Atwood machine - AbsoluteAstronomy.com

The Atwood machine was invented in 1784 by Rev. George Atwood

George Atwood

George Atwood was an English mathematician who invented a machine for illustrating the effects of Newton's first law of motion...

as a laboratory experiment to verify the mechanical laws of motion

Newton's laws of motion

Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

with constant acceleration

Acceleration

In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...

. Atwood's machine is a common classroom demonstration used to illustrate principles of classical mechanics

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

.

The ideal Atwood Machine consists of two objects of mass m₁ and m₂, connected by an inextensible massless string over an ideal massless pulley

Pulley

A pulley, also called a sheave or a drum, is a mechanism composed of a wheel on an axle or shaft that may have a groove between two flanges around its circumference. A rope, cable, belt, or chain usually runs over the wheel and inside the groove, if present...

.
When m₁ = m₂, the machine is in neutral equilibrium regardless of the position of the weights.

When m₁ ≠ m₂ both masses experience uniform acceleration.

Equation for constant acceleration

We are able to derive an equation for the acceleration by using force analysis.
If we consider a massless, inextensible string and an ideal massless pulley, the only forces we have to consider are: tension force (T), and the weight of the two masses (W₁ and W₂). To find an acceleration we need to consider the forces affecting each individual mass.
Using Newton's second law (with a sign convention

Sign convention

In physics, a sign convention is a choice of the physical significance of signs for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of...

) we can derive a system of equations

Simultaneous equations

In mathematics, simultaneous equations are a set of equations containing multiple variables. This set is often referred to as a system of equations. A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations...

for the acceleration (a).

As a sign convention, we assume that a is positive when downward for

, and that a is positive when upward for

. Weight of

and

is simply

and

respectively.

Forces affecting m₁:

Forces affecting m₂:

and adding the two previous equations we obtain

,

and our concluding formula for acceleration

Conversely, the acceleration due to gravity, g, can be found by timing the movement of the weights, and calculating a value for the uniform acceleration a:

.

The Atwood machine is sometimes used to illustrate the
Lagrangian method

Lagrangian mechanics

Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....

of deriving equations of motion.

Equation for tension

It can be useful to know an equation for the tension in the string. To evaluate tension we substitute the equation for acceleration in either of the 2 force equations.

For example substituting into

, we get

The tension can be found in using this method.

Equations for a pulley with inertia and friction

For very small mass differences between m₁ and m₂, the rotational inertia

Moment of inertia

In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

I of the pulley of radius r cannot be neglected. The angular acceleration of the pulley is given by the no-slip condition:

where

is the angular acceleration. The net torque

Torque

Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

is then:

Combining with Newton's second law for the hanging masses, and solving for T₁, T₂, and a, we get:

Acceleration:

Tension in string segment nearest m₁:

Tension in string segment nearest m₂:

Should bearing friction be negligible (but not the inertia of the pulley and not the traction of the string on the pulley rim), these equations simplify as the following results:

Acceleration:

Tension in string segment nearest m₁:

Tension in string segment nearest m₂:

Practical implementations

Atwood's original illustrations show the main pulley's axle resting on the rims of another four wheels, to minimise friction forces from the bearings

Rolling-element bearing

A rolling-element bearing, also known as a rolling bearing, is a bearing which carries a load by placing round elements between the two pieces...

. Many historical implementations of the machine follow this design.

An elevator with a counterbalance approximates an ideal Atwood machine and thereby relieves the driving motor from the load of holding the elevator cab — it has to overcome only weight difference and inertia of the two masses. The same principle is used for funicular

Funicular

A funicular, also known as an inclined plane or cliff railway, is a cable railway in which a cable attached to a pair of tram-like vehicles on rails moves them up and down a steep slope; the ascending and descending vehicles counterbalance each other.-Operation:The basic principle of funicular...

railways with two connected railway cars on inclined tracks.

Equation for constant acceleration

Equation for tension

Equations for a pulley with inertia and friction

Practical implementations

See also