Question

The Compton generator is a beautiful demonstration of the Coriolis force due to the earth's rotation, invented by the American physicist A. H. Compton (1892-1962, best known as author of the Compton effect) while he was still an undergraduate. A narrow glass tube in the shape of a torus or ring (radius \(R\) of the ring \(\gg\) radius of the tube) is filled with water, plus some dust particles to let one see any motion of the water. The ring and water are initially stationary and horizontal, but the ring is then spun through \(180^{\circ}\) about its east-west diameter. Explain why this should cause the water to move around the tube. Show that the speed of the water just after the \(180^{\circ}\) turn should be \(2 \Omega R \cos \theta,\) where \(\Omega\) is the earth's angular velocity, and \(\theta\) is the colatitude of the experiment. What would this speed be if \(R \approx 1 \mathrm{m}\) and \(\theta=40^{\circ} ?\) Compton measured this speed with a microscope and got agreement within 3\%.

Short answer

The water speed is approximately \(1.12 \times 10^{-4}\) m/s.

Step-by-step solution

Understanding the Problem

We are considering a toroidal tube filled with water, spun about its east-west diameter by 180 degrees. The goal is to determine the speed of the water inside the tube caused by the Coriolis effect due to Earth's rotation.

Identify Forces

The Coriolis force acts on a mass moving in a rotating frame, such as the Earth. For a mass moving with velocity \( v \), the Coriolis force \( F_c \) is given by \( F_c = 2 m v \Omega \sin \phi \), where \( m \) is the mass, \( \Omega \) is the Earth's angular velocity, and \( \phi \) is the latitude.

Determine Latitudinal Component

Because the ring is turned about its east-west axis, the component of the Coriolis force that affects the water is along the north-south direction, which corresponds to \( 2 m v \Omega \cos \theta \), using colatitude \( \theta = 90^{\circ} - \phi \).

Calculate Motion Due to Coriolis Force

When the ring is flipped, the Coriolis force causes a relative motion in the water, as if it is moving northward. This results in a circulation around the ring. Adjusting for the tube's geometry, the effective speed \( v \) induced is \( v = 2 \Omega R \cos \theta \), accounting for the ring being parallel to the surface of the Earth.

Substitute Values for Specific Speed

Given \( R \approx 1 \) m and \( \theta = 40^{\circ} \), calculate the speed \( v \). Use \( \Omega \approx 7.2921 \times 10^{-5} \) rad/s (Earth's angular velocity). Hence, \( v = 2 \times 7.2921 \times 10^{-5} \times 1 \times \cos(40^{\circ}) \approx 1.12 \times 10^{-4} \text{ m/s} \).

Analyze Result

The calculated speed, about \(1.12 \times 10^{-4}\) m/s, is the velocity at which the water moves around the tube just after it is turned 180 degrees. This small value aligns well with experimental observations.

Key concepts

Compton generator

The Compton generator is an ingenious setup designed to illustrate the Coriolis effect—the force experienced by objects moving within a rotating frame, such as the Earth. This apparatus, devised by physicist A. H. Compton, employs a toroidal glass tube that is filled with water and small dust particles.
By rotating this tube horizontally by 180 degrees about its east-west axis, one can visually observe the motion of water within the tube induced by the rotation of the Earth.
The Coriolis force causes the water to circulate in the tube, offering a tangible demonstration of Earth's rotation's effect on moving fluids.
Such an apparatus beautifully exemplifies how the rotational dynamics of our planet can influence even simple systems, providing insight into broader geophysical phenomena.

Toroidal tube dynamics

In the context of the Compton generator, the toroidal tube serves as a crucial component. It is essentially a ring-like structure in which the radius of the ring greatly exceeds that of the tube itself. This design ensures that when the tube is rotated, there is enough room for fluid movement that can be influenced by external forces like the Coriolis effect.
As the tube is spun 180 degrees, the water inside begins to move. This is because the rotation introduces a change in frame dynamics around the torus, creating a scenario where the water experiences forces that cause it to move around the tube.
The dynamics of the fluid within the toroidal structure reveal how changes in rotational orientation can induce motion in a seemingly stationary liquid, capturing the nuanced interactions between geometry and motion in fluid mechanics.

Earth's angular velocity

Earth's angular velocity, denoted as \( \Omega \), is a fundamental parameter approximately equal to \( 7.2921 \times 10^{-5} \) radians per second. It defines the rate at which Earth rotates around its axis and is essential to calculating the Coriolis force's effect on a moving object.
In the experiment with the Compton generator, Earth's angular velocity serves as a key variable in determining the speed at which water moves within the toroidal tube after it is rotated.
Because the Earth is constantly rotating, objects moving on or above its surface experience apparent forces that are a result of this motion, crucially impacting systems ranging from atmospheric ocean currents to simple experimental setups like the Compton generator.

Water motion in rotating frames

The movement of water within rotating frames, like the Compton generator setup, is a vivid illustration of how the Coriolis force operates. When the toroidal tube filled with water is spun 180 degrees, the water does not remain stationary. Instead, it moves in response to the Coriolis force because, even though the ring is immobile initially, the Earth itself is rotating.
In such scenarios, any movement results from forces acting in a rotating frame. Here, water turns around the toroid due to the changes in motion as the system is rotated beneath the influence of Earth's rotation.
The specific motion can be quantified by understanding the relationship between the speed of rotation, Earth's angular velocity, and the cosine of the colatitude angle. Such demonstrations illuminate the broader principles governing atmospheric and oceanic movements, offering a microcosmic view of how major geophysical processes are driven by Earth's continual spin.