View
229
Download
4
Category
Preview:
Citation preview
CENTROID
CENTRE OF GRAVITY
Centre of gravity : of a body is the point at which the whole weight of the body may be assumed to be concentrated.
It is represented by CG. or simply G or C.
A body is having only one center of gravity for all positions of the body.
1
Contd.www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 1
CENTRE OF GRAVITY
Consider a three dimensional body of any size and shape, having a mass m.
If we suspend the body as shown in figure, from any point such as A, the body will be in equilibrium under the action of the tension in the cord and the resultant W of the gravitational forces acting on all particles of the body.
2
Contd.www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 2
Resultant W is collinear with the Cord
Assume that we mark its position by drilling a hypothetical hole of negligible size along its line of action
Cord
Resultant
CENTRE OF GRAVITY
3
Contd.www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 3
To determine mathematically the location of the centre of gravity of any body,
Centre of gravity is that point about which the summation of the first moments of the weights of the elements of the body is zero.
we apply the principle of moments to the parallel system of gravitational forces.
CENTRE OF GRAVITY
6
Contd.www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 4
We repeat the experiment by suspending the body from other points such as B and C, and in each instant we mark the line of action of the resultant force.
For all practical purposes these lines of action will be concurrent at a single point G, which is called the
centre of gravity of the body.
CENTRE OF GRAVITY
4 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 5
w
A
A
w
B
AG
B
A
w
AB
CGB
A
C
C
B
Example:CENTRE OF GRAVITY
5 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 6
if, we apply principle of moments, (Varignon’s Theorem) about y-axis, for example,
The moment of the resultant gravitational force W, about any axis
=the algebraic sum of the moments about the same axis of the gravitational forces dW acting on all infinitesimal elements of the body.
dWxWhere W =
dW
Wx
The moment of the resultant about y-axis =
The sum of moments of its components about y-axis
CENTRE OF GRAVITY
7 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 7
where = x- coordinate of centre of gravity
x
x
x
W
dWxx
Similarly, y and z coordinates of the centre of gravity are
W
dWyy
W
dWzz and ----(1)
CENTRE OF GRAVITY
8 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 8
x
W
dWxx
W
dWyy
W
dWzz
With the substitution of W= m g and dW = g dm
m
dmxx
m
dmyy
m
dmzz
----(1)
----(2),,
,,
(if ‘g’ is assumed constant for all particles, then )
the expression for the coordinates of centre of gravity become
CENTRE OF MASS
9
Contd.www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 9
dV
dVxx
dV
dVyy
dV
dVzz
and ----(3)
If ρ is not constant throughout the body, then we may write the expression as
,
CENTRE OF MASS
The density ρ of a body is mass per unit volume. Thus, the mass of a differential element of volume dV becomes dm = ρ dV .
10
Contd.www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 10
m
dmxx
m
dmyy
m
dmzz ----(2),,
This point is called the centre of mass and clearly coincides with the centre of gravity as long as the gravity field is treated as uniform and parallel.
CENTRE OF MASS
Equation 2 is independent of g and therefore define a unique point in the body which is a function solely of the distribution of mass.
11 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 11
When the density ρ of a body is uniform throughout, it will be a constant factor in both the numerators and denominators of equation (3) and will therefore cancel.The remaining expression defines a purely geometrical property of the body.
dV
dVxx
dV
dVyy
dV
dVzz
and, ----(3)
CENTROID
12 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 12
When speaking of an actual physical body, we use the
term “centre of mass”.
Calculation of centroid falls within three distinct
categories, depending on whether we can model the
shape of the body involved as a line, an area or a
volume.
The term centroid is used when the calculation concerns
a geometrical shape only.
13
Contd.www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 13
LINES: for a slender rod or a wire of length L, cross-sectional area A, and density ρ, the body approximates a line segment, and dm = ρA dL. If ρ and A are constant over the length of the rod, the coordinates of the centre of mass also becomes the coordinates of the centroid, C of the line segment, which may be written as
L
dLxx
L
dLyy
L
dLzz
The centroid “C” of the line segment,
,,
14
Contd.www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 14
AREAS: when the density ρ, is constant and the body has a small constant thickness t, the body can be modeled as a surface area. The mass of an element becomes dm = ρ t dA.
