### A Ring of Water Shows the Earth’s Spin

Around 1913, while still an undergraduate, American physicist Arthur Compton described an experiment to demonstrate the rotation of the Earth using a simple laboratory apparatus.

Compton (1892–1962) won the Nobel Prize in Physics in 1927 for his work on scattering of EM radiation. This phenomenon, now called the Compton effect, confirmed the particle nature of light.

Compton’s Generator

Compton’s apparatus comprises a hollow circular tube filled with water [the figure above is from Compton, 1915]. The water is allowed to reach equilibrium such that it has no motion relative to the tube. The water and tube are stationary relative to Earth, but are rotating in an inertial frame. When the tube is flipped over through ${180^\circ}$, the water continues to rotate in the original direction, so there is a circulation relative to the tube. The speed of the water is small, but is sufficient to be measurable in the lab and gives a value for the Earth’s rotational speed.

Compton’s method is more versatile than Foucault’s pendulum: it enables the latitude to be determined, since ${\boldsymbol{\Omega}}$, the vector angular velocity of Earth, can be measured, giving the latitude ${\phi}$ in addition to ${\Omega}$. The flip can be about any axis, so that all three components of the Earth’s vector rotational speed can be ascertained.

Mean Drift Speed

We use a geographical frame of reference, with the ${x}$-axis eastward, ${y}$-axis northward and ${z}$-axis vertically upward. The unit orthogonal triad is ${(\mathbf{i, j, k})}$. A straightforward mechanical analysis (outlined below) shows that the mean speed of the flow around the tube after flipping about the east-west axis is ${\bar u = fR}$, where ${f = 2\Omega\sin\phi}$ is the Coriolis parameter and ${R}$ is the radius of the tube. In middle latitudes ${f\approx 10^{-4}\,s^{-1}}$ so if the tube has a one metre radius, ${\bar u\approx 10^{-4}\,}$m/s or one millimetre every ten seconds. A microscope is used to follow tracers in the fluid and to measure the speed.

If the flip is about the north-south axis, the resulting mean velocity is ${\bar v = 2\Omega R\cos\phi}$. Thus, ${\bar u / \bar v = \tan\phi}$ which yields the latitude. Thus, from simple measurements, both the latitude and Earth rotation rate can be deduced. This contrasts with Foucault’s pendulum, where only one of these quantities can be inferred.

We estimated that ${\bar u\approx 10^{-4}\,}$m/s or one millimetre every ten seconds. However, in Compton’s apparatus, the glass section of tube where the flow was observed was reduced in diameter to enhance the drift speed. He noted (in Compton, 1915) that:

“With the apparatus here described the motion was usually about as fast as that of the minute hand of a watch, and could easily be seen through the microscope.”

Outline of the Analysis

We take the Earth’s rotation to be about the polar axis, although this is not an essential assumption. We analyse the case where the tube is rotated about the east-west axis (the case of rotation about the north-south axis is similar). The angular velocity of Earth is

$\displaystyle \boldsymbol{\Omega} = (0, \Omega\cos\phi, \Omega\sin\phi) = \Omega\cos\phi\mathbf{j} + \Omega\sin\phi\mathbf{k} \,.$

The Coriolis acceleration in the frame of reference fixed to the Earth and rotating with it is

$\displaystyle \mathbf{a} = - 2\boldsymbol{\Omega\,\times}\mathbf{V} \,.$

We consider a parcel of fluid in the tube, located by an angle ${\alpha}$ relative to north, and assume the tube is rotated through an angle ${\beta}$ from horizontal about the ${\mathbf{i}}$-axis. Then the acceleration of the parcel due to the Coriolis force is

$\displaystyle 2 R \omega \Omega \cos\alpha\cos(\phi+\beta)$

where ${\omega = \dot\beta}$ is the rotation rate of the tube. To get the component of acceleration tangential to the tube, we multiply by another factor ${\cos\alpha}$. If we now integrate over the duration of the flip (in effect, over ${\beta}$) the quantity ${\omega}$ disappears from the resulting expression. We then average around the tube (over ${\alpha}$) to get the mean drift speed around the tube, ${\bar u = fR = 2 \Omega R \sin\phi}$. This gives us ${\Omega}$ if the latitude ${\phi}$ is known.

Sources

${\bullet}$ Compton, A.~H., 1913: A laboratory method of demonstrating the earth’s rotation. Science, 37, 960, 803–806. DOI: 10.1126/science.37.960.803

${\bullet}$ Compton, Arthur Holly, 1915: Watching the Earth revolve. Scientific American Supplement, No.~2047, 196-197.

${\bullet}$ Hand, Louis N. and Janet D. Finch, 1998: Analytical Mechanics. Cambridge Univ. Press.

${\bullet}$ Siboni, S., 2014: The Compton generator revisited, Eur. J. Phys., 35 (5), 055014. doi = 10.1088/0143-0807/35/5/055014.