ES 101 Study Guide Module G 2021 Sept 17
Hi, the ES 101 lecture instructors are collaborating to bring the Study Guides to you.
”Gyradius” is our original coined [short] word to stand for “radius of gyration” or
“gyration radius” of a volume of rigid mass. Gyration basically means spinning.
What are the main ideas in this module when applied to a Rigid Body whose mass is
distributed over a certain shape? (Compare these with a mass that is assumed to be
lumped at a Particle.)
(1) Centroid is the effective location of the mass* as if concentrated in a particle,
being also the effective point of action of the weight.*
(2) Gyradius is the effective radius of an imaginary circle, where the mass* seems to
be concentrated at the perimeter when the body spins about an axis at the center
of that imaginary circle.
(3) In most cases -- but not all -- we are interested in spinning or gyration about an
axis that passes through the centroid; hence concepts (1) and (2) are frequently
related.
(4) *mass and weight, unless otherwise described, refer to the total mass and total
weight of the body, respectively.
1
ES 101 Study Guide Module G 2021 Sept 17
2
ES 101 Study Guide Module G 2021 Sept 17
3
ES 101 Study Guide Module G 2021 Sept 17
For each letter of CENTROID, do you think the red dot fairly approximates the location
of the centroid of that letter?
4
ES 101 Study Guide Module G 2021 Sept 17
ALERT:
The basic definition of CENTROID as stated above is as a property of SHAPE.
Later in this module we shall explain how and why CENTROID is frequently – but not
always – associated with Center of Gravity or Center of Gravitational Force.
5
ES 101 Study Guide Module G 2021 Sept 17
Can you plot approximately (with a red dot) the centroid of each shape?
6
ES 101 Study Guide Module G 2021 Sept 17
7
ES 101 Study Guide Module G 2021 Sept 17
REFFLECT ON:
Table look-up method and composite-shape method are required in ES 101. You are
REQUIRED to remember the formulas for those basic shapes that are presented or
used in this Study Guide, the Sample Problems, and the Assignment Problems.
Integration method is optional in ES 101; yet with your background in calculus or
elementary analysis of Math 22, you may use integration method to derive the
formulas for the first moment of any defined shape, or the second moment, with
respect to any specified axis.
8
ES 101 Study Guide Module G 2021 Sept 17
9
ES 101 Study Guide Module G 2021 Sept 17
10
ES 101 Study Guide Module G 2021 Sept 17
Observe that, when there is a plane of symmetry the centroid lies in that plane.
The table here is for the coordinates that cannot be inferred on the basis of
symmetry.
11
ES 101 Study Guide Module G 2021 Sept 17
Which (red dot) of three choices best approximates the centroid of the shape?
12
ES 101 Study Guide Module G 2021 Sept 17
Which (red dot) of three choices best approximates the centroid of the shape?
13
ES 101 Study Guide Module G 2021 Sept 17
Which (red dot) of three choices best approximates the centroid of the shape?
14
ES 101 Study Guide Module G 2021 Sept 17
Which (red dot) of three choices best approximates the centroid of the shape?
15
ES 101 Study Guide Module G 2021 Sept 17
Which (red dot) of three choices best approximates the centroid of the shape?
16
ES 101 Study Guide Module G 2021 Sept 17
Which (red dot) of three choices best approximates the centroid of the shape?
17
ES 101 Study Guide Module G 2021 Sept 17
Would you agree?
18
ES 101 Study Guide Module G 2021 Sept 17
19
ES 101 Study Guide Module G 2021 Sept 17
20
ES 101 Study Guide Module G 2021 Sept 17
21
ES 101 Study Guide Module G 2021 Sept 17
REFFLECT ON:
Integration method is optional in ES 101; yet with your background in calculus or
elementary analysis of Math 22, you may use integration method to derive the
formulas for the first moment of any defined shape, or the second moment, with
respect to any specified axis.
