george lungu - calculating the moment of force

8/11/2019 George Lungu - Calculating the Moment of Force

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Calculating the moment of force

by George Lungu

- This tutorial presents a few ways of calculating the moment of force or torque. It also

proves the torque of a couple is the same regardless of the position of the point about which

the torque is calculated. The implications in modeling aircraft dynamics are significant.

The cross product:

The cross product or vector product is a binary operation on two vectors

in three-dimensional space. It results in a vector which is normal to the

plane containing the first two vectors. The sense of the product vector isthe sense of advancement of a right hand screw turned as to overlap the

first vector to the second one on the shortest path. The magnitude is

equal to the area of the parallelogram defined by the two vectors.

The definition of moment of force:

The moment of force (sometimes named

torque) about a reference point is vector

equal to the cross product between the

position vector of the origin of the force (as

measured from the reference point) and the

force vector.

O r F

OM

FrMO

sin FrMO

P
http://en.wikipedia.org/wiki/File:Cross_product_parallelogram.svg


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- The three formulas for calculating the momentof force F about point O are perfectly equivalent.

- The equivalency results from basic trigonometryin the right triangles ONP and FCP. P F

O

N

TF

CF

P

F

xxO xP

yP

yyO

O

yF

xF

Three equivalent ways of calculating the moment of force:

TO FOPM

sin FOP

MO

FONMO

C

F

T

- Most of the times our setup is found in a Cartesian systemof coordinate where we already have the coordinates of the

points and the x-y components of the forces involved.- Thats why the following formulas are very useful (we addthe moment contributions of the x and y force components).

A useful moment-of-force calculation formula in a 2D Cartesian reference:

yOPxOPO FxxFyyM

An important application:

- If a force acts on a body in an arbitrary point P, the bodywill experience a linear acceleration of center of gravity andan angular acceleration around the center of gravity both incompliance with Newtons second law.

- If the body has a moment of inertia I, we can write theangular form of Newtons law (where is the angular

acceleration- the rate of change of angular speed):

F

FT

)sin(OP

ON)sin(

CG

P

Fa

IMO

td

d

dt

d

2

2

a


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The Couple:

-A Couple is a system of parallel, opposite and equal forceswith a resultant moment but no resultant force. Anotherterm for a couple is a pure moment. Its effect is to createrotation without any acceleration of the centre of mass.

-

The resultant moment of a couple is called a torque. This isnot to be confused with the term torque as it is used inphysics, where it is merely a synonym of moment. Instead,torque is a special case of moment. Torque has specialproperties that moment does not have, in particular theproperty of being independent of reference point aboutwhich it is being calculated. - Wikipedia

The end.

F

F

Fd

- The moment of a couple, called torque,

is independent on

the reference point

and its magnitude is equal to:

d

F

F

x

y

O

1r

2r

21 rr

Demo:

- If we calculate the resultant momentum of the twoopposite forces about an arbitrary point O we can write:

221121 FrFrMMM OOresultant

- Since the forces are equal, parallel but opposite we have:

FrrFrFr

2121

- But the vector difference is only dependent on the relative position of the force origins and not on theposition of point O. It is proves that the torque is independent on the position of the point with respectto which it is calculated.

-As an important implication is the fact that the aerodynamic moment of the wing or stabilizerwill directly add to the total moment of an airplane without any scaling.

george lungu - calculating the moment of force

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