# Atwood's fall machine

Functional diagram of the fall machine

The Atwood drop machine was developed by George Atwood in 1784 . It was designed as proof of the laws of uniformly accelerated motion . With it you can get an arbitrarily reduced acceleration instead of the gravitational acceleration with simple means .

## Observation of a uniformly accelerated movement with a <g

Drawing of the Atwoodian fall machine

Two mass parts ( and ) are connected via a rotatable roll connected to a cord. The reel and line are considered to be mass and frictionless . ${\ displaystyle M_ {1}}$${\ displaystyle M_ {2}}$

For further consideration, the same mass is assumed for both pieces of mass , so there is initially an equilibrium of forces . Then you hang another piece of mass on one of the two pieces of mass , the result is a uniformly accelerated movement. The value of the acceleration can be calculated as follows: ${\ displaystyle M}$${\ displaystyle m}$

${\ displaystyle a = {\ dfrac {m} {2M + m}} g}$

To justify this formula, one considers the weight forces on both sides, which can be calculated as the product of the respective mass and the acceleration due to gravity . On one side (on the left in the sketch on the right) one receives the amount of force , on the other side (on the right in the sketch on the right) the amount of force . Since the forces act in opposite directions, the amount of the total force results from subtraction : ${\ displaystyle g}$${\ displaystyle F_ {1} = (M + m) g}$${\ displaystyle F_ {2} = Mg}$

${\ displaystyle F = (M + m) g-Mg = mg}$.

Since the mass is accelerated overall , it follows from Newton's second law${\ displaystyle 2M + m}$

${\ displaystyle (2M + m) a = mg}$,

which confirms the above formula for the acceleration.

## Systematic errors

The formulas given above only apply exactly under idealized conditions. A real setup has a number of deviations that affect the accuracy of a measurement of the acceleration due to gravity.

• The pulley is not massless, so it has a moment of inertia . When the masses accelerate, the wheel is also accelerated, absorbs kinetic energy and thus brakes the acceleration of the masses.
• Real ropes stretch when loaded, with the stretch being roughly proportional to the load. The rope is stretched differently on the two sides of the machine. While the fall machine is in operation, more and more rope is shifted to the side of the heavier weight. This means that the total length of the rope increases in the course of operation. In addition, the additional stretching of the rope absorbs potential energy.
• The bearing has a certain amount of static friction . This static friction must be overcome by the torque that the different masses exert on the roller. This means a lower limit for the difference in weights with which the machine can still function.
• The roller bearing is not completely free of friction even when it is in motion. The friction is approximately proportional to the angular speed of the roller. Another source of friction is the stretching of the rope as it rotates on the pulley. The energy consumed by this friction is no longer available to accelerate the masses.
• When the machine is not operated in a vacuum , energy is converted. The air friction increases approximately with the square of the speed. This energy is also no longer available for the movement of the masses and thus leads to a lower acceleration.
• The two distances to the earth's surface change and thus the gravitational force changes, because in the vicinity of the earth's surface g decreases by about 3.1 µm / s² per increased meter, because the acceleration of gravity decreases proportionally to the square of the distance from the center of the earth.

## Oscillating Atwood machine

Movement of a vibrating Atwood machine with a mass ratio M / m = 4.5
Oscillating Atwood machine (SAM)

A vibrating Atwood machine (abbreviated also SAM) is constructed in such a way that one of the two masses can vibrate in the common plane of the masses. With certain proportions of the masses involved, chaotic behavior results . The oscillating Atwood machine has two degrees of freedom of motion, and . ${\ displaystyle r}$${\ displaystyle \ theta}$

The Lagrange function of an oscillating Atwood machine is:

${\ displaystyle L (r, \ theta) = TV = {\ frac {1} {2}} M {\ dot {r}} ^ {2} + {\ frac {1} {2}} m ({\ dot {r}} ^ {2} + r ^ {2} {\ dot {\ theta}} ^ {2}) - gr (Mm \ cos (\ theta)),}$

Here referred to the acceleration due to gravity , and the kinetic and potential energy of the system. ${\ displaystyle g}$${\ displaystyle T}$${\ displaystyle V}$

## literature

George Atwood: A treatise on the rectilinear motion and rotation of bodies; with a description of original experiments relative to the subject . Cambridge 1784, doi : 10.3931 / e-rara-3910 (British English).