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# Atwood Machines

An Atwood machine consists of two weights, of mass and , connected by a light inextensible cord of length , which passes over a pulley of radius , and moment of inertia . See Figure 32.

Referring to the diagram, we can see that this is a one degree of freedom system whose instantaneous configuration is specified by the coordinate . Assuming that the cord does not slip with respect to the pulley, the angular velocity of pulley is . Hence, the kinetic energy of the system is given by

 (624)

The potential energy of the system takes the form
 (625)

It follows that the Lagrangian is written
 (626)

The equation of motion,
 (627)

thus yields
 (628)

or
 (629)

Consider the dynamical system drawn in Figure 33. This is an Atwood machine in which one of the weights has been replaced by a second Atwood machine with a cord of length . The system now has two degrees of freedom, and its instantaneous position is specified by the two coordinates and , as shown.

For the sake of simplicity, let us neglect the masses of the two pulleys. Thus, the kinetic energy of the system is written

 (630)

whereas the potential energy takes the form
 (631)

It follows that the Lagrangian of the system is
 (632)

Hence, the equations of motion,
 (633) (634)

yield
 (635) (636)

The accelerations and can be obtained from the above two equations via simple algebra.

Next: Sliding down a Sliding Up: Lagrangian Dynamics Previous: Motion in a Central
Richard Fitzpatrick 2011-03-31