Equilibrium of Concurrent Forces

Equilibrium of concurrent forces refers to a condition where a system of forces acting at a single point results in no net force on that point. For a system of concurrent forces to be in equilibrium, the vector sum of all the forces must be zero.

In this article, we will discuss about equilibrium of concurrent forces, its formula, examples, types of equilibrium of concurrent forces along with some examples based on it.

Equilibrium of Constructor Forces means that at a definite point vector sum of all the forces applied to an object are equal in magnitude and they are opposed to each other. In other words, the forces that act on the object have a balance i.e. equal and opposite forces due to which there is no change in velocity.

This principle was initially established on Newton’s first law of motion which implies that an object at rest will stay at rest and an object moving will continue moving with an unbalanced force unless another force arises.

Mathematically, the equilibrium of concurrent forces can be expressed as:

ΣF = 0

Here, ΣF is the vector sum (also referred to as the resultant) of all the forces that the object is being subjected to. The balance of the forces is “zero” when the sum is zero, the object is set to equilibrium.

Definition of Equilibrium of Concurrent Forces

A system of concurrent forces is said to be in equilibrium if the vector sum of all the forces acting at a common point is zero.

Formula of Equilibrium of Concurrent Forces

For the equilibrium of concurrent forces in two dimensions, the following vector equation must hold:

[Tex]\sum \vec{F} = \vec{F}_1 + \vec{F}_2 + \ldots + \vec{F}_n = 0 [/Tex]

Where:

[Tex]\sum \vec{F} [/Tex]means the sum of all forces in a vectorial way.

[Tex]\sum \vec{F}_1, \vec{F}_2, \ldots, \vec{F}_n [/Tex] are vectors of individual forces forcibly acting on the object.

Examples of Equilibrium of Concurrent Forces

• Tug-of-War: In a tug-of-war game, if the two opposing teams apply equal forces of equal magnitudes but in opposite directions, then they will not be able to move the rope. When this happens, the system is said to be in equilibrium. Net forces can go either way, and this will lead to a motion towards that side.
• Stationary Car: In the case of a car that is resting on a flat surface, the forces acting upon it which include gravity (that pulls everything downward) and the normal (upward) force that the ground pushes against it have a zero-net force when they are taken together. As a result, we see the car is at rest.

Equillibrium of concurrent forces can be further classified into two types namely:

• Static Equilibrium
• Dynamic Equilibrium

Static Equilibrium

Static equilibrium occurs when an object is at rest and remains at rest because the net force acting on it is zero. In this state, all the forces acting on the object are balanced, and there is no movement.

The fact is that when the object is in a state of static equilibrium, forces and torques adds up to zero in the absence of any external force. The object remains unchanged unless some external force interrupts its harmonious state.

Such an equilibrium is normally observed among stationary objects like a book lying on the table, a car parked at the parking, etc.

Dynamic Equilibrium

Dynamic Equilibrium, on the other hand, is the stage when an object possesses a constant velocity, and all the forces acting on it are balanced, summing up to zero.

On the contrary, dynamic equilibrium does not take place while objects are at rest and hence only when the object is in uniform motion without any acceleration.

Dynamic equilibrium is typically experienced with objects that are undertaking uninterrupted motion and, generally, the objects shown moving at a constant velocity, such as a car slowly moving straight along the road at a specific speed.

Coplanar forces are forces that are directed to remain within the same plane. It is a situation where multiple forces lie in the same plane and their vector sum equals zero, resulting in no net force acting on the object.

This condition ensures that the object remains in a state of rest or moves with constant velocity, implying both static and dynamic equilibrium.

Conditions for Equilibrium of Coplanar Forces

Following are the Conditions for Equilibrium of Coplanar Forces

1. The Sum of Forces in the X-Direction Must Be Zero

∑F_x = 0

This means the horizontal components of all the forces must balance each other out.

2. The Sum of Forces in the Y-Direction Must Be Zero

∑F_y = 0

This means the vertical components of all the forces must balance each other out.