Table look-up method and composite-shape method are required in ES 101. You are
REQUIRED to remember the formulas for those basic shapes that are presented or
used in this Study Guide, the Sample Problems, and the Assignment Problems.
22
ES 101 Study Guide Module G 2021 Sept 17
23
ES 101 Study Guide Module G 2021 Sept 17
24
ES 101 Study Guide Module G 2021 Sept 17
25
ES 101 Study Guide Module G 2021 Sept 17
ALERT:
In ES 101, we assume that you have learned the necessary calculus to locate the
centroid of a shape by integration; but we prefer that you need not perform the
calculus from scratch, and instead use known results from Tables and also combine
such tabulated results in locating the centroid of a so-called composite shape (see
sample problems below).
26
ES 101 Study Guide Module G 2021 Sept 17
27
ES 101 Study Guide Module G 2021 Sept 17
ALERT:
A void or hole in a composite shape is treated as a NEGATIVE AREA (or length or
volume, depending on the dimensions).
Depending on the chosen axes, which may be chosen to take symmetry into
consideration (as in the rightmost figure above), NEGATIVE COORDINATE/s may be
assigned to certain points.
28
ES 101 Study Guide Module G 2021 Sept 17
Center of Gravity
Center of Gravitational Force
Either of the above terms is the label for the point where the force of gravity is
effectively acting.
Center of gravity coincides with centroid when:
(1) The body is rigid in shape.
(2) The body is homogeneous in mass density.
(3) The body is surrounded by uniform gravity field.
29
ES 101 Study Guide Module G 2021 Sept 17
ALERT:
Note 3 significant figures (SIG FIGs) in each final answer.
SIG FIG is discussed in detail in Module A.
30
ES 101 Study Guide Module G 2021 Sept 17
31
ES 101 Study Guide Module G 2021 Sept 17
Can you confirm by table look-up the location (coordinates) of the centroid of each
component area, particularly the triangle and the semicircle?
32
ES 101 Study Guide Module G 2021 Sept 17
NEGATIVE:
The circle is a negative area (the contribution of the area has to be subtracted)
because it is a VOID or HOLE.
The ybar of the triangle is a negative coordinate because it is BELOW THE ORIGIN.
(By the way, for a triangle, the 1/3*height applies both ”vertically from a horizontal
base” and also “horizontally from a vertical base” as long as “height” is taken as
perpendicular distance from base to apex.)
33
ES 101 Study Guide Module G 2021 Sept 17
Note 3 significant figures (SIG FIGs) in each final answer.
SIG FIG is discussed further in Module A.
EVALUATE AND REFLECT ON:
Is it reasonable that xbar is less than 60 mm?
What if the y-axis origin is located at the lower point on the left (60 mm lower)? How
would the computations change? How would the coordinates of the centroid of the
composite area change?
34
ES 101 Study Guide Module G 2021 Sept 17
35
ES 101 Study Guide Module G 2021 Sept 17
Can you confirm by table look-up the location (coordinates) of the centroid of each
component volume, particularly the hemisphere and the cone?
For example, see Fig. 5.21 (Slide 11).
36
ES 101 Study Guide Module G 2021 Sept 17
NEGATIVE:
The cone is a negative volume (the contribution of the volume has to be subtracted)
because it is a VOID or HOLE.
The xbar of the hemisphere is a negative coordinate because it is TO THE LEFT OF THE
ORIGIN O.
37
ES 101 Study Guide Module G 2021 Sept 17
SEE SYMMETRY?
x is an axis of symmetry.
The centroid lies on the axis of symmetry. Check.
The centroid may be inside or outside the volume.
"Pwede naman na nasa outside ng object ang centroid" INDEED: in this case, the
centroid of the COMPOSITE volume is 15.00 mm to the right of origin O, "lumulutang
sa conical void" di ba? J
See the same figure in Slide 18 earlier.
ALERT:
Note 4 significant figures (SIG FIGs) in the final answer whose first digit is a ”1”
SIG FIG is discussed in detail in Module A.
