Just A Moment!**

 Moments Forces    Moments of Forces   Moments of Couples    Equilibrium

Definitions:

Center of Resistance (C.res): the point on a body (tooth) where a single force would produce translation (similar to the “center of mass”)

Force: any action or influence that accelerates an object. Force is a vector, which means it has direction and magnitude

Moment: the tendency to cause rotation around a point or axis

Moment of Force: with regard to a line or point – the turning effect of the force with respect to that line or point: a tendency to rotate.  A tooth that receives an applied  force not acting through the center of resistance “feels” a tendency to rotate, or a “moment from the applied force”.

Couple: two equal and opposite forces separated by a perpendicular distance.  Equal and opposite forces that are not collinear.

Moment of a Couple:  the rotational tendency produced by a “couple”

All force systems applied to a tooth are composed of either single forces and /or couples. The application of a force through the center of resistance of a tooth will result in translation of the tooth. The application of a force to act at points other than through the center of resistance will produce different tendencies for rotation.  Tooth rotation resulting from the application of a force always creates a simultaneous tendency to move the center of resistance of a tooth in the direction the force is acting.  In contrast, the location of a couple on a tooth is irrelevant to the resulting tooth movement. A couple can never move the center of resistance, and with a couple the center of rotation and the center of resistance will always be coincident. The equilibrium forces, associated with a moment of a couple, also are single point forces and can produce different tooth movements depending on where they are applied.  All tooth movement must be either translation and/or rotation as defined at the tooth’s center of resistance.  (1955 by Saunder’s Company)

 CENTER OF RESISTANCE

Translation (bodily movement): If a tooth were a motionless free body is space, any point acting through the center of mass would cause “all points on the tooth to move the same amount in the same direction”.  The in vivo tooth is not a free  body and is attached to the supporting tissues. Therefore the "center of resistance"  (C.res) not the term "center of mass" is used.  Any force acting through the center of resistance causes the tooth to translate.

A force acting through the center of resistance will cause will cause all points on the tooth to move the same amount in the same direction. This is termed translation and is possible in any direction.

MOMENT OF FORCE

The rotational tendency, or moment produced by a force not acting through the Center of Resistance is expressed as the Moment of  Force (M.F.).

First-order rotation                        Second-order rotation                    Third-order rotation

The magnitude of the MF is measured as the magnitude of force (F) multiplied by the perpendicular distance (d) between the line of force and the center of resistance (MF= F x D). In orthodontic applications the unit of moments of force are expressed in terms of the force multiplied by the distance or g mm.

A tooth that receives a force not acting through the C.res. feels a moment or tendency to rotate/ the magnitude is measured as the magnitude of the force times the perpendicular distance from the line of the force to the C.res and is expressed in force . distance units Mf = F.d

A COUPLE AND CENTER OF RESISTANCE

An arch wire may send a signal to the periodontium for tooth movement via a pair of equal and opposite noncollinear forces.  This force system of a couple is the sum of the force systems of the two equal and opposite forces that comprise the couple.

Note that alone each force of a couple would move the C.res in the direction as describe by a single-point force.

Therefore because the forces are equal and opposite, each force tends to move to move the C.res in an equal and opposite direction. So, no movement of the C.res can ever result from the application of a couple, no matter where the couple is placed on the tooth.

Moment of a couple resulting in first-order rotation. The two forces fo the couple on each bracket are located equidistant from the C.res  The C.res and the center of rotation will be coincident.  Mc = F x d	The moment of a couple in a second-order rotation where the two forces of the couple are not located equidistant for the Cres. No matter where the  couple is located on the tooth the C.res is always coincident with the center of rotation.

The moment of a couple in a third-order rotation where the two forces of the couple are not located equidistant for the C.res. No matter where the  couple is located on the tooth the C.res is always coincident with the center of rotation.	The moment of a couple in a third-order rotation where the two forces of the couple are not located equidistant for the C.res. No matter where the  couple is located on the tooth the C.res is always coincident with the center of rotation.

ADDITIVE AND SUBTRACTIVE COUPLES

Alone each force of a couple would move the C.res in the directions describe by a single-point force.  When the line of force of each of the two forces of the couple are located equidistant from the C.res both forces of the couple tend to rotate the tooth in the same direction around the C.res

Moment of a couple resulting in first-order rotations.  The two forces of the couple on each bracket are located equidistant from the C.res.  The C.res and the center of rotation will be coincident. Mc = F x d.	Moment of a couple resulting in second-order rotations.  The two forces of the couple on each bracket are located equidistant from the C.res.  The C.res and the center of rotation will be coincident. Mc = F x d.

When the lines of force of each of the two forces of the couple are not located equidistant from the C.res they still produce exactly the same rotational tendency. In the figure shown below  a third order couple in the bracket shows lines of acting at different distances from the C.res (d1 & d2). The force closer to the C.res produces a smaller moment in a clockwise direction (F x d1).  The force located further from the C.res (d2) produces and larger moment in an opposite (counterclockwise) direction.  When the moment for clockwise rotation (labial root torque) is subtracted from the moment for counterclockwise rotation, the remainder is a moment in a counterclockwise direction (lingual root torque exactly as if the couple were positioned with the two forces equidistant from the C.res, as shown above.

Moment of a couple resulting in a third-order rotation where  the two forces of the couple are not located equidistant from the C.res. No matter where the couple is located on the tooth, the C.res is always coincident with the center of rotation.	Moment of a couple resulting in a third-order rotation where  the two forces of the couple are not located equidistant from the C.res. No matter where the couple is located on the tooth, the C.res is always coincident with the center of rotation.

EFFECTS OF FIRST PERMANENT MOLAR ROTATIONAL CORRECTION

Note that the mesial aspect of the molar moves distally around the center of resistance (CR). However, the molar is not moved distally in a bodily fashion.

Key question: Examine the study casts from the lingual. Is the molar occlusion Class I on the lingual, but rotated to Class II on the buccal If the molar occlusion is Class II on the lingual, molar rotation will not result in a Class I occlusion.

** The above work was inspired by a lecture given by Gerald Samson, DDS,  at the 103 annual session of the American Association of Orthodontists 2003 -- On Hawaiisliand (Volcano Island) at the Wiakiloa Hilton Hotel  about 10 miles North of Kialua Kona, Hawaii where my brother lives.

Gerald Samson  (g)nathos, inc. gnathosconted.com  Email: geraldsamson@mindspring.com  Atlanta, Ga 770 850 0631