Rheonomous

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A mechanical system is rheonomous if the equations of constraints contain the time as an explicit variable^[1]. Such constraints are called rheonomic constraints.

[edit] Example: pendulum

A simple 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.

A simple pendulum with oscillating pivot point

Refer to figure at right, Assume the top end of the string is attached to a pivot point undergoing a simple harmonic motion

$x_t=x_0\cos\omega t\,\!$ ,

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

$\sqrt{(x - x_0\cos\omega t)^2+y^2} - L=0\,\!$ .