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.
Physics / Uniform circular motion: Conical pendulum
FREE TO EXPLOREEXPERIMENT 22 · PHYSICS LABORATORY
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. Explore a virtual apparatus, collect readings and analyse a graph.
CHANGE · OBSERVE · MEASURE
Your virtual workbench.
TIME FOR 20 REVOLUTIONS—
MEASUREMENT STATUS
Ready for a virtual observation.
Adjust the apparatus, then take a reading.
Timing is generated from the model; a short on-screen trial represents a longer physical interval. Animation is illustrative.
Simulation assumptions and limits
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.
Know your apparatus.
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.
FOLLOW THE METHOD
A reason behind every step.
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. Suggested independent settings: 20.0, 30.0, 40.0, 50.0, 60.0 °.
Suspension Length Metrology and Pivot Leveling
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.
Why this step matters
Cord length L directly governs the geometry of the cone; any flexing or slipping at the pivot alters the effective depth h = L cos θ.
Floor Scale Centering and Plumb Alignment
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.
Why this step matters
Accurate radial alignment guarantees that circular orbits maintain a constant radius r centered on the vertical axis.
Orbital Orbit Initiation and Apex Angle Stabilization
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.
Why this step matters
A purely tangential launch prevents radial wobbling or elliptical precession, establishing true uniform circular motion.
Photogate Synchronization and 20-Revolution Timing
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.
Why this step matters
Timing 20 consecutive revolutions reduces the fractional uncertainty of individual gate trigger latency by a factor of 20.
Systematic Angular Sweep Across Five Apex Angles
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.
Why this step matters
Sweeping over a wide angular range from 20° to 60° maps the relationship between period squared and vertical depth across a 2:1 depth variation.
Coordinate Transformation, Regression, and Gravity Determination
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.
Why this step matters
Linear regression over the transformed coordinate h filters experimental noise and verifies the theoretical prediction of centripetal dynamics.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into a result.
Points are your observations. The fit uses SI x and y values before display scaling.
LINEAR FIT · y = mx + b
T² = (4π² / g) · h ⇔ g = 4π² / S
| # | Apex half-angle θ (°) | Time for 20 revolutions (s) | Vertical axial depth h = L cos(θ) (m) | Period squared T² (s²) | Remove |
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