Uniform circular motion: Conical pendulum lab report

1. Title and aim

Uniform circular motion and centripetal dynamics of a conical pendulum. Investigate the dynamics of uniform circular motion for a conical pendulum of fixed suspension length L = 1.000 m across varying apex half-angles θ, measuring the period of revolution T, verifying the dynamic governing relation T² = (4π² / g) · (L cos θ), and extracting the local acceleration due to gravity g from the linear regression slope.

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 9.8038 m s⁻², differing by 0.029% 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

• Rigid Overhead Swivel Pivot and Low-Friction Bearing: A sturdy ceiling/gantry clamp supporting a miniature ball-bearing swivel that permits smooth 360° circular motion without twisting the cord.
• Dense Spherical Brass Bob and Monofilament Cord: A precision-machined 200.0 g spherical brass bob suspended by a lightweight, low-stretch monofilament cord of calibrated length L = 1.000 m.
• Concentric Circular Floor Scale and Radius Grid: A graduated circular mat marked with concentric circles from r = 0.20 m to 0.90 m to visually guide and verify the circular orbital radius r = L sin(θ).
• Optical Infrared Photogate and Digital Revolution Counter: An infrared beam sensor mounted tangent to the orbital path, coupled to a digital quartz chronometer that records 20 complete revolutions to 0.01 s.
• Fixed suspension cord length L = 1.000 m; bob mass m = 0.200 kg (200.0 g).
• Digital photoelectric revolution counter timing 20 consecutive revolutions to 0.01 s.
• Reference gravitational acceleration g = 9.80665 m s⁻².

4. Theory and graph choice

The conical pendulum provides an exceptional experimental platform for investigating the fundamental principles of uniform circular motion, centripetal acceleration, and dynamic equilibrium. First analyzed mathematically by Dutch polymath Christiaan Huygens in his 1673 treatise Horologium Oscillatorium, the conical pendulum consists of a dense mass suspended from a fixed overhead pivot by a lightweight cord of constant length L, revolving with uniform angular velocity in a horizontal circle rather than swinging back and forth in a plane. When the bob moves steadily at an apex half-angle θ to the vertical, its trajectory sweeps out the surface of a right circular cone. Because the motion takes place entirely in a horizontal plane with zero vertical acceleration, the upward vertical component of the suspension string tension T cos(θ) exactly balances the downward gravitational weight mg. Simultaneously, the inward horizontal radial component of the tension T sin(θ) supplies the necessary centripetal force m ω² r required to hold the bob in a circular path of radius r = L sin(θ). Combining these two dynamic conditions eliminates the string tension, yielding the classic relationship tan(θ) = ω² r / g. Substituting r = L sin(θ) and expressing angular velocity in terms of the orbital period T = 2π / ω reveals a remarkable property: the period of a conical pendulum is given by T = 2π sqrt[(L cos θ) / g]. The quantity h = L cos(θ) represents the vertical axial depth of the cone—the vertical distance from the overhead pivot bearing down to the horizontal plane of rotation. Consequently, the period of a conical pendulum of length L inclined at angle θ is mathematically identical to the small-amplitude period of a simple planar pendulum whose effective length equals the vertical depth h. Plotting the square of the period T² against the vertical depth h linearizes the governing relationship into the standard form T² = (4π² / g) · h. By performing an ordinary least-squares regression over varying apex angles, students can directly verify centripetal dynamics and extract the local acceleration due to gravity g from the slope S = 4π² / 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 s² m⁻¹.

5. Method

1. Measure the suspension cord length L with a precision steel rule from the swivel bearing pivot center to the center of mass of the spherical brass bob. Adjust the clamping collar until L equals exactly 1.000 m (1000.0 mm). Verify that the pivot mount is rigid and plumb.
2. Allow the pendulum bob to come to complete rest at its equilibrium position. Center the zero point of the concentric circular floor grid directly beneath the resting bob using a plumb line. Tape the floor grid securely to the floor to prevent shifting during testing.
3. Draw the bob outward to the radius corresponding to the target apex half-angle θ = 20.0° (r = L sin 20° = 0.342 m). Give the bob a smooth tangential impulse at right angles to the cord, launching it into a steady horizontal circular orbit directly over the reference circle.
4. Position the optical photogate sensor so that the equator of the orbiting bob cleanly interrupts the infrared beam once per revolution. Arm the digital counter and record the total elapsed time for 20 complete revolutions (20T) to 0.01 s. Repeat three times to confirm period stability.
5. Repeat the launch and timing procedure for apex angles of 30.0°, 40.0°, 50.0°, and 60.0°, corresponding to vertical depths h = 0.866 m, 0.766 m, 0.643 m, and 0.500 m. Note how the period decreases systematically as the cone becomes flatter.
6. For each angle, compute the mean period T = (20T) / 20, period squared T², and vertical axial depth h = L cos θ. Construct a graph of T² versus h. Perform an ordinary least-squares regression to extract the slope S, and compute the experimental gravity as g = 4π² / 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 = 0.0292%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 2.6284e-07; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 8.4246e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 6.3971e-04.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.00205 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.

Timing resolution over twenty revolutions (±0.01 s), apex angle determination and floor circle parallax (±0.5°), aerodynamic drag on the cord and spherical bob causing orbital decay, and slight elliptical orbital precession (eccentricity) contribute uncertainty. The OLS inverse-slope standard error reflects random period scatter across angle stations, but does not include shared cord stretch under tension or suspension pivot bearing friction.

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 that the suspension cord length L is measured precisely from the swivel bearing pivot axis to the center of mass of the spherical brass bob.
• Ensure the bob moves in a stable, true horizontal circle without elliptical wobble or vertical oscillation before beginning timer recording.
• Check that the overhead swivel bearing rotates with negligible friction to prevent twisting and knotting of the suspension cord.
• Shield the circular flight path from ventilation drafts or air conditioner currents to eliminate aerodynamic orbital distortion.

This virtual experiment assumes a point-mass bob, a massless and perfectly inextensible suspension cord, zero pivot bearing friction, and zero aerodynamic drag. It assumes a perfectly circular orbit without elliptical precession or nutation. Generated data represent ideal uniform circular motion; physical experiments will exhibit small random variations from manual launch eccentricity. Theoretical period of revolution: T = 2π sqrt(h / g), where h = L cos(θ). Linearized relation: T² = (4π² / g) · h. The ordinary least-squares regression extracts the slope S = 4π² / g, from which gravity is recovered as g = 4π² / 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 9.8038 m s⁻². A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.