Short Answer:

A two-force body is a body on which only two forces act. For the body to remain in equilibrium, these two forces must be equal in magnitude, opposite in direction, and collinear (acting along the same line).

A three-force body is a body acted upon by three forces. For such a body to be in equilibrium, the lines of action of all three forces must either be concurrent (meet at a single point) or be parallel. These types of bodies are commonly found in machines and structural systems such as beams, levers, and links.

Detailed Explanation :

Two-Force and Three-Force Body

In Engineering Mechanics, all physical bodies are subjected to external forces and moments. The number of forces acting on a body determines its classification and the method of analysis required for equilibrium. Two important types of bodies often studied in statics are the two-force body and the three-force body.

Both these concepts are fundamental for analyzing simple mechanical elements like beams, links, pins, and trusses. They help engineers understand how forces act on various machine components and structural members to maintain equilibrium.

Two-Force Body

A two-force body is a body on which only two forces act. These two forces may act at different points, but for the body to remain in equilibrium, they must satisfy specific conditions.

Definition:
A two-force body is a body acted upon by two forces only, which are equal in magnitude, opposite in direction, and collinear in line of action.

Explanation:
When a body is subjected to only two forces, these forces must balance each other to keep the body in a stable condition (equilibrium). If the forces are not collinear or not equal and opposite, the body will either move or rotate.

Therefore, for equilibrium of a two-force body:

The two forces must be equal in magnitude so that their effects cancel out.
The two forces must be opposite in direction to prevent motion in one direction.
The two forces must be collinear, meaning their lines of action should lie along the same straight line to prevent rotation.

Mathematical Condition:

Example 1:
A straight bar or link in tension or compression (such as a connecting rod or a truss member) is a typical example of a two-force body. The two forces act at the ends of the member, one pulling (tension) or pushing (compression) the body.

Example 2:
A rope with a load hanging on it is also a two-force body, where one force acts at the point of suspension and the other at the load, both acting along the same line.

Diagram Description:
If you imagine a straight bar AB with forces F1 and F2 acting at its ends, both must be equal, opposite, and along the line AB for equilibrium.

Application:

Used in the analysis of truss members where each member carries only axial load (tension or compression).
Used in machine links, such as connecting rods, which transmit force between moving parts.

Three-Force Body

A three-force body is a body on which exactly three forces act. The equilibrium of such a body depends on the relative directions and points of application of these forces.

Definition:
A three-force body is a body acted upon by three forces, and for equilibrium, either the three forces must be concurrent (their lines of action intersect at a common point) or they must be parallel.

Explanation:
When three forces act on a body, they can produce translation and rotation. For the body to remain in equilibrium, both these tendencies must be eliminated. This is possible only if:

The lines of action of the three forces meet at a single point (concurrent).
or
The three forces are parallel, meaning they all act in the same or opposite direction but balance each other.

In such conditions, the vector sum of the three forces equals zero, and their moments also cancel out, keeping the body at rest.

Mathematical Condition:
For equilibrium of a three-force body,

This condition ensures that there is no linear or rotational motion.

Example 1:
A beam supported by two supports and carrying a load at the center is a three-force body. The three forces are:

The load acting downward, and
The two reaction forces at the supports acting upward.

Example 2:
A ladder leaning against a wall is another example.

The weight of the ladder acts vertically downward at its center.
The reaction from the wall acts horizontally.
The reaction from the ground acts vertically upward.
The three forces meet at a single point, making it a concurrent three-force system.

Example 3:
A crane boom supporting a load can also be considered a three-force body, where the boom experiences three forces: the weight, the cable tension, and the hinge reaction.

Conditions for Equilibrium of Three-Force Body

For a three-force body to remain in equilibrium, the following two main conditions must be satisfied:

The lines of action of all three forces must be concurrent or parallel.
Each force must be in magnitude and direction such that their vector sum equals zero.

Graphically, the three forces can be represented using the triangle law of forces. According to this law, if three forces acting on a body are in equilibrium, they can be represented in magnitude and direction by the three sides of a triangle taken in order.

Difference between Two-Force and Three-Force Bodies

Feature	Two-Force Body	Three-Force Body
Number of Forces	Two	Three
Condition for Equilibrium	Forces must be equal, opposite, and collinear	Lines of action must be concurrent or parallel
Resultant Force	Zero if forces are balanced	Zero if triangle law of forces is satisfied
Example	Link in tension or compression	Ladder leaning against a wall
Moment	No moment (collinear forces)	Moment balanced by the other two forces

Applications in Engineering

Truss Analysis: Truss members are two-force bodies as they carry only axial loads.
Machine Design: Levers, cranes, and lifting devices often behave as three-force bodies.
Structural Stability: Determining reactions in beams and frames uses these principles.
Statics Problems: Simplifies complex systems into manageable parts for equilibrium analysis.

Conclusion

In conclusion, a two-force body is one that has only two forces acting on it, and for equilibrium, these forces must be equal, opposite, and collinear. A three-force body, on the other hand, has three forces acting on it, and for equilibrium, their lines of action must be concurrent or parallel. Understanding these two types of bodies is essential in mechanical and structural engineering as they form the basis of analyzing systems like trusses, beams, and machines for stability and balance.