Department of Mechanical & Manufacturing Engineering

Faculty of Engineering, University Putra Malaysia

EMM 3802

Engineering Materials and Statics Laboratory

Title : Report of CENTER OF GRAVITY

Group Members :

1) Kok Yin Hui 1670272) Muhammad Hakim Bin Arif Fadzilah 1674033) Fatin Najihah Binti Mat Ali 1684354) Michelle Anne Anak Jimun 1687995) Hamdi Bin Nik Musa 1689056) Noraini Binti Md Isa 169697

Lecturer : Dr. Che Nor Aiza binti Jaafar

Demonstrator : Ms. Siti Ujila Masuri

Submission Date : 10th October 2012

1.0 INTRODUCTION AND THEORITICAL BACKGROUND

Center of gravity is defined as a geometric property of any object where all the weight of the objects can be considered to be concentrated. A body is composed of an infinite number of particles of differential size and is located within a gravitational field. Then, each of these particles will have a weight .A parallel force system will form and the result of this system is the total weight of the body which passes through a single point which is the center of gravity. In this experiment, we learn two methods in determining the center of gravity of several asymmetric shapes, which are through experimental measurements and theory calculations.

The first method is through plum line method, which also known as experimental method. The body is held by a pin inserted at a point near the body’s parameter so as to let it freely rotate around the pin. Then, the plumb line is dropped from the pin. Trace the plumline on the body and then step is repeat begin with the new point until the intersection of the lines is found which is the center of gravity.

Figure 1: Steps in Plumb line method

Another way is by using geometric decomposition which is used to locate the centroid of the body.

χ1=∑ χ i Ai

Aiy1=

∑ y i A i

A i

Where, x: coordinate of x-axisy: coordinate of y-axisA: area

Asymmetrical set

Plane board

Plumb line

Plumb bob

2.0 EXPERIMENTAL OBJECTIVES

The objectives of this experiment are to measure the center of gravity for asymmetrical shape using different methods which is the experiment measurements and theory calculations. Through experiment measurement, plumb line method is applied whereas for theory calculations, geometric decomposition is used.

Besides that, another objective is to relate between the effect of density and shapes in determining the center of gravity. It is important that both density and shape should be even so that we can get the precise value of the center of gravity of the asymmetrical shape.

3.0 LAB APPARATUS

Marker pen (was used for the outlining the line on the three different shapes). Ruler, paper and eraser (were used for the drawing of the three different shapes). Asymmetrical set (was used as the object samples for measuring the center of

gravity). Plumbline apparatus (were used for measuring the center of gravity).

Figure 2: Experimental apparatus

4.0 EXPERIMENTAL PROCEDURE

1. The plumbline apparatus was set up for the experiment. (as shown in Figure 2)2. The center of the gravity was measured on three different shapes using the

plumbline method.3. For the theory section, the center of the gravity was calculated for the three

different shapes from the previous step using the equation below.

χ1=∑ χ i Ai

Aiy1=

∑ y i A i

A i

4. The centroid coordinate from both methods were compared.

5.0 RESULT, ANALYSIS AND CALCULATION

Bodies 1:

The coordinate of center of gravity found using plumbline method: (103mm, 93mm).

The coordinate of center of gravity found using calculation method: (100mm,100mm)

χ1=∑ χ i Ai

Ai=

(100 mm ) (200 mm ) (200 mm )−(100 mm ) ( ԉ )(40 mm) ²(200 mm ) (200 mm )−( ԉ )(40 mm) ²

=100 mm

y1=∑ y i A i

A i=

(100 mm ) (200 mm ) (200 mm )− (100 mm ) ( ԉ)(40 mm) ²(200 mm ) (200 mm )− ( ԉ )(40 mm) ²

=100 mm

The different of center of gravity in x-coordinate: 103mm-100mm=3mm

The different of center of gravity in y-coordinate: 100mm-93mm=7mm

Bodies 2:

The coordinate of center of gravity found using plumbline method: (38mm, 0mm).

The coordinate of center of gravity found using calculation method: (100mm,100mm)

χ2=4 r3 ԉ

=4 (94)3 ԉ

=39.89 mm

y2=0 mm

The different of center of gravity in x-coordinate: 39.89mm-38mm=1.89mm

The different of center of gravity in y-coordinate: 0mm-0mm=0mm

Bodies 3:

The coordinate of center of gravity found using plumbline method: (100.5mm, 61.5mm).

