ENG 222 Dr. Krstic
STATICS OF A RIGID BODY
In this section, you will learn about the effects of concentrated forces acting on a rigid
body.
MAIN CONCEPTS
The following main concepts will be covered in this section.
Rigid Body
A rigid body is an object that does not experience deformations when being loaded
(pressed or pulled). In reality, all objects are deformable; however, rigid body
idealization is suitable in many problem-solving situations in which the deformations
are relatively small.
Loads
Loads are the forces and moments that a rigid body must support to perform its
function. Loads will tend to cause translation and/or rotation of the rigid body under
consideration.
Reactions
Reactions are the forces and moments that hold or constrain a rigid body in equilibrium.
They are called reactions because they react when other forces on the rigid body change.
If a load on a rigid body increases, the reaction forces will automatically increase in
response to maintain equilibrium.
Internal Forces
Internal forces are forces that hold the parts of an object together.
Free Body Diagram (FBD)
A free body diagram is the tool used to visually show the rigid body under consideration,
and all loads, reactions, and internal forces that influence the object.
Example
Consider the free body diagram for a truck where W represents the weight of the rigid
body and R1 and R2 represent the reactions that support the load (weight) and maintain
the equilibrium.
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ENG 222 Dr. Krstic
W
R2
R1
Principle of Transmissibility
The principle of transmissibility states that a force acting on a rigid body could be
replaced by an equal (magnitude and direction) force acting at a different point located
on the line of action, without having a different effect on the rigid body.
Compared to a particle that represents a unique point of application,
when acting on rigid bodies, forces are sliding vectors.
F
A

B
F
Force F is a sliding vector. The effect on the rigid body
is identical regardless of the point of application.
If you are ever confused, return to this section to revisit the main concepts.
VECTOR (CROSS) PRODUCT OF TWO VECTORS
The vector (cross) product is a multiplication operation applied to two vectors which
produces a third vector that is mutually perpendicular to both as a result.
Consider two concurrent vectors A and B contained within a plane .
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ENG 222 Dr. Krstic
C=AxB
A

B

The cross product operation is denoted by C = A x B, where C represents a vector
perpendicular to the plane .
The magnitude of a cross product (vector C) is defined as
|C| = |A| |B| sin 
The direction of a cross product is specified by the right-hand rule. There are two ways
to apply the right-hand rule: the three-finger method and the point-and-curl method.
In the three-finger method, align your index finger with the first vector and your middle
finger with the second, then your thumb will point in the direction of the cross product.
In the point-and-curl method, align your right hand with the direction of the first
vector. Then curl your fingers to follow the sense of rotation that will bring the first
vector in line with the second. The thumb defines the direction of the cross product.
Rotate 1st vector until it
aligns with the 2nd vector.
Consider this interactive tool to visualize the cross product of two vectors.
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ENG 222 Dr. Krstic
Note that the commutative property of the cross product is not preserved.
AxBBxA
C=AxB
A
A


B
Bring A in line with B.
B
C=BxA
Bring B in line with A.
Try applying the right-hand rule to determine the direction of the cross product (vector
C) in both cases.
RECTANGULAR COMPONENTS OF A CROSS PRODUCT
Cross Product of Unit Vectors
By using the point-and-curl method
y
ixj=k
j
i
z
x
Bring i in line with j.
k
However, when attempting to calculate the cross
product i x i, the result is zero.
|i x i| = |i| x |i| sin 0 = 0
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ENG 222 Dr. Krstic
We can conclude that mixed cross products will “survive”, and identical cross products
will “die”. To remember the order of the mixed cross products, use the cross-product
circle.
For example, to calculate i x j, you are going in the
positive (counterclockwise) direction around the blue
inner circle from i towards j to reach the answer k. But to
calculate j x i, you will have to go in the negative
(clockwise) direction around the red circle and thus get a
-k. Try using the circle to calculate other unit vector
cross products.
Cross Product of Arbitrary Vectors
For two vectors A = Ax i + Ay j + Az k and B = Bx i + By j + Bz k, the cross product A x B
can be calculated as
A x B = (Ax i + Ay j + Az k) x (Bx i + By j + Bz k) =
= AxBx (i x i) + AxBy (i x j) + AxBz (i x k) +
+ AyBx (j x i) + AyBy (j x j) + AyBz (j x k) +
+ AzBx (k x i) + AzBy (k x j) + AzBz (k x k)
Using the cross-product circle (mixed cross products will “survive”, identical will “die”),
the result can be reduced to
A x B = (AyBz – AzBy) i + (AzBx – AxBz) j + (AxBy – AyBx) k
where quantities in parentheses represent scalar components of the cross product vector
C:
Cx = AyBz – AzBy
Cy = AzBx – AxBz
Cz = AxBy – AyBx
It is important to remember that in the above equations, all scalar components of
vectors A and B should be accounted for with their correct signs (positive or negative).
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ENG 222 Dr. Krstic
A more streamlined method of calculating the cross product is by evaluating the
determinant of this 3x3 matrix:
Note that the first row contains the unit vectors, the second row contains the scalar
components of A, and the third row contains the scalar components of B. The
determinant of this matrix is calculated using the method of cofactors, as follows:
Each term contains a 2x2 determinant, which is evaluated with the formula:
Leading to the previously derived cross product result
A x B = (AyBz – AzBy) i + (AzBx – AxBz) j + (AxBy – AyBx) k
In practice, the easiest way to remember this equation is to use the augmented
determinant, where the first two columns have been copied and placed after the
determinant. The cross product is then calculated by adding the product of the red
diagonals (add when going from left to right) and subtracting the product of the blue
diagonals (subtract when going from right to left).
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You can use the following tools to assist you in calculating the cross product:
Cross Product Calculator
3x3 Determinant Calculator
Cross products are used in statics to find the moment of a force about a point.
MOMENT OF A FORCE ABOUT A POINT
What is a Moment?
When a force acts on a body, it can potentially produce two effects: translation of the
body in the direction of the force and rotation of the body about an axis. For a body in
equilibrium, we say that forces have a tendency to produce translation or rotation since
no actual motion occurs.
The rotational tendency of a force is called a moment, short for “moment of a force.” You
may remember using the term torque for the same quantity in physics. To an engineer,
torque means a moment about the long axis of an object that produces twisting and
torsional stresses. You will learn more about the torsion in your Strength of Materials
class.
Consider an example in which a wrench is used to tighten a nut at point A.
MA
Force F applied to the handle of the wrench creates a moment MA about the axis
perpendicular to the page through the center of the nut at point A. The M is bold
because moments are vector quantities, and the subscript A indicates the axis or center
of rotation. The direction of the moment can be either clockwise or counterclockwise,
depending on how the force is applied. If the nut is frozen, no actual rotation occurs, but
the force still produces a tendency to rotate (moment). This tendency increases with the
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ENG 222 Dr. Krstic
magnitude of the force, and also with the distance between the line of action of the force
and the center of rotation.
A moment of force is a measure of the tendency of that force to rotate a
body about a selected point or axis.
Consider this interactive tool to visualize the change in magnitude and direction of the
moment.
Moment of a Force
Consider points O and A located on a rigid body, with point A being a point on the line
of action of force F. Point O is called the point of interest and will be identified in the
problem as the center of rotation. Since force F is a sliding vector that can act at any
point along its line of action, we can assume that it is applied at point A. Point A is
referred to as the point of application. The location of point A with respect to point O is
defined by a position vector r.
The position vector r has its tail point located at the point of interest and its
head point located at the point of application.
y
M
Point of
O
Interest
x
r

