Moments of ForceMoments of Force

D. Gordon E. Robertson, PhD, FCSB

Biomechanics Laboratory,

School of Human Kinetics,

University of Ottawa, Ottawa, Canada

D. Gordon E. Robertson, PhD, FCSB

Biomechanics Laboratory,

School of Human Kinetics,

University of Ottawa, Ottawa, Canada

Moment of a Force (when F is at 90º to d)

• turning effect of a force, also called torque

• product of force (F) and moment arm (d) of the force from the axis (A) of rotation

• moment arm is the perpendicular distance from the axis of rotation to the line of the force

M = F d

force (F)

axis (A)

line of action of force

momentarm (d)

Moment of a Force (when F is at 90º to d)

M = F d• direction (+ / − sign) of

the moment of force depends on the right-hand rule

• i.e., counter-clockwise is positive

• units are newton metres or N.m

force (F)

axis (A)

line of action of force

momentarm (d)

Moment of a Force (r, F, )

• if moment arm length is difficult to compute use:

M = r F sin • r is length of line from

axis to line of force

• theta is angle between line of force and line of r

force (F)

axis (A)

line of action of force

line from Ato force (r)

angle betweenr and F ()

Moment of a Force (r, F, )

M = r F sin • direction (sign) of

moment follows right-hand rule

• i.e., if force “turns” line r counter-clockwise about axis at A then moment is positive

force (F)

axis (A)

line of action of force

line from Ato force (r)

angle betweenr and F ()

Moment of a Force (r, F, )

• to simplify and clarify positive directions of moments and forces, add reference axes to each figure

force (F)

axis (A)

line of action of force

line from Ato force (r)

angle betweenr and F ()

+

• positive directions are defined by the arrows

• add axes labels (X, Y) for additional clarity

Y

X

Moment of a Force (r, F, )

Factors that increase moment of force

• increase force (F)• increase lever arm

length (r)• increase angle ()

between lever and line of force to perpendicular

M = r F sin

force (F)

axis (A)

line of action of force

line from Ato force (r)

angle betweenr and F ()

Moment of a Force (cross-product)

• if components of force and line connecting axis to line of force are known, use vector cross-product:

M = r x F

force (F)

axis (A)

r

Moment of a Force (cross-product)

• first resolve vectors r and F into their rectangular coordinates

• then apply:

M = r x F

= ( rx Fy − ry Fx ) k

• k is the unit vector about the Z-axis

axis (A)

force (F)

axis (A)

r

Moment of a Force (cross-product)

• first resolve vectors r and F into their rectangular coordinates

• then apply:

M = r x F

= ( rx Fy − ry Fx ) k

• k is the unit vector about the Z-axis

force (F)

axis (A)

r

Fy

Fx

ry

rx

Moment of a Force (cross-product)

• put components at their original points of application

force (F)

axis (A)

Fy

Fx

ry

rx

Moment of a Force (cross-product)

• put components at their original points of application

• next slide force vectors along their lines of action and multiply

force (F)

axis (A)

r

Fy

Fx

ry

rx

Moment of a Force (cross-product)

• if only scalar part of the moment is wanted, use this notation:

M = [ r x F ]z

M = ( rx Fy − ry Fx )

force (F)

axis (A)

r

Fy

Fx

ry

rx

Example

Given:

r = (20.0, −65.0) cm

F = (220, 150.0) N

M = [ rx Fy − ry Fx ]z

=[ (20.0 150.0) −

(-65.0 220) ]

= 3000 + 14 300

= 17 300 N.cm

= 173.0 N.m

force (F)

axis (A)

r

Fy

Fx

ry

rx