If ρ and t are constant over entire area, the coordinates of the ‘centre of mass’ also becomes the coordinates of the centroid, C of the surface area and which may be written as
A
dAxx
A
dAyy
A
dAzz
The centroid “C” of the Area segment,
,,
15Contd.www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 15
VOLUMES: for a general body of volume V and density ρ, the element has a mass dm = ρ dV .
If the density is constant the coordinates of the centre of mass also becomes the coordinates of the centroid, C of the volume and which may be written as
V
dVxx
V
dVyy
V
dVzz
The centroid “C” of the Volume segment,
, ,
16
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 16
Centroid of Simple figures: using method of moment ( First moment of area)
Centroid of an area may or may not lie on the area in question.
It is a unique point for a given area regardless of the choice of the origin and the orientation of the axes about which we take the moment.
17 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 17
Moment of Total area ‘A’ about y-axis
=Algebraic Sum of moment of elemental ‘dA’ about the same axis
where (A = a1 + a2 + a3 + a4 + ……..+ an)
(A) x = (a1) x1 + (a2) x2 + (a3) x3 +
……….+(an) xn
= First moment of area
The coordinates of the centroid of the surface area about any axis can be calculated by using the equn.
18 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 18
If an area has an axis of symmetry, then the centroid must lie on that axis.If an area has two axes of symmetry, then the centroid must lie at the point of intersection of these axes.
AXIS of SYMMETRY:
It is an axis w.r.t. which for an elementary area on one side of the axis , there is a corresponding elementary area on the other side of the axis (the first moment of these elementary areas about the axis balance each other)
19Contd.www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 19
For example:
The rectangular shown in the figure has two axis of symmetry, X-X and Y-Y. Therefore intersection of these two axes gives the centroid of the rectangle.
B
DD/2
D/2
B/2 B/2
X X
Y
Y
xx
dada
da × x = da × x
Moment of areas,da about y-axis cancel each other
da × x + da × x = 020
Contd.www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 20
AXIS of SYMMETYRY
‘C’ must lie at the intersectionof the axes of symmetry
‘C’ must lie on the axis
of symmetry
‘C’ must lie on the axis of symmetry
21 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 21
EXERCISE PROBLEMS
Locate the centroid of the shaded area shown
Problem No.1:
Ans: x=12.5, y=17.5
22
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 22
Locate the centroid of the shaded area shown
Problem No.2:
D=600
Ans: x=474mm, y=474mm23
EXERCISE PROBLEMS
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 23
Locate the centroid of the shaded area w.r.t. to the axes shown
Problem No.3:
Ans: x=34.4, y=40.324
EXERCISE PROBLEMS
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 24
Locate the centroid of the shaded area w.r.t. to the axes shown
Problem No.4:
Ans: x= -5mm, y=282mm
10
25
EXERCISE PROBLEMS
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 25
Locate the centroid of the shaded area w.r.t. to the axes shown
Problem No.5
Ans:x =38.94, y=31.46
30
26
EXERCISE PROBLEMS
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 26
Locate the centroid of the shaded area w.r.t. to the axes shown
Problem No.6
x
y
Ans: x=0.817, y=0.2427
EXERCISE PROBLEMS
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 27
Problem No.7
Locate the centroid of the shaded area w.r.t. to the axes shown
Ans: x= -30.43, y= +9.5828
EXERCISE PROBLEMS
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 28
Problem No.8
Locate the centroid of the shaded area.
Ans: x= 0, y= 67.22(about base)
20
29
EXERCISE PROBLEMS
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 29
Problem No.9
Locate the centroid of the shaded area w.r.t. to the base line.
Ans: x=5.9, y= 8.17
2
30
EXERCISE PROBLEMS
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 30
Problem No.10
Locate the centroid of the shaded area w.r.t. to the axes shown
Ans: x=21.11, y= 21.1131
EXERCISE PROBLEMS
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 31
Problem No.11
Locate the centroid of the shaded area w.r.t. to the axes shown
Ans: x= y= 22.2232
EXERCISE PROBLEMS
www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 32
Recommended