Equilibrium of a Particle and Force System Resultants

Equilibrium of a Particle 

Equilibrium:

A particle is said to be in equilibrium if it remains at rest or moving with constant velocity and does not rotate about any axis.
This condition of rest or moving with constant velocity is also called Static Equilibrium.

Coplanar Force Systems:

If a particle is subjected to a system of coplanar forces that lie in the x-y plane then every forces can be resolved into its components.

Since, it is 2D so the equilibrium equations are:

Σ Fx = 0

Σ Fy = 0

3D Force System:

In case of three-dimensional force system, we can resolve the forces into their respective rectangular components.

So the equations of equilibrium are:

Σ Fx = 0

Σ Fy = 0

Σ Fz = 0

Force System Resultants 

When a force is applied to a body it will produce a tendency of rotation of body about a point which us not on the line of action of force.

Moment of a Force (Scalar Formulation):

In this sense, the momen is considered as scalar and us calculated as:

Moment = (Force) × (Moment Arm)

M = Fd

Cross Product Formulation  of Moment of A Force:

The moment of a force will be formulated using Cartesian vectors is given by:

Principle of Moment:

Principle of moment or Varignon's Theorem states that:

The moment of a force about a point is equal to the sum of the moments of the components of the force about that point.

This method is generally easier than finding the same moment using M = Fd.

Moment of a Force about a Specified Axis:

To determine the turning effect of a force, we can use scalar or vector analysis.

1. Scalar Analysis:

By using the scalar analysis of Turning effect of a force, we have:

M = Fd

2. Vector Analysis:

The vector analysis of moment of a force about an axis is given by:

Couple:

It is defined as:

Two parallel forces that have the same magnitude bit opposite directions and are separated by a distance (d). 

Moment of Couple:

The moment produced by a couple is called Moment of a Couple.

We determine the moment of Couple of a system by the algebraic sum of moments acted on the object, which is given by:

M = Ma + Mb

Couple moment is a free vector, so it only depends on the position vector between the forces.

There are two analysis of formulating couple moment:

1. Scalar Formulation:

The moment of Couple by using scalar Formulation is given by :

M = Fd

2. Vector Formulation:

The moment of Couple by using vector Formulation  is given by:

M = r × FΣ

Equivalent Couples:

If two couples produce a moment with the same magnitude and direction, then these two couples are equivalent. 

Resultant Couple Moment:

Since, couple moments are vectors, their resultant can be determined by vector addition.

M = Σ(r × F)

Or 

Resultant Force = ΣF 

Resultant Moment = ΣM at origin + ΣM

Or 

Resultant Force along x = ΣFx

Resultant Force along y = ΣFy

Resultant Moment at O = ΣMo + ΣM

Reduction of a Simple Distributed Loading:

A uniformly distributed load is a load that is distributed or spread across the whole region of an element. In other words, the magnitude of the load remains uniform throughout the whole element.

Magnitude of Resultant Force:

The Magnitude of resultant force Fr is equivalent to the sum of all the forces in the system, which is given by:

Location of Resultant Force:

The location of force or the geometric center or centroid of the area under the distributed loading is given by:

And, the location of centroid of area under distributive loading for different shapes are:

References:

• www.youtube.com
• www.wikipedia.com
• http://web.aeromech.usyd.edu.au/statics/doc/friction/Friction1.htm
• From Book Engineering Mechanics statics and dynamics by R.C. Hibbeler