38
ES 101 Study Guide Module G 2021 Sept 17
39
ES 101 Study Guide Module G 2021 Sept 17
REFFLECT ON:
Integration method is optional in ES 101; yet with your background in calculus or
elementary analysis of Math 22, you may use integration method to derive the
formulas for the first moment of any defined shape, or the second moment, with
respect to any specified axis.
Table look-up method and method of composite shape are required in ES 101. You
are REQUIRED to remember the formulas for those basic shapes that are presented
or used in this Study Guide, the Sample Problems, and the Assignment Problems.
40
ES 101 Study Guide Module G 2021 Sept 17
41
ES 101 Study Guide Module G 2021 Sept 17
A semicircular thin slab with differential thickness (thinness?) dx is just like a
semicircular area multiplied by the differential length dx.
By table look-up, can you confirm the xbar and ybar (coordinates of centroid) of such
slab element (still in terms of [local] radius r)?
Then, when [local radius] r is replaced in terms of x, the integration becomes fully in
terms of one variable, x.
42
ES 101 Study Guide Module G 2021 Sept 17
43
ES 101 Study Guide Module G 2021 Sept 17
REFLECT ON:
See the same figure in Slide 17 earlier.
Can you visualize the coordinates xbar and ybar?
Is xbar the same or equivalent as for a full (not half) cone? See the full cone in Slide
11 earlier (caution: distinguish the origin of measurement of xbar). What is local r at
that location of xbar (expressed in terms of a)?
Is ybar less than the local r (at the location of xbar); that is, is the centroid located
within the volume?
44
ES 101 Study Guide Module G 2021 Sept 17
REFLECT ON:
The centroid of a volume may coincide with the center of mass and center of gravity
of a rigid homogenous body in a unifiorm gravity field.
The centroid of an area or volume may correspond to the effective point of action of
distributed forces (or force components), if considering only external effects on a
body. See, for example, Module H.
The centroid of an area or volume may correspond to the effective point of action of
distributed stresses on a cross-section of a body; see, for example, ES 102 next sem.
:)
45
ES 101 Study Guide Module G 2021 Sept 17
”Gyradius” is our original coined [short] word to stand for “radius of gyration” of a
volume of rigid mass.
46
ES 101 Study Guide Module G 2021 Sept 17
FLASHBACK:
Which red dots approximate well the respective centroids of the individual letters of
CENTROID?
Now, if each letter is initially “SPINNED” about an axis perpendicular to this page and
passing through the red dot, which situation do you think will take more effort,
spinning about an axis through the centroid or spinning about another axis that is
parallel but non-centroidal?
47
ES 101 Study Guide Module G 2021 Sept 17
48
ES 101 Study Guide Module G 2021 Sept 17
49
ES 101 Study Guide Module G 2021 Sept 17
REFLECT ON:
The centroid of a volume may coincide with the center of mass and center of gravity
of a rigid homogenous body in a uniform gravity field.
In this lesson we focus on properties of the rigid MASS as distributed over the
volume.
50
ES 101 Study Guide Module G 2021 Sept 17
51
ES 101 Study Guide Module G 2021 Sept 17
Mass moment of inertia =
Moment of inertia of mass =
Second [-degree] moment of mass
“Second” because of the second power [or degree] on the distance r [from the
specified axis].
52
ES 101 Study Guide Module G 2021 Sept 17
53
ES 101 Study Guide Module G 2021 Sept 17
54
ES 101 Study Guide Module G 2021 Sept 17
55
ES 101 Study Guide Module G 2021 Sept 17
RECALL:
From Slide 1 earlier: Gyradius is the effective radius of an imaginary circle, where the
[total] mass seems to be concentrated at the perimeter when the body spins about
an axis at the center of that imaginary circle.
56
ES 101 Study Guide Module G 2021 Sept 17
REPEAT:
Gyradius is the effective radius of an imaginary circle, where the [total] mass seems
to be concentrated at the perimeter when the body spins about an axis at the center
of that imaginary circle.
