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

Example: pendulum

As shown at right, a simple pendulum

Pendulum

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...

 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

,

where is the position of the weight and 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

Simple harmonic motion

Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and molecular vibration....

,

where is amplitude, is angular frequency, and 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 .