Constrained motion is a fundamental concept in physics where an object's movement is restricted by external conditions. Unlike free motion, where an object can move in any direction, constrained motion limits the possible paths. A perfect example is a pendulum, where the bob is constrained to move along a circular arc due to the string connection.
There are two main types of constraints in physics. Holonomic constraints depend only on position and time, like a bead sliding on a wire where the bead must always stay on the wire's path. Non-holonomic constraints involve velocity relationships, such as a wheel rolling without slipping where the constraint relates the wheel's rotation to its translation.
Constraint forces are the forces that maintain the restrictions on motion. These include normal forces from surfaces, tension forces from strings or rods, and friction forces. In ideal constraints, these forces do no work because they act perpendicular to the allowed motion. For example, on an inclined plane, the normal force constrains the block to move along the surface.
Mathematically, constraints are described by equations that relate the coordinates of the system. For example, a pendulum of length L has the constraint equation x squared plus y squared equals L squared. This constraint equation reduces the degrees of freedom, forcing the particle to move only along the circular path rather than freely in three-dimensional space.
Constrained motion has numerous applications in engineering and science. Mechanical systems like gears have rotational constraints that ensure synchronized motion. Robotic linkages use constraints to control end-effector positions. Even planetary motion follows orbital constraints. Understanding these principles helps engineers design efficient mechanisms and predict system behavior accurately.