Atwood machine

The Atwood 's machine was invented in 1784 by George Atwood. It was designed as a proof of the laws of uniformly accelerated motion. With this one can get with simple means instead of the acceleration an arbitrarily reduced acceleration.

Operation

Two mass parts ( ET) are connected via a rotatable roller with a string. The role and the cord are considered as mass and smoothly.

For the further consideration of the same mass is assumed in both mass pieces, so it prevails initially balance. Then you hang on one of the two mass pieces another piece of mass mass, it results in a uniformly accelerated motion. The value of the acceleration can be calculated as follows:

In support of this formula, we consider the weight forces on both sides, which can be calculated as the product of the respective mass and the acceleration of gravity. On one side ( in the right sketch on the left ) to get the amount of force on the other side ( in the right drawing) the amount of force. Since the forces opposed to the amount of the total force is given by subtraction:

Since the total mass is accelerated, it follows from Newton's second law

Thus confirming the above formula for the acceleration.

Systematic errors

The above formulas are exact only under idealized conditions. A real structure has a number of discrepancies that go into the accuracy of a measurement of the gravitational acceleration.

  • The idler pulley is not massless, so it has a moment of inertia. At an acceleration of the masses, the wheel is also accelerated, absorbs kinetic energy and thus slows down the acceleration of the masses.
  • Real rope stretch under load, the strain is proportional to stress about. The rope is stretched to different extents on both sides of the machine. During the event the machine is in operation, more rope is always shifted to the side of the higher weight. That is, the total length of the rope is greater than in the course of operation. In addition, the additional elongation of the rope takes up potential energy.
  • The bearing has a certain friction. This friction must be overcome by the torque which exert the different masses of the roll. This means a lower limit for the difference in the weights with which the machine is still working.
  • The camp of the role is not completely free of friction in movement. The friction is approximately proportional to the angular velocity of the roll. Another source of friction is the elongation of the rope, while it rotates on the reel. The power consumed by this friction energy is no longer accelerating masses available.
  • When the engine is not operated in a vacuum, energy is lost due to air friction. The air friction increases approximately with the square of the velocity. Also, this energy is no longer accelerating masses available and thus leads to a lower acceleration.

Swinging Atwood's machine

A vibrating machine Atwood ( abbreviated SAM) is constructed so that one of the two masses to oscillate in the common plane of the masses. In certain conditions the masses involved there is a chaotic behavior. The oscillating Atwood machine has two degrees of freedom of movement, and.

The Hamiltonian of a swinging Atwood's machine is:

This means the acceleration of gravity, and the kinetic and potential energy of the system.

Sheave Degrees of freedom (physics and chemistry)
87112
de