Short Answer:

A constrained body is a body whose motion is restricted or limited by external supports, connections, or boundaries. These constraints prevent free movement in one or more directions and create reaction forces to maintain equilibrium. On the other hand, an unconstrained body is free to move or rotate in any direction since there are no restrictions or supports acting on it. In mechanics, understanding the difference between constrained and unconstrained bodies helps in analyzing structures and determining the required supports to maintain stability.

Detailed Explanation :

Constrained and Unconstrained Bodies

In Engineering Mechanics, the study of motion and equilibrium of bodies requires understanding how forces and supports affect movement. A body can either be free to move in all directions or be restricted by certain conditions such as supports, hinges, or connections. These restrictions are called constraints.

A constraint is any limitation that prevents a body from moving freely under the action of forces. Depending on whether constraints are present or not, bodies are classified into constrained and unconstrained bodies.

This classification is very important in the analysis of mechanical systems, structures, and machines, where forces and reactions depend on the type of constraints applied.

Constrained Bodies

A constrained body is one whose movement is restricted by some external means such as supports, joints, or connections. These restrictions or constraints prevent the body from moving or rotating freely in certain directions.

When a body is constrained, external supports develop reaction forces or moments to resist motion. As a result, the body may either remain completely stationary (in static equilibrium) or move in a specific manner allowed by the constraints.

Definition:
A constrained body is defined as “a body whose motion is restricted in one or more directions due to the presence of external supports or connections.”

Example:

A beam resting on supports.
A door hinged to a wall (can rotate but cannot translate).
A car moving on a road (movement restricted to surface).

Types of Constraints in Constrained Bodies

There are several ways in which a body can be constrained. The main types of constraints include:

Linear Constraint:
- Restricts motion along a straight line.
- Example: A piston moving inside a cylinder — it can only move up and down.
Angular Constraint:
- Restricts rotation about a particular axis.
- Example: A door hinge prevents translation but allows rotation about its hinge axis.
Point Constraint:
- Restrains motion at a single point, often allowing rotation but not translation.
- Example: A beam supported at one end by a pin or hinge.
Surface Constraint:
- Restricts movement of a body to a surface, allowing sliding but preventing motion perpendicular to the surface.
- Example: A block sliding on a table — it cannot move vertically but can move horizontally.

Degrees of Freedom in Constrained Bodies

The degree of freedom (DOF) of a body represents the number of independent movements it can perform.

A completely free body in a plane has three degrees of freedom: translation in x-direction, translation in y-direction, and rotation about an axis perpendicular to the plane.
When constraints are added, each constraint removes one or more degrees of freedom.

Example:

A hinged (pinned) body can rotate but cannot move horizontally or vertically, so it has one degree of freedom.
A fixed body has all its motions restricted, so it has zero degrees of freedom.

Examples of Constrained Bodies in Engineering

Fixed Beam:
Both ends are fixed; hence it cannot move or rotate — fully constrained.
Simply Supported Beam:
The beam can rotate but cannot translate vertically — partially constrained.
Shaft in Bearings:
The shaft can rotate but not move axially — constrained in translation but free in rotation.
Door Hinged on Wall:
The door can rotate but cannot move linearly — partially constrained.

These examples show how constraints restrict certain motions while allowing others.

Unconstrained Bodies

An unconstrained body is one that is completely free to move or rotate in any direction. There are no supports, joints, or restrictions acting on it, so it can translate or rotate when any external force is applied.

Definition:
An unconstrained body is defined as “a body that can move freely in any direction under the action of forces, as it has no external supports or restrictions.”

Example:

A ball freely floating in space.
A free block on a frictionless surface.
A satellite orbiting freely in space.

In such bodies, no reaction forces develop because no external constraints exist.

Degrees of Freedom in Unconstrained Bodies

An unconstrained body in two-dimensional motion has three degrees of freedom:

Translation along the x-axis.
Translation along the y-axis.
Rotation about an axis perpendicular to the plane.

In three-dimensional motion, a body can have six degrees of freedom, which include translation along x, y, z axes and rotation about x, y, z axes.

Because there are no constraints, the body is free to move or rotate independently in all these directions.

Comparison Between Constrained and Unconstrained Bodies

Property	Constrained Body	Unconstrained Body
Definition	Motion restricted by supports or connections	Free to move in any direction
Presence of Supports	Yes	No
Reaction Forces	Developed at supports	Not developed
Motion	Limited or controlled	Free and unrestricted
Examples	Hinged beam, car on road, pendulum	Ball in space, block on smooth surface

Importance in Engineering

Understanding constrained and unconstrained bodies is crucial in engineering design and analysis because:

It helps determine the type and number of supports needed to maintain equilibrium.
It ensures structural stability and prevents unwanted motion.
It allows the accurate calculation of reaction forces at supports.
It provides the basis for designing machines and mechanisms with specific allowed movements.
It aids in identifying whether a structure is statically determinate (solvable) or indeterminate (requires advanced methods).

Practical Applications

Bridges: Designed as constrained systems to carry loads safely.
Cranes: Use partial constraints to allow specific motion directions.
Engines: Pistons are linearly constrained within cylinders.
Robotics: Control of constraints defines how robotic arms move and operate.
Vehicles: Constrained to move along roads but free to roll and steer.

Conclusion

In conclusion, constrained bodies are those whose motion is restricted by external supports or connections, while unconstrained bodies are free to move or rotate in any direction. Constraints play a crucial role in maintaining equilibrium and stability in mechanical systems. Depending on the degree of restriction, bodies can have varying degrees of freedom. Understanding these concepts helps engineers design efficient, stable, and safe mechanical systems and structures that perform desired motions without losing balance or control.