The coordinate of center of gravity found using calculation method: (100mm,100mm)

χ3=∑ χ i Ai

Ai=

(100 mm ) (32 mm ) (170 mm )+(100 mm ) (200 mm )(30 mm)(32 mm ) (170 mm )+(200 mm )(30mm)

=100 mm

y3=∑ y i A i

A i=

(115 mm ) (32 mm ) (170 mm )+(15 mm ) (200 mm )(30 mm)(32 mm ) (170 mm )+ (200 mm )(30 mm)

=62.55 mm

The different of center of gravity in x-coordinate: 100mm-100.5mm=0.5mm

The different of center of gravity in y-coordinate: 62.55mm-61.5mm=1.05mm

Object

6.0 DISCUSSION

1. Is there a difference between the centroid coordinate obtained from these two methods? If yes, please discuss why.

There is a difference between the results obtained from calculation method and experimental method (Plumbline method) because experimental method subjects to environmental influences and affect its accuracy. The environmental influences include the friction between the bodies and the plane board, and the friction between the bodies and hanging knot will affect the bodies from perfect free hanging, thus results in different center of gravity. Besides, the air movement during conducting the experiment will create minor imbalance forces act on bodies’ surface, this will also gives an inaccurate center of gravity. Moreover, the human error in drawing the straight line on the bodies during free hanging is very skill dependent, moreover, the course marker pen’s tip is also reducing the accuracy of center of gravity.

2. What are the other methods available to determine the centroid for asymmetrical shapes? Please discuss in details with example for ONE method.

Another method in determining the center of gravity of an object is through simple balancing method. It is carried out by balancing the object on another object with small contact surface between them. The contact surface is where the center of gravity located. For example like placing an object onto the flat surface of a small cylindrical part (as shown below). For more precision, we should note down the area of surface contact and continue the experiment trying with smaller and smaller contact surface. However, this experiment is encouraged to be carried out in a closed room with minimum air movement disturbance for better result accuracy.

3. In what situation will the plumbline method be more precise in determining the centre of gravity compared to the equation calculation?

Suspected center of gravity

The plumline method will be more precise if applied on object with uneven thickness or uneven density distribution. If the object measured is designed in uneven thickness of uneven density, more consideration have to be included for the calculation method and thus will comes out complicated calculation as well, thus, chances of making mistake is greater. Besides, some complex shape’s object is also encouraged to use plumbline method in finding the center of gravity to avoid complicated calculation.

4. What is the correlation between density and shape in calculating the center of gravity?

Density means mass per unit volume. If the density of an object is uneven distribute, the center of gravity will be located more toward the denser part because that particular part results in higher mass value. Similarly, the center of gravity will also locate at the side of larger shape object. For example of object drawn below (assumed top view of object with 10mm thickness), the center of gravity will locate toward the right because it has larger shape of size which results in heavier mass value. In conclusion, center of gravity grows towards denser and larger shape’s side of object.

5. Discuss any possible error occurred during experiment.

Parallax error: This error may occur when the eye of the observer is not perpendicular to the object. It may cause them to accidently constructing the line when the shapes still hanging on the plumbline incorrectly.Object error: The marker pen used for the experiment is more likely to have thick tip which may cause the constructed line for center of gravity to be in disproportion. This can cause the center of gravity to be inaccurately measured when the constructed lines are already misallocated.Nature error: This error may happen during the experiment when the wind from the surrounding blows the hanging shape on the plumbline apparatus. This can cause the center of gravity slightly misplace and cause the observer to measure it imperfectly.

7.0 CONCLUSION

Based on the experiment, we can conclude that center of gravity is the point at where the entire weight of the object concentrated, in another word, it is the point in balancing the object as well. Locating the center of gravity can be done in two methods which are plumbline method and geometry decomposition calculation.

Both the methods have its pros and cons, and the results obtained are slightly different. This can be explained as the result of surrounding disturbance during plumbline method, which included air movement, friction between plane board and object, friction between hanging pin and object and some human errors. For the calculation method, it is more precise and efficient as it does not require any apparatus. However, the calculation method is limited to simple objects, in which complex shape objects or objects with uneven thickness and density may cause difficulty in calculate.

In engineering field, center of gravity is important as to ensure an object design with high stability. For example, stability of an automobile is extremely important to the driver especially during making a sharp corner or emergency brake in order to ensure the safety of the user. Thus mastering the basic knowledge about the center of gravity is achieved in this lab.