2 · S.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0102%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 1.1750e-08; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 4.8525e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 6.5642e-05.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.00194 m s⁻². This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Chronometer timebase jitter and trigger contact latency (±0.0001 s), millimeter scale parallax and zero-reference indexing error (±0.5 mm), residual electromagnetic core remanence delay, aerodynamic form drag at higher velocities, and ambient room air currents contribute uncertainty. The OLS regression standard error reflects random timing scatter across stations, but does not capture shared scale zero offsets or geographic latitude differences from the 9.80665 m s⁻² standard reference.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Verify the absolute vertical plumb of the drop tower using a dual-axis spirit level to ensure the plummet falls completely freely without rubbing against guide rails or photogate brackets.
• Ensure the release electromagnet coil is fitted with a fast-quench flyback diode to prevent inductive arcing and residual magnetic holding delay upon circuit interruption.
• Shield the drop path from ambient drafts, ventilation air streams, and air conditioner currents to eliminate lateral aerodynamic disturbances.
• Maintain a resilient, energy-absorbing landing cushion at the base of the column to catch the plummet without mechanical rebound or deformation of its precision reference edges.

This virtual experiment assumes uniform one-dimensional gravitational acceleration g = 9.80665 m s⁻² without aerodynamic drag, air buoyancy, or mechanical friction. It neglects electromagnetic release hysteresis delay, photogate optical beam transit latency, and finite bob radius offsets. The generated fall times represent an idealized vacuum trajectory; virtual data cannot reproduce physical air turbulence or building floor vibrations. Theoretical fall time: t = sqrt(2h / g). Linearized relation: h = (1/2 g) · t². The ordinary least-squares fit extracts the slope S = 1/2 g, from which g = 2 · S. Reference standard gravity g = 9.80665 m s⁻².

12. Conclusion

The worked example illustrates how to obtain Acceleration due to gravity from measurements, graph variables and a fitted coefficient. The estimate is 19.6153 m s⁻². A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.