SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

Moment of a force, Couple  



ole.gif

Moment of a force (or torque) about an axis. In Fig. 1 a body that is free to rotate about an axis AB is acted on by a force F. We wish to find the moment of F about the axis AB. In Fig. 2 is a cross-section S taken perpendicular to the axis of rotation AB with O indicating the axis. The force Fs shown in the figure is the component of F lying in the cross-sectional plane S (F is decomposed into a component parallel to the plane and a component perpendicular to the plane). The distance d from the axis O to the line of action of Fs is called the force arm or moment arm.


Def. 1 Moment of a force (or torque) about an axis. The moment of force F about the axis AB is defined as the product of the moment arm and force Fs i.e. 


ole1.gif

            Moment of F = d × Fs


The moment of force about an axis is a measure of the effectiveness of the force in producing rotation about the axis. If the effect of the force is to produce counterclockwise rotation, the moment is considered positive. If the effect is to produce clockwise rotation, the moment is negative. The moment is often considered to be a vector quantity. In this case the moment of F is defined as


            M = ole2.gif × Fs


where ole3.gif is the vector from the axis O to point C in Fig. 2. It is a vector perpendicular to the plane S.


 

Def. 2 Moment of a force about a point. The moment of a force F about a point O is the vector quantity r × F where r is the position vector from the moment center O to any point on the line of action of the force F. See Fig. 3. Its magnitude is the product of the magnitude of F and the perpendicular distance from O to the line of action of F. Its direction is perpendicular to the plane determined by O and the line of action of F.


Denote the moment of F about O by MO(F) and let

 

1)        r = xi + yj + zk,          F = Fx i + Fy j + Fz k,            MO(F) = Mx i + My j + Mz k


ole4.gif

Then, by definition


ole5.gif                                                                          

                                                                        

Expanding the determinant we get


MO(F) = i(Fzy - Fyz) + j(Fxz - Fzx) + k(Fyx - Fxy)


Comparing this expression with that for MO(F) in 1) above, we have


Mx = Fzy - Fyz,          My = Fxz - Fzx,          Mz = Fyx - Fxy


It can be shown that Mx, My, and Mz are the respective moments of the force F about the x, y, and z axes as defined by Def. 1 above (moment of force about an axis). It is this fact that makes the moment of force about a point, M = r×F, so very useful.

ole6.gif

                                                                                                  

Def. Moment center. The point about which moments are computed. Symbolizing moment as r × F, the moment center is the origin from which the position vector r to any point on the line of action of the force F is measured.


Def. Couple. A couple is a pair of parallel forces, equal in magnitude and opposite in direction. See Fig. 4. Two such forces cannot be combined into a single force since their sum in every direction is zero. Their sole effect is that of producing torque, a tendency towards rotation.


Moment of a couple. The moment of a couple about some moment center is given by the sum of the moments of the two forces constituting the couple about that moment center. Fig. 4 shows a couple consisting of two forces in the plane. The moment of the couple about point O (an axis of rotation perpendicular to the plane) is given by

            MO = (a + d)F - aF

or

            MO = Fd .


To find the moment of a couple in three-dimensional space we can employ the concept of the moment about a point. Consider Fig. 5. The moment C of the couple shown there, consisting of two opposing parallel forces F separated by a distance d, is given by


            C = Mo(F) + Mo(-F) = r1 × F + r2 × -F = (r1 - r2) × F


or


            C = a × F


where a = r1 - r2


By definition of the cross product, the magnitude of C is |a × F| = aFsin θ and a sin θ is equal to d,

ole7.gif

the perpendicular distance between the forces.


The moment M of a couple is a vector perpendicular to the plane of the two forces. Its magnitude is Fd where F is the magnitude of the forces and d is the perpendicular distance between them. Its direction is given by the right-hand convention: Curl the fingers of the right hand in the direction in which the couple tends to cause rotation and the thumb points in the direction of the couple.


Couples obey all the laws of vectors. The magnitude of a couple is the same for all moment centers.

ole8.gif

Resolution of a force into a force and a couple. Consider a body with a force F acting at point A as shown in Fig. 6a. At any other point B two equal and opposite forces F may be applied with no external effect on the body as shown in Fig. 6b. If these forces are parallel to the original force F, then a couple M = Fd is formed by the original F and the force F in the opposite direction at B. In Fig. 6c we see how the original force F at point A has been replaced by a force of the same magnitude and direction at point B and a couple. The magnitude of the couple is Fd, the product of the magnitude of F and the distance through which its line of action has been shifted. A force can always be replaced by an equal force having any parallel line of action and the corresponding couple. We now state this with vector notation in the following theorem:

 

Theorem1. A single force F acting at point A may be replaced by an equal and similarly directed force acting at any other point B and a couple C = r×F where r is the vector from B to A.  

It follows that a given force and couple which act in the same plane may be combined to yield a single equal force having a unique line of action.

 

 

References.

 McLean, Nelson. Engineering Mechanics. (Schaum)

 Meriam. Mechanics, Part I - Statics.



More from SolitaryRoad.com:

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together

Television

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

Sizing up people

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


[ Home ] [ Up ] [ Info ] [ Mail ]