Physics · Mechanics · Grade 10

Non-inertial frames of reference

Why does the plane of a pendulum nobody touches turn over several hours?

The situation

A 67-metre wire hangs from the dome of the Panthéon in Paris, with a heavy bob on the end swinging slowly. As hours pass, the plane of oscillation slowly turns, leaving a star-shaped trace in the sand — as though someone were turning the bob with an invisible hand. In fact nothing turns the bob: the floor beneath it — the Earth itself — is rotating. How fast will this turning be in Tashkent, and what about at the equator?

Topic

Non-inertial frames of reference. Foucault pendulum and the Coriolis force

Key ideas

Coriolis force
Foucault pendulum
Rotating frame of reference

Formulas

{\vec{a}}_{K} = - 2 \vec{Ω} \times \vec{v}

Coriolis acceleration: acts on a body moving in a rotating frame of reference.

\ddot{x} = - ω_{0}^{2} x + 2 Ω \sin φ \dot{y}, \ddot{y} = - ω_{0}^{2} y - 2 Ω \sin φ \dot{x}

Foucault pendulum (small oscillations, $x$ — east, $y$ — north, $ω_{0}^{2} = g / L$ ).

Ω_{p} = Ω \sin φ \approx {15.04}^{\circ} / h \cdot \sin φ

The rate of turning of the plane of oscillation, $φ$ is the latitude.

T_{p} = \frac{T_{⋆}}{\sin φ}, T_{⋆} = 23 h 56 min

The period of one full turn; $T_{⋆}$ is the sidereal day.

d \approx ω v t^{2}

On a carousel: the deflection, in the rotating frame, of a ball thrown for a time $t$ .

Theory

Non-inertial frame of reference

A rotating floor — the Earth or a carousel — is a non-inertial frame of reference. For Newton's laws to hold in it, inertial forces are added: the centrifugal force and the Coriolis force $- 2 m \vec{Ω} \times \vec{v}$ . The Coriolis force acts only on a moving body and is perpendicular to the velocity: for anticlockwise rotation (the northern hemisphere) it turns the body to the right of its direction of motion. In a frame moving with translational acceleration (a braking bus, a lift) the only inertial force is $- m {\vec{a}}_{0}$ ; weightlessness and overload follow from this — in detail in the “Newton's laws” experiment (the lift).

Foucault pendulum

The plane of oscillation of a pendulum stays almost unchanged relative to the stars. The vertical component of the Earth's rotation vector is $Ω \sin φ$ , so the floor turns beneath the pendulum at this angular velocity. At the pole the plane makes a full turn in one sidereal day (23 h 56 min), at the equator it does not turn at all, and in Tashkent ≈ 9.9°/h — a full turn takes ≈ 36 hours.

Baer's law: the right bank of a river

On any body moving horizontally on the Earth's surface the horizontal component of the Coriolis force, $2 m Ω v \sin φ$ , acts — the very force that turns the Foucault pendulum. River water too is pressed to the right of the direction of flow in the northern hemisphere and to the left in the southern — whichever way the flow is directed. Baer's law (named after the naturalist Karl Baer): rivers in the northern hemisphere erode the right bank more, those in the southern the left bank. The force is very small: at $v = 2$ m/s, $φ = 41 °$ , $a_{K} \approx 1.9 \cdot 10^{- 4}$ m/s², i.e. ≈ 0.002 % of $g$ ; in a river 500 m wide the water level is only ≈ 1 cm higher at the right bank. But it acts ceaselessly for hundreds and thousands of years. In real rivers the soil type and the bends of the channel often influence the shape of the banks more strongly, so the law is not visible in every river. Work it out in the “Baer's law” section of the panel.

Carousel: one motion — two pictures

For an observer standing on the ground, a thrown ball (without air resistance) flies along a straight line in the horizontal plane; the rotational velocity of the thrower is added to the ball's. The child on the carousel, though, sees the ball flying along a curved path — in the child's frame the Coriolis and centrifugal forces “work”. To hit the target you must calculate the deflection in advance and throw a little to the left: the first estimate is $α \approx arctg (ω D / v)$ .

Model and simplifications

The Foucault pendulum is solved for small oscillations ( $A ≪ L$ ) by the RK4 method; the centrifugal force is taken to be included in the gravitational force. If “Air resistance” is on, the amplitude falls by a factor of $e$ in about 4 hours (real pendulums are kept going by an electromagnetic kick) — the rate of turning does not change because of this. The trace in the sand is drawn from one extreme of each swing to the other. Time is speeded up — in the chosen mode 1 s on screen equals 1 min, 10 min or 1 hour. There is no air resistance on the carousel.

Did you know?

In 1851 Léon Foucault demonstrated the rotation of the Earth for everyone to see, using a 28 kg pendulum hung from a 67 m wire in the Panthéon in Paris.
In the northern hemisphere cyclones rotate anticlockwise and in the southern hemisphere clockwise — this too is a result of the Coriolis force.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Momentum and its conservation — Grade 10
Banked turn — Grade 10
Centre of gravity and equilibrium — Grade 10
Aeroplane wing — Grade 10
Dynamics of a rigid body — Grade 10
Inclined plane and Atwood machine — Grade 10
Free fall and air — Grade 10
Jet propulsion and the rocket — Grade 10
Viscosity and flow — Grade 10
Gyroscope and precession — Year 1