Moment of a Force about a line
The capstan was used in the days of square-rigged sailing ships:
Crewman turned it by pushing on the handles (Case a), providing
power for such tasks as raising anchors & hoisting yards
A vertical force F (case 2) applied to one of the handles does not
cause the capstan to turn, even though the magnitude of the
moment about point P is d|F| in both cases.
The measure of the tendency of a force to cause rotation about a
line or axis is called the moment of the force about the line
Consider a line L & force
:
Let
be the moment of about an arbitrary point P on L, the
moment of about L is the component of parallel to L, which
is denoted by , the magnitude of the moment of about L is
| |and when the thumb of the right hand is pointed in the
direction of
, the arc of the fingers indicates the direction of
the moment about L
| |=
=
)
In terms of a unit vector
=(
=[
)
)]
along L,
is given by:
| |=
=
)
The mixed triple product in this expression is given in terms of the components of the 3
vectors by:
|
|=
Notice that the value of the scalar
direction of
The absolute value of
=
) determines both the magnitude &
is the magnitude of
If
is positive,
points in the direction of
If
is negative,
points in the direction opposite to
The result obtained doesn’t depend on which point on L is chosen to determine
Whether we use point P or P’ to determine the moment of
result
[
)]
={
) ]}
because
=[
)+
=[
)]
is perpendicular to
)]
=
about L, we get the same
 Important Notes:
1.
When the line of action of → is perpendicular to a plane containing L, the
magnitude of the moment of → about L is equal to the product of the
magnitude of → and the perpendicular distance D from L to the point
where the line of action intersects the plane: | |= |→|D
2.
When the line of action of → is parallel to L, the moment of → about L is
zero:
= 0. Since
= →  → is perpendicular to →,
L and the vector component of
3.
is perpendicular to
parallel to L is zero
When the line of action of → intersects L, the moment of → about L is zero.
Since we can choose any point on L to evaluate
, we can use the point
where the line of action of → intersects L. The moment
is zero, so its vector component parallel to L is zero
about that point
Summary
In summary, determining the moment of a force about a point P requires 3 steps:
1. Determine a vector :
Choose any point P on L and determine the components of a vector from P to any
point on the line of action of .
2. Determine a vector :
Determine the components of a unit vector along L (either direction).
3. Evaluate :
Calculate
= 
& determine
using
=[
)]
Example:
determine the moment of force
about line L.
Example:
What is the moment of the force
about the axis of the bar BC?
Example:
The force
of this force about the x-axis and a-axis
acts at point A as shown. Determine the moments