Chapter 2 Review of Forces and Moments - Brown University

2.2 Moments
The moment of a force is a measure of its tendency to rotate an object about some point. The physical
significance of a moment will be discussed later. We begin by stating the mathematical definition of the
moment of a force about a point.
2.2.1 Definition of the moment of a force.
To calculate the moment of a force about some point, we need to know
three things:
A r-r A
F
1. The force vector, expressed as components in a basis
j r A
( Fx , Fy, F
z)
, or better as F= Fxi+ Fyj + Fzk
r
2. The position vector (relative to some convenient origin) of the
point where the force is acting ( x, yz , ) or better
r = xi+ yj+
zk
i
3. The position vector of the point (say point A) we wish to take moments about (you must use the
( A) ( A) ( A)
( A) ( A) ( A)
same origin as for 2) ( x , y , z ) or rA = x i+ y j+
z k
The moment of F about point A is then defined as
M = ( r− r ) × F
We can write out the formula for the components of
product
M = [( x− x ) i+ ( y− y ) j+ ( z− z ) k] × [ Fi+ F j+
Fk]
A A A A x y z
i j k
= ( x−xA) ( y− yA) ( z−zA)
F F F
x y z
A
A
M
A
in longhand by using the definition of a cross
{( y yA) Fz ( z zA) Fy} {( z zA) Fx ( x xA) Fz} {( x xA) Fy ( y yA)
Fx}
= − − − i + − − − j+ − − − k
The moment of F about the origin is a bit simpler
MO = r×
F
or, in terms of components
M = [ xi+ yj+ zk] × [ Fi+ F j+
Fk]
O x y z
i j k
= x y z
F F F
x y z
{ yFz zFy} { zFx xFz} { xFy yFx}
= − i + − j+ − k