Eas 151 Cog, Com, Centroid.pdf

1. Understand the concept of CoG, CoM and the Centroid; 2. Locate the location of CoG, CoM and the Centroid for a system of discrete particles and a body;

1. Discuss the concept of CoG, CoM and the Centroid. 2. Determine the location of CoG, CoM and the Centroid of a body. 3. Determine the location of CoG, CoM and the Centroid of composite bodies.

To design the structure for supporting a water tank, we will need to know the weight of the tank and water as well as the locations where the resultant forces representing these distributed loads act. How can we determine these resultant weights and their lines of action?

A water tank

One concern about a sport utility vehicle (SUV) is that it might tip over while taking a sharp turn.

One of the important factors in determining its stability is the SUV’s center of mass. Should it be higher or lower to make a SUV more stable? How do you determine the location of the SUV’s center of mass?

To design the ground support structure for the goal post, it is critical to find total weight of the structure and the center of gravity’s location. Integration must be used to determine total weight of the goal post due to the curvature of the supporting member.

The goal post

How do you determine the location of overall center of gravity?

1. The_________ is the point defining the geometric center of an object. A) Center of gravity

B) Center of mass

C) Centroid

D) None of the above.

1. The_________ is the point defining the geometric center of an object. A) Center of gravity

B) Center of mass

C) Centroid

D) None of the above.

2. To study problems concerned with the motion of matter under the influence of forces, i.e., dynamics, it is necessary to locate a point called____________________.

A) Center of gravity

B) Center of mass

C) Centroid

D) None of the above.

1. The_________ is the point defining the geometric center of an object. A) Center of gravity

B) Center of mass

C) Centroid

D) None of the above.

2. To study problems concerned with the motion of matter under the influence of forces, i.e., dynamics, it is necessary to locate a point called____________________.

A) Center of gravity

B) Center of mass

C) Centroid

D) None of the above.

• A body is composed of an infinite number of particles. • If the body is located within a gravitational field, then each of these particles will have a weight dW. • The center of gravity (CoG) is a point, denoted G, which locates the resultant weight of a system of particles or a solid body. • From the definition of a resultant force, the sum of moments due to individual particle’s weights about any point is the same as the moment due to the resultant weight located at G. • [Note that the sum of moments due to the individual particle’s weights about point G is equal to zero].

• The location of the CoG, measured from the y axis, is determined by equating the moment of W about the y axis to the sum of the moments of the weights of the particles about this same axis. • If dW is located at point 𝑥 , 𝑦, 𝑧 , then 𝒙𝑾 = • Similarly; 𝒚𝑾 =

𝒙𝒅𝑾

𝒚𝒅𝑾and 𝒛𝑾 =

𝒛𝒅𝑾;

• Therefore, the location of the center of gravity G with respect to the x, y, z axes becomes:

By replacing the W with mg in the equations, the coordinates of the CoM can be found.

Similarly, the coordinates of the centroid of volume, area, or length can be obtained by replacing W by V, A, or L, respectively.

• The centroid, C, is a point defining the geometric center of an object. • The centroid coincides with the CoM or the CoG only if the material of the body is homogenous (density or specific weight is constant throughout the body). • If an object has an axis of symmetry, then the centroid of object lies on that axis. • In some cases, the centroid may not be located on the object.

Centroid (Volume) Centre of Gravity Centroid (Area)

Centre of Mass Centroid (Line)

Differential elements 1. Choose an appropriate differential element dA at a general point (x,y). 2. Generally, if y is easily expressed in terms of x (e.g., y = x2 + 1), use a vertical rectangular element. 3. If the converse is true, then use a horizontal rectangular element. 4. Express dA in terms of the differentiating element dx (or dy).

5. Determine coordinates 𝑥 , 𝑦 of the centroid of the rectangular element in terms of the general point (x,y). Integrations 6. Express all the variables and integral limits in the formula using either x or y depending on whether the differential element is in terms of dx or dy, respectively, and integrate. [Note: Similar steps are used for determining the CoG or CoM.].

Determine the centroid for the area shown below.

• Differential elements: • Horizontal element; • Vertical element. to determine 𝒙, 𝒚 . • Or, consider: • Horizontal/ vertical element to solve for 𝒙; • Vertical/ horizontal element to solve for 𝒚.

Determine the centroid for the area shown below.

Horizontal element

Determine the centroid for the area shown below.

Vertical element

Determine the centroid for the area shown below by considering horizontal differential element.