57
ES 101 Study Guide Module G 2021 Sept 17
58
ES 101 Study Guide Module G 2021 Sept 17
Derivation of Eq (a):
Replace non-centroidal coordinates (y,z) in terms of centroidal coordinates (y,’ z’) plus
the coordinates of the centroid G itself relative to the non-centroidal origin O, (ybar,
zbar).
Expand the terms in the square bracket, and then distribute the integration over the
four terms.
Of the four integrals, recognize the second and third integrals as identically zero.
(Why? These two are now FIRST moments; they involve the FIRST power on distance.
Consider the second integral: this is the first moment with respect to the z’x’ plane;
but the z’x’ plane contains the centroid G, hence the particular first moment is
identically zero (similarly, see Slide 21). Similarly for the third integral.)
Of the four integrals, the first integral is just like definition of second moment with
respect to the centroidal axes y’z’ while the fourth integral is simply the total mass m.
Hence the final (simple) form of Eq (a) with only two terms.
Derivation of Eq (b) or Eq (c) is by procedure analogous to that for Eq (a). SO, WAIT A
SECOND, why is parallel-axes “theorem” so useful?
59
ES 101 Study Guide Module G 2021 Sept 17
IG (I sub-G) is alternative symbol for I bar.
Do you recall our question about spinning each letter of the word CENTROID? See
Slide 47.
If each letter is “spinned” about an axis perpendicular to the page and passing
through the red dot, which situation do you think takes more effort, spinning about
an axis through the centroid or spinning about a non-centroidal parallel axis? (Hint:
that effort is proportional to the I.)
60
ES 101 Study Guide Module G 2021 Sept 17
IG (I sub-G) is alternative symbol for I bar.
kG (k sub-G) is alternative symbol for k bar.
G is commonly used as label or subscript associated with the centroid of the volume
(of rigid mass).
The centroid is also the center of gravity when:
*the body is rigid;
*the body is homogeneous; and
*the gravity field is uniform.
Hmmm… do you wonder why this is Module G? J
61
ES 101 Study Guide Module G 2021 Sept 17
62
ES 101 Study Guide Module G 2021 Sept 17
63
ES 101 Study Guide Module G 2021 Sept 17
For the circular cone, see later Slide 79 with SP 9.11 for centroidal moment of inertia
of the mass about an axis perpendicular to x axis.
64
ES 101 Study Guide Module G 2021 Sept 17
Every axis x, y or z in this table is through the centroid.
65
ES 101 Study Guide Module G 2021 Sept 17
”DECOMPOSITE” SHAPE: A case of table “look-into”
Consider again the thin circular disk D as in this table, BUT WITH NEW NOTATION of
M for mass and R for radius: IxD = (1/2)MR2 which is centroidal.
Shall we look into the IxS of just the upper semicircular thin disk S, with respect to
exactly the same axis, which is non-centroidal? (We need to recall the definition of
the property being described in the formula.)
Let mass = m = (1/2) M and radius r = R. (We encounter this situation in SP 9.13.)
The said IxS of the semicircular thin disk (which happens to be non-centroidal) is HALF
of IxD of the thin disk (which happens to be centroidal):
IxS = (1/2) (1/2) MR2 = (1/2) (1/2) (2m) r2 = (1/2) m r2
The above result for a semicircular thin disk may look puzzling in form, as if
symbolically the same as for a circular thin disk; but the mass m now stands for the
mass of the semicircular disk (not as if the mass M of a whole circular disk of same
density and radius, which would be double in value: M=2m).
Our main reference book explains this (with different notations) in relation with SP
9.13, on its p. 519.
66
ES 101 Study Guide Module G 2021 Sept 17
”DECOMPOSITE” SHAPE: Another case of table “look-into”
67
ES 101 Study Guide Module G 2021 Sept 17
Every axis x, y or z in this table is through the centroid.
We associate a very small diameter for the rod, very small thickness (thinness?) for
the plate, …. and consider the centroid to be in the middle of that diameter or
thickness.
