Rheonomous

A mechanical system is rheonomous if the equations of constraints contain the time as an explicit variablecite book |last=Goldstein|first=Herbert|title=Classical Mechanics|year=1980| location=United States of America | publisher=Addison Wesley| edition= 3rd| isbn=0201657023 | language=English| pages=pp. 13] . Such constraints are called rheonomic constraints.

Example: pendulum

As shown at right, a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string’s length is a constant. Therefore, this system is scleronomous; it obeys scleronomic constraint: $sqrt\{x^2+y^2\}\; -\; L=0,!$ ,

where $(x,\; y),!$ is the position of the weight and $L,!$ is length of the string.

Refer to figure at right, Assume the top end of the string is attached to a pivot point undergoing a simple harmonic motion:$x\_t=x\_0cosomega\; t,!$ ,

where $x\_0,!$ is amplitude, $omega,!$ is angular frequency, and $t,!$ is time.

Although the top end of the string is not fixed, the length of this inextensible string is still a constant. The distance between the top end and the weight must stay the same. Therefore, this system is a rheonomous; it obeys rheonomic constraint:$sqrt\{(x\; -\; x\_0cosomega\; t)^2+y^2\}\; -\; L=0,!$ .

ee also

*Pendulum
*Lagrangian mechanics
*Holonomic
*Scleronomous

References