z
x
F
A
Point of
Application
The moment M of a force F about point O is defined as the vector product of the
position vector r and force F:
M=rxF
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The direction of moment M is determined by the right-hand rule (follow the sense of
rotation that will bring r in line with F). Since r x F  F x r, it is important always to
calculate the moment as r x F to obtain a direction that complies with the direction
indicated by the right-hand rule.
The most direct way of calculating the moment is by solving the determinant.
First Row: Unit Vectors
Second Row: Position Vector (scalar components)
Third Row: Force Vector (scalar components)
Always include unit vectors in the first row, scalar components of the position vector in
the second row, and scalar components of the force vector in the third row.
The magnitude of a moment is calculated as
|M| = |r| |F| sin 
If we draw a line d perpendicular to the line of action of force F, we can define a right
triangle as shown in the figure:
y
M
Point of
O
Interest
x
r
d
z
x


F
A
Point of
Application
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The distance d represents the shortest distance between the point of interest and the
line of action of force F. Using basic trigonometry, distance d can be calculated as
d = |r| sin 
Substituting this into the equation for the magnitude of the moment will define the
minimum distance equation.
M=Fd
Distance d is called the arm of the moment.
The moment of a force has units representing any combination of the force and length
units. For example: N-m, lb-ft, lb-in, N-mm, etc.
Principle of Transmissibility
As stated before, point A can be located anywhere on the line of action of force F. Sliding
force F to a different point on its line of action has no effect on the moment and can
sometimes simplify the solution.
y
F
A2
F
r2
A1
r1
x
O
MO = r1 x F = r2 x F
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ENG 222 Dr. Krstic
Moment of a Force About a Point
y
B
F
u
P
r
A
O
x
z
To calculate the moment of a force F about point P, apply this strategy:
1. Identify the coordinates of the point of interest P (Px, Py, Pz). Note that the point
of interest doesn’t have to be located at the origin of the coordinate system.
2. Identify the coordinates of the point of application A (Ax, Ay, Az).
3. Draw the position vector r in the problem sketch starting from the point of
interest P (tail) and ending at the point of application A (head).
4. Resolve vectors r and F into rectangular components.
(A-P): r = rx i +ry j + rz k
F = F u = Fx i + Fy j +Fz k
Where u is a unit vector along the line of action of force F:
(B-A): u = AB / |AB|
5. Calculate the moment about point P as a vector product:
MP = MPx i + MPy j + MPz k
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6. If required, calculate the magnitude:
MP =  MPx2 + MPy2 + MPz2
7. If required, calculate the minimum distance between the point of interest P and
the line of action of force F:
MP = F d  d = MP / F
Try this interactive tool to reinforce the concept of a moment about a point.
Varignon’s Theorem
Varignon’s Theorem is a method used to calculate a moment about a point due to
concurrent forces developed in 1687 by French mathematician Pierre Varignon (1654 –
1722). It states that the sum of the moments of several concurrent forces about a point is
equal to the moment of the resultant of those forces. In other words, the moment of a
force about a point equals the sum of the moments of its components.
y
Q
P
r
F
A
O
z
x
MP = r x (F + Q) = r x F + r x Q
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y
Fy
r
P
O
z
F
A
Fx
Fz
x
MP = r x F = r x Fx + r x Fy + r x Fz
Example and Teamwork
Be prepared for class. Download and print the class problems handout from Canvas or
write directly in a PDF file. Alternatively, sketch the problem on a sheet of paper or in
your notebook before coming to class. There will be no time allotted for sketching the
problem in class.
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