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Mechanics: Statics




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Moment of a force in 2D about a point

When an application is subjected to coplanar forces, forces in a plane and the effect of the depth on structure analysis is negligible, the problem can be simplified into two dimensional analysis. Because the direction of moment of a force is always normal to the plane joining the reference point and the line of action, the moment of forces in a plane about a point can therefore refer to the sense of the moment about a point.

image 

According to the right hand rule, a counterclockwise moment about a reference point can be taken as positive, while a clockwise moment about a reference point can be taken as negative. 

image 

Moment Addition of Coplanar Forces

The total moment of forces about a reference point can be determined by adding up all moments of forces.

image 

Therefore, the magnitude of the effective moment , M, can be expressed as:

image 

Moment of Concurrent Forces

For concurrent forces, the moment of the resultant of concurrent forces is also equal to the resultant force times the prependicular distance between the reference point and the line of action of the rusultant force. The effective moment can also be computed by adding up all individual moment of concurrent forces.

image 

Therefore, the magnitude of the moment , M, can be expressed as:

image 

Moment Decomposition

Since the effective moment of concurrent forces about a reference point is equal to the sum of the moments of the concurrent forces as stated in Varignon's Theorem. In other words, moment of a force can also be decomposed through force decomposition.

A force, F in a plane can be decomposed into two retangular components, Fx and Fy. The moment, M of force, F can therefore be decomposed into MFx and MFy

image 

Therefore, the magnitude of the moment , M, can be expressed as:

image 

Moment Vector

When the force vector is decomposed into two rectangular components, the corresponding perpendicular distance components are also equal to the magnitudes of the two rectangular components of the postition vector, r of the point of application. And the moment vector, M becomes the third dimension of the rectangular coordinate system.

image 

Therefore the force vector, F and position vector, r can be expressed as:

image  and  image 

where the minus sign indicates the direction as usual

The moment of a force, M about a reference point can be defined as the vector product of r and F.

image

Imply

image     

Since the magnitude of vector product is the product of the magnitudes of the two vectors and the sine of the angle formed by the vectors, imply

image

Therefore, the vector product of a unit vector with itselt is equal to a zero vector, imply

image  and  image 

For the sense of the direction, the vector product of two different unit vectors is equal to the unit vector in the third dimension according to the right hand rule, imply

image  and  image 

Substitute all variables into the moment vector equation, imply

image     


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References

  1. I.C. Jong; B.G. rogers, 1991, Engineering Mechanics: Statics and Dynamics, Saunders College Publishing, United States of America
  2. F.P. Beer; E.R. Johnston,Jr.; E.R. Eisenberg, 2004, Vector Mechanics for Engineers: Statics, McGraw-Hill Companies, Inc., New York
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ID: 110600001 Last Updated: 6/9/2011 Revision: 1 Ref:

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