Circular motion · A steady cone

Conical Pendulum

Why does a wider cone complete a turn sooner? A wider steady cone needs a higher rotation rate at the same cord length and gravity. The cord must still support the weight, and its growing horizontal component turns the bob more strongly. The increase in speed more than offsets the longer circular path, so one turn takes less time.

Grab the blue bob and widen its horizontal orbit. Watch the cord trace a cone, then compare its radius, period and tension.

Orbit radius $0$

One turn $0$

Cord tension $0$

Cone angle $0$

Inward net force $0$

Azimuth rate $0$

Grab a blue ball

Widen the cone.

Cone angle

\theta

50\,{}^\circ

Cord length

L

1\,\mathrm m

Gravity

g

9.81\,\frac{m}{s^2}

Mass of each bob

m

0.5\,\mathrm{kg}

Positions use metres, forces newtons and time seconds. Right, up and toward you are positive. Azimuth increases from the right toward you; its rate is positive in the calculations and CSV.

The cord's vertical force balances the bob's weight. Its horizontal force turns the velocity inward. A wider steady cone needs a higher rotation rate at the same length and gravity.

The initial tangential velocity is chosen to give a steady circular orbit. Dragging the bob or changing a control prepares another such orbit. A general spherical-pendulum swing, nutation and preparation work are not simulated.

The blue line is a short motion trail. The separate curve shows the computed steady-orbit relation over the control range, with the current setting marked. CSV contains one complete calculated orbit; no measured acquisition is implied.

Bob mass changes tension and inward force, but not the steady angle-speed relation. The cord is massless, inextensible and always taut. Air drag, bearing friction, elasticity and the bob's rotational inertia are omitted.

Depth is physical: the bob travels around the vertical axis. The stand, bearing and bob radius are schematic. Cord length is measured to the bob's centre; the visible blue sphere represents a point mass.

The bob follows an exact steady horizontal circle on a massless, inextensible cord in uniform gravity. Changing a control prepares a new steady orbit. Air drag, transients, bob size and cord elasticity are omitted; hardware thickness is schematic.

Start with the essentials

Focus question: At fixed cord length, does a wider orbit take longer to complete?
One-sentence intuition: A wider steady cone needs more inward force per unit weight. The required increase in speed more than offsets the longer circular path, so its period is shorter.

Core mathematical model

The two components of tension

T\cos\theta=mg,\qquad T\sin\theta=m\omega^2r

There is no vertical acceleration. The horizontal component supplies the inward acceleration.

Radius, angular speed and period

r=L\sin\theta,\quad \omega=\sqrt{\frac{g}{L\cos\theta}},\quad P=2\pi\sqrt{\frac{L\cos\theta}{g}}

At the same length and gravity, the cosine decreases as the cone opens. The period therefore decreases.

Common difficulties

The bob has no outward real force

Typical misconceptionAn outward centrifugal force balances the inward tension in the laboratory frame.

Better mental modelIn the laboratory frame the real horizontal force points inward and continuously turns the velocity. Centrifugal force belongs to a rotating-frame description.

New settings prepare a new steady motion

Typical misconceptionThe instantaneous change of angle is a prediction of how the real apparatus accelerates.

Better mental modelThe model compares exact steady configurations. It does not integrate the forces during preparation, so no settling time or transient power is reported.

Run the experiment

01
Grab the ball
Pause, then drag a blue ball outward or upward in the main scene.
What to observe: The cone opens, the radius grows and the calculated period falls.
02
Compare narrow and wide cones
Select Narrow cone, then Wide cone, and compare a few revolutions.
What to observe: The wide cone turns faster even though its circular path is longer.
03
Change gravity and mass
Try Moon gravity, then change Mass in the collapsed group.
What to observe: Weaker gravity slows a given cone. More mass increases tension but leaves its period unchanged.

Scientific sources

Richard Fitzpatrick · The conical pendulum · J. C. Maxwell · On Governors (1868) · TecQuipment · Flyballs and sleeve