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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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The Brief and Tragic Life of Évariste Galois

On the morning of 30 May 1832 a young man stood twenty-five paces from his friend. Both men fired, but only one pistol was loaded. Évariste Galois, a twenty year old mathematical genius, fell to the ground. The cause of Galois’s death is veiled in mystery and speculation. Whether both men loved the same woman or had irreconcilable political differences is unclear. But Galois was abandoned, mortally wounded, on the duelling ground at Gentilly, just south of Paris. By noon the next day he was dead [TM169 or search for “Galois” at irishtimes.com].

Galois-Stamp

French postage stamp issued in 1984.

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Stokes’s 200th Birthday Anniversary

Next Tuesday, the 30th of August, is the 200th anniversary of the birth of George Gabriel Stokes. This extended blog post is to mark that occasion. See also an article in The Irish Times.

Navier-Stokes-Equations

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Algorithms: Recipes for Success

The impact of computing on society is ever-increasing. Web-based commerce continues to grow and artificial intelligence now pervades our lives. To make wise choices, we need to understand how computers operate and how we can deploy them most constructively. Listen to any computer scientist and soon you will hear the word “algorithm” [TM168 or search for “thatsmaths” at irishtimes.com].

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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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Learning Maths without even Trying

Children have an almost limitless capacity to absorb knowledge if it is presented in an appealing and entertaining manner. Mathematics can be daunting, but it is possible to convey key ideas visually so that they are instantly accessible. Visiting Explorium recently, I saw such a visual display demonstrating the theorem of Pythagoras, which, according to Jacob Bronowski, “remains the most important single theorem in the whole of mathematics” [TM167 or search for “thatsmaths” at irishtimes.com].

Explorium-Banner

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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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