The Foucault pendulum demonstrates fundamental physics principles about rotating reference frames and inertia.
INERTIAL vs NON-INERTIAL REFERENCE FRAMES
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Inertial Frame: A frame where Newton's laws work without modification
(approximately: distant stars, deep space)
Non-Inertial Frame: A frame that's accelerating or rotating
(Earth's surface, a merry-go-round)
The pendulum "knows" about the inertial frame because inertia
is defined relative to it. The pendulum resists changes to its
swing plane in the inertial frame, not in Earth's frame.
ANGULAR MOMENTUM CONSERVATION
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The pendulum bob has angular momentum:
L = r × p = r × mv
This angular momentum is conserved relative to fixed space.
Earth's rotation cannot affect it (ignoring friction).
THE CORIOLIS FORCE
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In a rotating frame, moving objects experience a "fictitious" force:
F_coriolis = -2m(Ω × v)
Where:
• m = mass
• Ω = Earth's angular velocity vector
• v = velocity in the rotating frame
This force is perpendicular to the velocity, causing curved paths.
• The Coriolis effect causes the precession of a Foucault pendulum.
• Inertia keeps the pendulum swinging in a fixed plane relative to the stars.
• The formula T = 24/sin(φ) gives the precession period in hours at latitude φ.
• At 30° latitude, the precession period is 48 hours (24/0.5).
• The pendulum demonstrates that Earth is a non-inertial reference frame.
• Foucault pendulums are driven mechanisms - they need energy input to keep swinging.
• Without driving, air resistance would stop the pendulum within hours.
• The bob must be heavy (often hundreds of pounds) to minimize disturbances.
• Longer pendulum wires give longer swing periods, making observation easier.
• The Panthéon pendulum completes one swing cycle every 16.4 seconds.
PRECESSION RATE DERIVATION
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Earth's angular velocity: Ω = 2π / 24h = 7.27 × 10⁻⁵ rad/s
At latitude φ, the vertical component of Ω is:
Ω_vertical = Ω × sin(φ)
The pendulum precesses at this rate, giving:
Precession rate = 360° × sin(φ) / 24h
= 15° × sin(φ) per hour
Full rotation period:
T = 360° / (15° × sin(φ)) = 24h / sin(φ)
PERIOD OF SWING
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The pendulum's swing period (back and forth) is:
T_swing = 2π × √(L/g)
For a 67-meter pendulum (like the Panthéon):
T_swing = 2π × √(67/9.8) ≈ 16.4 seconds