...
Moment
...
Arm
...
| Excerpt |
|---|
...
Also |
...
called |
...
the |
...
"lever |
...
arm", |
...
the |
...
moment |
...
arm |
...
is |
...
the |
...
distance |
...
of |
...
closest |
...
approach |
...
between |
...
the |
...
...
...
...
of |
...
a |
...
force |
...
and |
...
the |
...
...
...
...
. |
...
It |
...
is |
...
used |
...
to |
...
compute |
...
the |
...
...
produced |
...
by |
...
the |
...
force |
...
about |
...
the |
...
axis |
...
of |
...
rotation. |
Geometrical Definition
Illustration of the Method
The moment arm of a force about a specific axis of rotation can be found geometrically by constructing the force's line of action and then finding the shortest distance between the line of action and the axis. The procedure is shown in the figures below.
Vertical Forces
|
|
|
Given forces. | Construct line of action for each. | Find the shortest distance (shortest distance |
Horizontal Forces
|
|
|
Given forces. | Construct line of action for each. | Find the shortest distance (shortest distance |
General Angles
|
|
|
Given forces. | Construct line of action for each. | Find the shortest distance. |
Key Points
Some key points to remember:
- The moment arm must be perpendicular to the force.
- The moment arm is never perpendicular to the position vector that gives the point of application of the force with respect to the axis of rotation.
- The moment arm for vertical forces (e.g. gravity) is always perfectly horizontal.
- The moment arm for horizontal forces is always perfectly vertical.
- The moment arm must always be the shortest distance between the line of action and the axis, so it will always be less than or equal to the distance from the force's point of application to the axis of rotation.
Utility
Calculating Torque
The moment arm is often given the symbol:
| Latex |
|---|
{excerpt} h3. Geometrical Definition h4. Illustration of the Method The moment arm of a force about a specific [axis of rotation] can be found geometrically by constructing the force's [line of action] and then finding the shortest distance between the line of action and the axis. The procedure is shown in the figures below. h5. Vertical Forces |!vertmarm1.png|width=200!|!vertmarm2.png|width=200!|!vertmarm3.png|width=200!| |Given forces.|Construct line of action for each.|Find the shortest distance (shortest distance \\ always perpendicular to line of action, which\\ implies horizontal moment arm for vertical force).| h5. Horizontal Forces |!hormarm1.png|height=160!|!hormarm2.png|height=175!|!hormarm3.png|height=175!| |Given forces.|Construct line of action for each.|Find the shortest distance (shortest distance \\ always perpendicular to line of action, which\\ implies vertical moment arm for horizontal force).| h5. General Angles |!oddmarm1.png|width=180!|!oddmarm2.png|width=300!|!oddmarm3.png|width=300!| |Given forces.|Construct line of action for each.|Find the shortest distance.| h4. Key Points Some key points to remember: * The moment arm must be perpendicular to the force. * The moment arm is *never* perpendicular to the position vector that gives the point of application of the force with respect to the axis of rotation. * The moment arm for vertical forces (e.g. gravity) is always perfectly horizontal. * The moment arm for horizontal forces is always perfectly vertical. * The moment arm must always be the shortest distance between the line of action and the axis, so it will always be less than or equal to the distance from the force's point of application to the axis of rotation. h3. Utility h4. Calculating Torque The moment arm is often given the symbol: {latex}\begin{large}\[ \mbox{moment arm = } r_{\perp}\]\end{large}{latex} |
When
...
the
...
moment
...
arm
...
for
...
a
...
given
...
force
...
F
...
about
...
a
...
chosen
...
...
...
...
is
...
known,
...
the
...
magnitude
...
of
...
the
...
...
due
...
to
...
F
...
about
...
the
...
axis
...
is:
| Latex |
|---|
}\begin{large}\[ |\tau| = Fr_{\perp} \]\end{large}{latex} {td} {tr} {table} {live-template:RELATE license} |