1. ES2501: Statics/Unit 9-1: Moment of Force (3D cases)
General:
- To extend the definition of the moment to 3D problems a vector
definition based on the cross-product of two vectors must be used.
The moment of a force is a vector whose direction represents the
axis of rotation and whose magnitude represents the moment of
the force about the axis.
Two definitions are equivalent but the vector definition is more
convenient for 3D problems.
- Need knowledge of the cross-product of two vectors

2. ES2501: Statics/Unit 9-2: Moment of Force (3D cases)
Cross-Product of two vectors:
Definition:
new vector
Note: The result
is a Vector
Applications:
Moment of a force
Angular momentum
Area of the parallelogram spanned by the two vectors

3. ES2501: Statics/Unit 9-3: Moment of Force (3D cases)
Cross-Product of two vectors:
Properties:
( Non-commutative)
( associative with multiplication
by a scalar))
( distributive)
( condition for parallel lines)
( cross-product of
coordinate unit
vectors)
Equivalence??
Expression in terms
of Cartesian Components

4. ES2501: Statics/Unit 9-4: Moment of Force (3D cases)
Cross-Product of two vectors(con’d):
Ezample:
Check:
The same results!

5. ES2501: Statics/Unit 9-5: Moment of Force (3D cases)
Moment of a force about a point (Vector Form):
Moment of about point O
Position vector from O to ANY point A on the line of action of the forceabout point O
General Comments:
Moment of a force about a point is represented by a VECTOR determined
by the cross product of a position vector from the point to any point on
the line action of the force and the force itself.
Direction: use the right-hand rule
Magnitude:
For a system of forces:
-A can be ANY point on the line action of the force
In consistent with
2D definition!
Additive
the same d !!

6. ES2501: Statics/Unit 9-6: Moment of Force (3D cases)
Calculation of Moment in term Cartesian Components:
Example: Fine the total moment of all three forces
about point O
easier
systematic

7. ES2501: Statics/Unit 9-7: Moment of Force (3D cases)
Moment of a Couple
Couple a pair of forces with the same magnitude and
opposite direction
Moments of a Couple:
Distance between lines of action of the couple of force A
Position vector from B to A
A --- Any point on the line of action of
B --- Any point on the line of action of
Moments of
a Couple B O’ O
Moments of a couple about ANY point is the same