Posts Tagged 'Mechanics'

The “extraordinary talent and superior genius” of Sophie Germain

When a guitar string is plucked, we don’t see waves travelling along the string. This is because the ends are fixed. Instead, we see a standing-wave pattern. Standing waves are also found on drum-heads and on the sound-boxes of violins. The shape of a violin strongly affects the quality and purity of the sound, as it determines the mixture of standing wave harmonics that it can sustain [TM179 or search for “thatsmaths” at irishtimes.com].

Sophie-Germain-Stamp

French postage stamp, issued in 2016, to commemorate the
250th anniversary of the birth of Sophie Germain (1776-1831).

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The Intermediate Axis Theorem

In 1985, cosmonaut Vladimir Dzhanibekov commanded a mission to repair the space station Salyut-7. During the operation, he flicked a wing-nut to remove it. As it left the end of the bolt, the nut continued to spin in space, but every few seconds, it turned over through {180^\circ}. Although the angular momentum did not change, the rotation axis moved in the body frame. The nut continued to flip back and forth, although there were no forces or torques acting on it.

Dzhanibekov-00

Flipping nut [image from Veritasium].

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An Attractive Spinning Toy: the Phi-TOP

It is fascinating to watch a top spinning. It seems to defy gravity: while it would topple over if not spinning, it remains in a vertical position as long as it is spinning rapidly.

There are many variations on the simple top. The gyroscope has played a vital role in navigation and in guidance and control systems. Many similar rotating toys have been devised. These include rattlebacks, tippe-tops and the Euler disk. The figure below shows four examples.

Spinning-Tops-4

(a) Simple top, (b) Rising egg, (c) Tippe-top, (d) Euler disk. [Image from website of Rod Cross.]

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The curious behaviour of the Wilberforce Spring.

The Wilberforce Spring (often called the Wilberforce pendulum) is a simple mechanical device that illustrates the conversion of energy between two forms. It comprises a weight attached to a spring that is free to stretch up and down and to twist about its axis.

Wilberforce-Spring

Wilberforce spring [image from Wikipedia Commons].}

In equilibrium, the spring hangs down with the pull of gravity balanced by the elastic restoring force. When the weight is pulled down and released, it immediately oscillates up and down.

However, due to a mechanical coupling between the stretching and torsion, there is a link between stretching and twisting motions, and the energy is gradually converted from vertical oscillations to axial motion about the vertical. This motion is, in turn, converted back to vertical oscillations, and the cycle continues indefinitely, in the absence of damping.

The conversion is dependent upon a resonance condition being satisfied: the frequencies of the stretching and twisting modes must be very close in value. This is usually achieved by having small adjustable weights mounted on the pendulum.

There are several videos of a Wilberforce springs in action on YouTube. For example, see here.

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Billiards & Ballyards

In (mathematical) billiards, the ball travels in a straight line between impacts with the boundary, when it changes suddenly and discontinuously We can approximate the hard-edged, flat-bedded billiard by a smooth sloping surface, that we call a “ballyard”. Then the continuous dynamics of the ballyard approach the motions on a billiard.

SAMSUNG

Elliptical tray in the form of a Ballyard.

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Boxes and Loops

We will describe some generic behaviour patterns of dynamical systems. In many systems, the orbits exhibit characteristic patterns called boxes and loops. We first describe orbits for a simple pendulum, and then look at some systems in higher dimensions.

SimplePendulum-PhasePortrait-Colour

Phase portrait for a simple pendulum. Each line represents a different orbit.

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Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph

The Greeks regarded the heavens as the epitome of perfection. All flaws and blemishes were confined to the terrestrial domain. Since the circle is perfect in its infinite symmetry, it was concluded by Aristotle that the Sun and planets move in circles around the Earth. Later, the astronomer Ptolemy accounted for deviations by means of additional circles, or epicycles. He stuck with the circular model [TM162 or search for “thatsmaths” at irishtimes.com].

Hodograph-AB

Left: Elliptic orbit with velocity vectors. Right: Hodograph, with all velocity vectors plotted from a single point.

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