Rheonomous
Encyclopedia
A mechanical system is rheonomous if the equations of constraints contain the time as an explicit variable. Such constraints are called rheonomic constraints.
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
,
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 .
Example: pendulum
As shown at right, a simple pendulumPendulum
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 .