The centroid of an area is given by:

Define an area dA: 𝒅𝑨 = 𝒙 𝒅𝒚

From: 𝒚 =

𝒉 𝒃

𝒃−𝒙

𝒙=𝒃 𝟏−

𝒚 𝒉

Hence, 𝒅𝑨 = 𝒃 𝟏 −

𝒚 𝒉

𝒅𝒚

𝒙=

𝒙=

𝒙=

𝑨 𝑨 𝒉 𝟎

𝒙𝒅𝑨 𝒅𝑨

𝒚 𝒙 𝟐 ∙𝒃 𝟏 − 𝒅𝒚 𝒉 𝒉 𝒚 𝒃 𝟏 − 𝒅𝒚 𝟎 𝒉

𝒉𝒃 𝟎 𝟐

𝟏−

𝒚 𝒚 ∙𝒃 𝟏− 𝒅𝒚 𝒉 𝒉 𝒃𝒉 𝟐

𝒃𝟐 𝒉 ∙ 𝒃 𝟐 𝟑 𝒙= = 𝒃𝒉 𝟑 𝟐

𝒚=

𝒚=

𝑨

𝒚𝒅𝑨 𝒅𝑨

𝑨 𝒉 𝒃 𝒚 𝟎 𝒉 𝒉𝒃 𝟎 𝒉 𝟐

𝒉 − 𝒚 𝐝𝐲 𝒉 − 𝒚 𝐝𝐲

𝒃𝒉 𝒚= 𝟔 𝒃𝒉 𝟐 𝒉 𝒚= 𝟑 The centroid of the area is 𝒃 𝒉 , 𝟑 𝟑

Determine the centroid for the area shown below by considering vertical differential element.

The centroid of an area is given by:

Define an area dA: 𝒅𝑨 = 𝒚 𝒅𝒙

From: 𝒚 =

𝒉 𝒃

𝒃−𝒙

Hence, 𝒉 𝒅𝑨 = 𝒃 − 𝒙 𝒅𝒙 𝒃

𝒙=

𝒙𝒅𝑨

𝑨 𝑨

𝒙=

𝒅𝑨

𝒃 𝒉 𝒙 ∙ 𝟎 𝒃 𝒃𝒉 𝟎 𝒃

𝒃 − 𝒙 𝒅𝒙

𝒃 − 𝒙 𝒅𝒙

𝒉𝒃𝟐 𝒃 𝟔 𝒙= = 𝒉𝒃 𝟑 𝟐

𝒚=

𝒚𝒅𝑨

𝑨 𝑨

𝒚=

𝒚=

𝒅𝑨

𝒃𝒚 𝟎 𝟐

𝒉 𝒃 − 𝒙 𝒅𝒙 𝒃 𝒃𝒉 𝒃 − 𝒙 𝒅𝒙 𝟎 𝒃 ∙

𝒃 𝒉 𝟎 𝟐𝒃 𝟐

𝒃−𝒙 ∙

𝒉 𝒃 𝒉 𝟔 𝒚= = 𝒉𝒃 𝟑 𝟐

𝒉𝒃

𝒉 𝒃 − 𝒙 𝒅𝒙 𝒃 𝟐

The centroid of the area is 𝒃 𝒉 , 𝟑 𝟑

To determine 𝒙, it is easy to consider a vertical element as a differential element.

To determine 𝒚, it is easy to consider a horizontal element as a differential element.

Determine the centroid for the area shown below.

• Differential elements, consider: • Vertical element to solve for 𝒙; • Horizontal element to solve for 𝒚.

Determine the centroid for the area shown below.

𝒅𝑨 = 𝒚 𝒅𝒙 = 𝒙𝟐 𝒅𝒙 𝟏

𝒅𝑨 = 𝑨

𝟎

𝒙𝟐 𝒅𝒙 =

𝟏 𝟐 𝐦 𝟑

𝒙𝒅𝑨 𝒙= 𝟏 𝟑 𝟏 𝒙 ∙ 𝒙𝟐 𝒅𝒙 𝟎 𝒙= 𝟏 𝟑 𝟏 𝟑 𝟒 𝒙 = = 𝐦 = 𝟎. 𝟕𝟓 𝐦 𝟏 𝟒 𝟑 𝑨

Determine the centroid for the area shown below.

