Tag Archives: Science
In physics, the Lorentz transformations (or transformation) are coordinate transformations between two coordinate frames that move at constant velocity relative to each other. The transformations are named after Hendrik Lorentz, a Dutch physicist in electrodynamics.
The most common form of the transformation is expressed as:
where (t, x, y, z) and (t′, x′, y′, z′) represent an event’s coordinates in two frames with relative velocity v, c is the speed of light, and the Lorentz factor:
This most attractive example is in the apparatus collection at Bowdoin College. It has no maker’s name and is about 40 cm high.
The gyroscope was invented in 1852 by the French experimental physicist Leon Foucault (1819-1868) as part of a two-pronged investigation of the rotation of the earth. The better-known demonstration of the Foucault pendulum showed that the plane of rotation of a freely-swinging pendulum rotated with a period that depends on the latitude of its location.
His gyroscope was a rapidly rotating disk with a heavy rim, mounted in low-friction gimbals. As the earth rotated beneath the gyroscope, it would maintain its orientation in space. This proved to be hard to do in practice because the frictional forces bring the spinning system to rest before the effect could be observed. The gimbal bearings also introduce unwanted torque. But, the principle is well-known to all children who move their toy gyroscopes about and observe that the spinning disk stays in the same orientation. Continue reading
How Gyroscopes Work
by M. Brain
Gyroscopes can be very perplexing objects because they move in peculiar ways and even seem to defy gravity. These special properties make gyroscopes extremely important in everything from your bicycle to the advanced navigation system on the space shuttle. A typical airplane uses about a dozen gyroscopes in everything from its compass to its autopilot. The Russian Mir space station used 11 gyroscopes to keep its orientation to the sun, and the Hubble Space Telescope has a batch of navigational gyros as well. Gyroscopic effects are also central to things like yo-yos and Frisbees!
If you have ever played with toy gyroscopes, you know that they can perform all sorts of interesting tricks. They can balance on string or a finger; they can resist motion about the spin axis in very odd ways; but the most interesting effect is called precession. This is the gravity-defying part of a gyroscope. Continue reading
In physics, torque can be thought of informally as “rotational force”. The concept of torque, also called moment or couple, originated with the work of Archimedes on levers. The force applied to a lever, multiplied by its distance from the lever’s fulcrum, is the torque. Torque is measured in units of newton metres, and its symbol is τ.
For example, a force of three newtons applied two metres from the fulcrum exerts the same torque as one newton applied six metres from the fulcrum. This assumes the force is in a direction at right angles to the straight lever. More generally, one may define torque as the cross product:
F is the vector of force.
r is the vector from the axis of rotation to the point on which the force is acting.
The rotational analogues of force, mass and acceleration are torque, moment of inertia and angular acceleration respectively. Continue reading
Primes, Modular Arithmetic, and Public Key Cryptography
(April 15, 2004)
Every cipher we have worked with up to this point has been what is called a symmetric key cipher, in that the key with which you encipher a plaintext message is the same as the key with which you decipher a ciphertext message. As we have discussed from time to time, this leads to several problems. One of these is that, somehow, two people who want to use such a system must privately and secretly agree on a secret key. This is quite difficult if they are a long distance apart (it requires either a trusted courier or an expensive trip), and is wholly impractical if there is a whole network of people (for example, an army) who need to communicate. Even the sophisticated Enigma machine required secret keys. In fact, it was exactly the key distribution problem that led to the initial successful attacks on the Enigma machine.
However, in the late 1970’s, several people came up with a remarkable new way to solve the Continue reading