Also Check,

• Scalars and Vectors
• Types of Vectors
• Vector Algebra

Example 1: A box is on the top of a table. The box has a weight of 100 N and two reciprocally opposite directed forces are applied horizontally with 50 N intensity each. Find out whether the current box condition is in equilibrium or not.

Solution:

Vertical Forces:

• Weight of the box (downward force): W=100N
• Upward force exerted by the table (normal force): Fn​=100N (to balance the weight)

Horizontal Forces:

• Force applied horizontally on the right: F1​=50N
• Force applied horizontally on the left: F2​=50N

Analysis:

• Vertically: W = Fn​, so the box is not moving up or down.
• Horizontally: F1​ = F2​, so the forces cancel each other out, and the box doesn’t move left or right.

Therefore,

The box is stationary. It’s neither moving up or down, nor left or right.

Example 2: Two wires hold the cross-street directional signal. In the first wire, the tension is 300 N, whereas the second wire is under the tension of 400 N. When this angle is equal to 60 degrees, find the weight of the traffic lights.

Solution:

Let’s denote:

T1​ as the tension in the first wire (300 N)

T2​ as the tension in the second wire (400 N)

W as the weight of the traffic lights (the force we want to find)

θ as the angle between the wires (60 degrees)

Using trigonometry, we can find the vertical components of the tensions:

For the first wire: T1y​=T1​sinθ

For the second wire: T2y​=T2​sinθ

Since the vertical components of the tensions are acting in opposite directions and balance the weight, we have:

T1y​+T2y​=W

Substituting the expressions for the vertical components of tensions:

T1​sinθ+T2​sinθ=W

300N×sin(60∘)+400N×sin(60∘)=W

(300×0.866)+(400×0.866)=W

259.8+346.4=W

606.2=W

Therefore,

the weight of the traffic lights is approximately 606.2 Newtons.

Example 3: A crate on a rough surface experiences a 300 N horizontal force to the right and a 400 N downward force. With a friction coefficient of 0.3, is it in equilibrium?

Solution:

The maximum frictional force: ?_max = ?? = 0.3×400 = 120 N Since the applied horizontal force exceeds this (300 N>120 N), the crate is not in equilibrium.

Problem 1: 10 N and 15 N act on an object at a point, are two forces. To them, the angle between them is 60 degrees. Calculation of the force’s magnitude and its direction is required.

Problem 2: Three forces pull a point in three different ways lying at 30 degrees above the x-axis, 45 degrees below the x-axis, and 60 degrees to the left of the y-axis with magnitudes of 20 N, 30 N, and 40 N respectively. Find the resultant force.

Problem 3: On an object three forces are here: 8 N upward on the x-axis, -6 N downwards on the same axis, and 10 N upward on the y-axis. Set the level of intensity and the direction.

Problem 4: An object at rest experienced four forces: 20 N in the segment of 0 degrees, 30 N in one-quarter of the turn, 40 N in a half turn, and 50 N in the three hundred and sixty-degree segment. Elaborate if the object is in an equilibrium state and, if so, then figure out the size and direction of the resultant forces.

What is the difference between static equilibrium and dynamic equilibrium?

Static equilibrium deals with fixed/inert objects having equal forces which in turn lead to neutral parallelism, while dynamic equilibrium deals with the moving objects which are having equal forces acting making velocity constant.

What is the method to find out if the object is unbalanced?

An object is in equilibrium when the sum of all forces acting on it is zero and in addition, the zero sum of torques about any given point is also zero.

What are coplanar forces?

Forces on coplanar are those which are present in the same plane. The forces in coplanar equilibrium, which means all the vectors lie in the same plane, satisfy the conditions of static equilibrium, and their resultant is zero.

How do you calculate equilibrium during multiple forces acting concurrently?

To analyze proper problems of concurrent forces, find out all the forces acting on the object, determine their ribs and directions, and then use the equations of static equilibrium to obtain unknowns quantities.

Are there some real-world case studies of equilibrium by concurrent forces?

Equilibrium of coexistent forces becomes a decisive factor in structural engineering, architecture, mechanics, and even such daily situations as when situation as when balancing objects on shelves and the forces on vehicles are considered.