𝒅𝑨 = 𝟏 − 𝒙 𝒅𝒚 = 𝟏 − 𝒚 𝒅𝒚 𝟏

𝒅𝑨 = 𝑨

𝟎

𝟏 𝟐 𝟏 − 𝒚 𝒅𝒚 = 𝐦 𝟑

𝒚𝒅𝑨 𝒚= 𝟏 𝟑 𝟏 𝒚 ∙ 𝟏 − 𝒚 𝒅𝒚 𝟎 𝒚= 𝟏 𝟑 𝟏 𝟑 𝟏𝟎 𝒚= = 𝐦 = 𝟎. 𝟑 𝐦 𝟏 𝟏𝟎 𝟑 𝑨

1. The steel plate with known weight, nonuniform thickness and density is supported as shown. Of the three parameters (CoG, CoM, and Centroid), which one is needed for determining the support reactions? Are all three parameters located at the same point?

A) B) C) D)

(Center of gravity, Yes) (Center of gravity, No) (Centroid, Yes) (Centroid, No)

1. The steel plate with known weight, nonuniform thickness and density is supported as shown. Of the three parameters (CoG, CoM, and Centroid), which one is needed for determining the support reactions? Are all three parameters located at the same point?

A) B) C) D)

(Center of gravity, Yes) (Center of gravity, No) (Centroid, Yes) (Centroid, No)

2. When determining the centroid of the area above, which type of differential area element requires the least computational work? A)

Vertical

C) Polar

B) Horizontal D) Any one of the above

1. The steel plate with known weight, nonuniform thickness and density is supported as shown. Of the three parameters (CoG, CoM, and Centroid), which one is needed for determining the support reactions? Are all three parameters located at the same point?

A) B) C) D)

(Center of gravity, Yes) (Center of gravity, No) (Centroid, Yes) (Centroid, No)

2. When determining the centroid of the area above, which type of differential area element requires the least computational work? A) Vertical

B) Horizontal

C) Polar

D) Any one of the above

Given

:The steel plate is 0.3 m thick and has a density of 7850 kg/m3.

Find

: The location of its center of mass. Also compute the reactions at A and B.

Plan

: Follow the solution steps to find the CoM by integration. Then use 2-dimensional equations of equilibrium to solve for the external reactions.

Solution for CoM: 1. Choose dA as a vertical rectangular strip. 2.

𝑑𝐴 = 𝑦2 − 𝑦1 𝑑𝑥 =

3.

𝑥=𝑥

4.

𝑦=

𝑦2 +𝑦1 2

=

2𝑥 − −𝑥 𝑑𝑥

2𝑥−𝑥 2

Solution for Support Reactions: 1. Place the weight of the plate at the centroid C. 2. Area, A = 4.667 m2;

3. Weight, W = (7850) (9.81) (4.667) 0.3 = 107.8 kN 4. Applying Equations of Equilibrium

+   FX = – Ax + 47.92 sin 45 = 0 AX = 33.9 kN FBD to find the reactions at A and B.

+   FY = Ay + 47.92 cos 45 – 107.8 = 0 AY = 73.9 kN

1. If a vertical rectangular strip is chosen as the differential element, then all the variables, including the integral limit, should be in terms of _____ . A) x

B) y

C) z

D) Any of the above.

1. If a vertical rectangular strip is chosen as the differential element, then all the variables, including the integral limit, should be in terms of _____ . A) x

B) y

C) z

D) Any of the above.

2. If a vertical rectangular strip is chosen, then what are the values of 𝑥 and 𝑦? 𝑥 𝑦 , 2 2

A)

𝑥, 𝑦

B)

C)

𝑥, 0

D) 𝑥,

𝑦 2

1. If a vertical rectangular strip is chosen as the differential element, then all the variables, including the integral limit, should be in terms of _____ . A) x

B) y

C) z

D) Any of the above.

2. If a vertical rectangular strip is chosen, then what are the values of 𝑥 and 𝑦? 𝑥 𝑦 , 2 2

A)

𝑥, 𝑦

B)

C)

𝒙, 𝟎

D) 𝑥,

𝑦 2

• A composite body consists of a series of connected “simpler” shaped bodies. • Such a body can often be sectioned or divided into its composite parts. • To determine the CoG for the entire body:

The gravity wall is made of concrete. Determine the location 𝑥 , 𝑦 of the CoG for the wall

Composite parts:

3 2

1

4

3

4

1

2

Sec.

A

1

9

2.1

1.9

18.9

17.1

2

1.44

1.8

0.2

2.592

0.288

3

-2.7

1.2

2.4

-3.24

-6.48

4

-0.9

3.4

1.4

-3.06

-1.26

Total

6.84

15.192

9.648

𝑥 = 2.22 m 𝑦 = 1.41 m