Short Answer:

A coplanar force system is a system in which all the forces acting on a body lie in the same plane. This means the lines of action of all the forces are contained within one plane, usually the x–y plane. Coplanar force systems are the most common type of force systems studied in Engineering Mechanics. They can be either concurrent, parallel, or non-concurrent, depending on whether the forces meet at a point, are parallel, or intersect at different points. These systems are used to analyze the equilibrium and motion of objects in two-dimensional conditions.

Detailed Explanation :

Coplanar Force System

In Engineering Mechanics, a coplanar force system refers to a condition where all the forces acting on a body lie within the same plane. It is one of the simplest and most common types of force systems encountered in mechanical and structural analysis. When forces are coplanar, they can be represented and analyzed on a two-dimensional (2D) plane, such as a sheet of paper or a coordinate system (x–y plane).

A coplanar force system is easier to study because only two components of forces — horizontal and vertical — need to be considered. These systems are widely used to study the equilibrium of beams, trusses, frames, and various mechanical components where all the forces act in one plane.

For example, the forces acting on a simply supported beam, a ladder resting against a wall, or a bridge structure are all examples of coplanar force systems.

Classification of Coplanar Force Systems

Coplanar force systems can be classified based on the point of intersection of the lines of action of the forces. There are mainly three types:

1. Coplanar Concurrent Force System
2. Coplanar Non-Concurrent Force System
3. Coplanar Parallel Force System

Let us understand each type in detail.

1. Coplanar Concurrent Force System

A coplanar concurrent force system is one in which the lines of action of all the forces intersect at a single common point in the same plane. In such a system, the resultant of all the forces also passes through the same point.

For example, if several strings are tied together at a knot and forces are applied along each string, the knot experiences a concurrent coplanar force system.

The equilibrium condition for a concurrent system is that the sum of all horizontal forces and the sum of all vertical forces must be zero, i.e.,

Such systems are simple to analyze because they involve only linear forces and no moments about the point of concurrency.

Examples:

• Tensions acting at the joint of a truss.
• Forces at the point of a crane hook or pulley.
• Forces acting on a ring connected by multiple strings.

2. Coplanar Non-Concurrent Force System

A coplanar non-concurrent force system is one in which the lines of action of the forces do not meet at a single point but still lie in the same plane. These forces act on different points of the body and can produce both translation and rotation.

In this system, the resultant of all forces does not pass through the same point as any of the forces, and moments are also produced due to different points of application.

To maintain equilibrium in a non-concurrent coplanar system, the following conditions must be satisfied:

Where:

• ΣFx = Sum of horizontal forces
• ΣFy = Sum of vertical forces
• ΣM = Sum of moments about any point

Examples:

• Forces acting on a simply supported or cantilever beam.
• Forces acting on a ladder resting on a wall and ground.
• Loads acting at various points of a bridge.

3. Coplanar Parallel Force System

A coplanar parallel force system is one in which all the forces act in parallel lines within the same plane. These forces may be in the same or opposite directions, but their lines of action never intersect.

Parallel forces are very common in mechanical and structural engineering, especially in systems subjected to uniformly distributed or concentrated loads.

For equilibrium in a parallel force system:

Examples:

• The weight of a beam supported at both ends.
• Forces acting on a table under multiple loads.
• Reactions at supports in structures with vertical loads.

Graphical and Analytical Representation

Coplanar force systems can be represented both graphically and analytically:

• Graphically:
  Using vector diagrams, such as the parallelogram law, triangle law, or polygon law of forces, the resultant of coplanar forces can be found visually.
• Analytically:
  By resolving all forces into horizontal and vertical components, the resultant can be calculated using:

and the direction of the resultant is given by:

This method is often preferred in engineering calculations for accuracy.

Applications of Coplanar Force Systems

Coplanar force systems are commonly used in the following engineering applications:

1. Structural Engineering:
   To analyze loads acting on beams, trusses, and bridges.
2. Mechanical Engineering:
   To calculate forces on levers, pulleys, and machine components.
3. Civil Engineering:
   Used in stability and design analysis of buildings, ladders, and retaining walls.
4. Statics and Dynamics:
   Helps in studying equilibrium and motion in two-dimensional systems.
5. Experimental Mechanics:
   Used in verifying laws of equilibrium and vector addition through force tables.

Importance of Coplanar Force System

Understanding coplanar force systems is crucial because most real-world engineering problems can be simplified into two-dimensional (coplanar) models. It provides the basis for analyzing equilibrium, finding reaction forces, and ensuring structural stability in engineering designs.

By mastering coplanar force systems, engineers can easily predict how a structure or machine will behave under various loading conditions, ensuring safety and efficiency in design.

Conclusion

In conclusion, a coplanar force system consists of all forces acting in the same plane. It is the most fundamental type of force system in mechanics and forms the basis for understanding equilibrium and motion in two-dimensional analysis. Depending on the intersection of force lines, it can be concurrent, non-concurrent, or parallel. Coplanar force systems are essential in analyzing beams, structures, and machines, making them a key concept in Engineering Mechanics and mechanical design.