Similarly for every thin disk, plate, or lamina in the other slides.
68
ES 101 Study Guide Module G 2021 Sept 17
69
ES 101 Study Guide Module G 2021 Sept 17
70
ES 101 Study Guide Module G 2021 Sept 17
71
ES 101 Study Guide Module G 2021 Sept 17
REFLECT ON:
See Slide 68 and apply parallel-axes theorem, to derive the same result as in the slide
above.
Integration method is optional in ES 101; yet with your background in calculus or
elementary analysis of Math 22, you may use integration method to derive the
formulas for the first moment of any defined shape, or the second moment, with
respect to any specified axis.
Table look-up method and method of composite shape are required in ES 101. You
are REQUIRED to remember the formulas for those basic shapes that are presented
or used in this Study Guide, the Sample Problems, and the Assignment Problems.
72
ES 101 Study Guide Module G 2021 Sept 17
73
ES 101 Study Guide Module G 2021 Sept 17
74
ES 101 Study Guide Module G 2021 Sept 17
REFLECT ON:
See Slide 64 for the same result.
75
ES 101 Study Guide Module G 2021 Sept 17
EVALUATE:
Is the gyradius about axis x equal to radius a, less than a, or greater than a?
76
ES 101 Study Guide Module G 2021 Sept 17
77
ES 101 Study Guide Module G 2021 Sept 17
REFLECT ON:
See Slide 64 for the same result.
78
ES 101 Study Guide Module G 2021 Sept 17
EVALUATE:
With Slide 64, compare centroidal (y’’) and non-centroidal (y) moments of inertia.
Which one is smaller?
79
ES 101 Study Guide Module G 2021 Sept 17
80
ES 101 Study Guide Module G 2021 Sept 17
81
ES 101 Study Guide Module G 2021 Sept 17
82
ES 101 Study Guide Module G 2021 Sept 17
83
ES 101 Study Guide Module G 2021 Sept 17
Remember Slide 66? ”DECOMPOSITE” SHAPE: A case of table “look-into”
84
ES 101 Study Guide Module G 2021 Sept 17
Remember Slide 68? Use it together with parallel axes theorem for Iz that is noncentroidal (the axis is at the edge).
You may do similarly for Iy that is also non-centroidal (the axis is at the edge).
REFLECT ON: Do you get the same result of Iy as the ”shortcut” polar equation
Iy=Ix+Iz in this slide?
The “polar” axis shortcut equation is discussed in our main reference book in Section
9.13 Moments of Inertia of THIN PLATES, equation (9.38).
85
ES 101 Study Guide Module G 2021 Sept 17
REFLECT ON:
The parallel-axes theorem is perfectly applicable even when the non-centroidal axis
involved is outside the volume.
Here, axes y and z are “outside” the “circular plate.”
Does the Polar Axis Shortcut (see previous slide) work here, too, for Iy=Ix+Iz?
86
ES 101 Study Guide Module G 2021 Sept 17
87
ES 101 Study Guide Module G 2021 Sept 17
88
ES 101 Study Guide Module G 2021 Sept 17
89
ES 101 Study Guide Module G 2021 Sept 17
90
ES 101 Study Guide Module G 2021 Sept 17
91
ES 101 Study Guide Module G 2021 Sept 17
The pages here may have been plenty; we marked some (on integration method) as
optional reading for ES 101. (Integration method is optional in ES 101; yet with your
background in calculus or elementary analysis of Math 22, you may use integration
method to derive the formulas for the first moment of any defined shape, or the
second moment, with respect to any specified axis.)
Solve as many of the Assignment Problems (in the main reference book B10) as you
can within the budgeted time.
One of those is specified by instructors for Handwritten Problem-Solving submission
during the specific week when you choose to do the module HPS; see the UVLe tile
“Module G.”
Congratulations for finishing reading this Module G study guide. Now, the Problems.
J
92
ES 101 Study Guide Module G 2021 Sept 17
93