A conical pendulum has a bob moving at constant speed in a horizontal circle while its cord sweeps out a cone. The cord’s vertical tension component balances weight; its horizontal component supplies the inward resultant. This calculator finds the geometry for a chosen revolution period, angle from vertical and gravitational acceleration.
Conical Pendulum Interactive Calculator
Choose a revolution period, angle from vertical and gravity. Calculate the required vertical drop, cord length, orbit radius and RPM. The bob follows the resulting horizontal circle at the selected period.
Equation Used
The article period relation says a conical pendulum depends on vertical height h, not directly on cord length L. This calculator inverts that relation to find the height needed for a target period, then uses the selected cone angle to calculate cord length and orbit radius.
- Ideal steady horizontal circular motion, with constant angle from vertical.
- Massless, inextensible cord measured from fixed pivot to bob center.
- Point-mass bob for the calculation; sphere size in the drawing is illustrative.
- No air drag, friction, transient start-up or driving mechanism is modeled.
The view automatically fits the calculated geometry. Use the labelled dimensions and scale bar when comparing physical sizes. Support and bob sizes are illustrative.
Constant height and circular motion
The support point is fixed. For the ideal steady motion shown here, the bob remains a vertical distance h below it and moves around a circle of radius r. The cord length L is constant, with h = L cos(theta) and r = L sin(theta).
The main view shows the bob traveling around the circle. The radial section below it shows the actual angle from vertical, without perspective foreshortening. The dashed circular path is a reference, not a track or a mechanical guide.
The separate force diagram shows tension along the cord and weight vertically down. Their vector sum points inward. The dashed resultant is not an additional physical force. With no resistance in this ideal model, no drive is needed to maintain the initialized steady motion. A real demonstration loses energy and may require a drive; its details are outside this calculation.
Studying uniform circular motion
The conical pendulum is a useful teaching example because the bob has no vertical acceleration while continuously accelerating inward. University physics treatments use it to connect circular motion, force resolution and geometry.
Rotating flyball devices and suspended rotating rides involve related ideas, but additional constraints and forces can change their equations. This free-bob calculator does not size a clock drive, governor, amusement ride or powered installation.
Geometry from the desired period
Let P be the time in seconds for one complete revolution, g gravitational acceleration and theta the cord angle from vertical. Vertical balance gives S cos(theta) = mg. The horizontal equation is S sin(theta) = m omega² r, where S is tension and omega = 2 pi/P.
Using r = L sin(theta), these equations give P = 2 pi sqrt(L cos(theta)/g) = 2 pi sqrt(h/g). Inverting the period equation gives h = g(P/(2 pi))². Then L = h/cos(theta), r = h tan(theta), and RPM = 60/P.
Mass cancels from the geometry calculation. The force diagram therefore reports ratios to weight: S/(mg) = 1/cos(theta) and net inward force/(mg) = tan(theta). It does not assign a load rating to a cord or support.
Example: a three-second revolution
For P = 3 s, theta = 25° and g = 9.80665 m/s², the vertical drop is 9.80665(3/(2 pi))² = 2.236 m. The required cord length is 2.236/cos(25°) = 2.467 m, and the orbit radius is 2.236 tan(25°) = 1.043 m.
The rotation rate is 60/3 = 20 RPM. Tension is approximately 1.103 times the bob’s weight; the inward resultant is 0.466 times its weight.
If the target period doubles while gravity and angle stay unchanged, h, L and r all become four times larger. The auto-fitted view preserves the shape, so compare the dimension labels and scale bar to see the physical size change.
Fixed period versus fixed cord length
This calculator solves for cord length from a specified period. At fixed period and gravity, h remains fixed when angle changes; L and r change. That does not mean an existing pendulum keeps the same period when its angle changes.
For a fixed physical cord length, P = 2 pi sqrt(L cos(theta)/g), so increasing angle shortens the period. The small-angle limit approaches 2 pi sqrt(L/g). At exactly zero angle the radius is zero and there is no finite circular orbit; the calculator starts at 1°.
The model excludes drag, pivot losses, cord stretch, support motion and the transition into steady motion. It provides no universal clock accuracy, surface-finish requirement or guaranteed useful angle range.
Conical pendulum questions
Why does the vertical drop stay the same when I change angle?
You have fixed the target period and gravity. Those determine h; the calculator changes L and r to accommodate the new angle.
Does bob mass change the period?
Not in the ideal point-mass model. Mass cancels from the equation, although actual tension scales with mass.
Is the dashed inward arrow another force?
No. It is the resultant of tension and weight. Adding it again as a third force would double-count the inward effect.
Why does the picture stay about the same size when the period changes?
The view fits the calculated dimensions to keep the bob visible. The labels and scale bar show the physical size.
Does this simulate starting or damping?
No. It shows an already-established steady circular orbit. Start-up, resistance and drive behavior require a different dynamic model.
Physics references
R. Field, University of Florida PHY 2053, September 20, 2011, page 4: Conical Pendulum. Force balance and the period relation for a bob moving in a horizontal circle.
University of Nevada, Las Vegas: Conical Pendulum. Geometry, tension components and the distinction between the inward resultant and an additional force.
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