COG Assignment - Practical report on Centre of Gravity along with l PDF

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Practical report on Centre of Gravity along with l...

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Introduction The aim of this experiment was to determine the position of the centre of gravity of a two-dimensional geometric shape using the theory of centre of gravity, as well as to compare the results from the experiment with the result obtained by calculations. Centre of gravity; according to R.C. Hibbeler, a body is composed of an infinite number of particles of differential size, and so if the body is located within a gravitational field, the each of these particles will have weight. These weights will approximately form a parallel force system and the resultant of this system is the total weight of the body which passes through a single point, which is called centre of gravity. The experiment’s requirements were to demonstrate the centre of gravity of a 2D geometric shape.

Table 1: Centre of Gravity for Individual Simple Shapes Simple Shapes 1.

2.

3.

Dimensions of shape

Area

COG of in x-directions

Area times COG for shape (AnXn)

d= 101mm

A 1=

X1=50.5mm A1X1 =

r= 50.5mm

4005.92 mm2

L=101mm

A 2=

B=100mm

10 100 mm2

d= 101mm

A 3=

r= 50.5mm

4005.92 mm2

202 298.96 mm2

Sum (A1 to A4) =

Sum (A1X1 to A 3X 3) =

Sum (A1Y1 to A 3Y 3) =

18 111.84mm2

914 647.92 mm2

905 592 mm2

202 298.96mm2

X2=50.5mm A2X2 = 510 050mm2

X3=50.5mm A3X3 =

COG of in Area times COG y-direction for shape (AnYn) Y 1=

A 1Y 1=

121.4mm

486 318.69mm2

Y 2=

A 2Y 2=

50mm

505 000 mm2

Y 3=

A 3Y 3=

-21.4mm

-85 726.69 mm2

Table 2: Centre of Gravity for the Whole Plate Complete shape (Draw the whole plate)

Measured COG for whole plate (X)

Calculated COG for whole plate(X)

% difference between measured and calculated

50.3mm

50.5mm

0.4%

Measured COG for whole plate(Y)

Calculated COG for whole plate(Y)

% difference between measured and calculated

51mm

50mm

1.96%

Theory and analytical calculations and results 1.Semi-Circle: Dimensions: diameter = 101 mm radius= 50,5 mm Area = πr2/2 =π(50)2/2 =4005,92 mm2 *measured value of C.O.G (centre of gravity) in the X-direction = 50,5 mm AnXn = 4005,92 mm2 × 50,5 mm =202 298,9 6mm3 *measured value of C.O.G in the Y-direction = 121,4mm AnYn = 4005,92 mm2 × 121,4 mm = 121,4 mm3

2. Rectangle: Dimensions: length = 101mm Breadth = 100mm Area = length × breadth = 101 mm × 100 mm = 10 100 mm2 *measured value of C.O.G in the X-direction = 50,5 mm AnYn = 10 100 mm2 × 50,5 mm = 510 050 mm3 *measured value of C.O.G in the Y-direction = 50mm AnYn = 10 100 mm2 × 50 mm = 505 000 mm3

3.Semi-Circle: Dimensions: diameter = 101 mm radius= 50,5 mm Area = πr2/2 = π(50)2/2 =4005,92 mm2 *measured value of C.O.G in the X-direction = 50,5 mm AnXn = 4005,92 mm2 × 50,5 mm =202 298,96 mm3 *measured value of C.O.G in the Y-direction = -21,4 mm AnYn = 4005,92 mm2 × -21,4 mm = -85 726,69 mm3

Experimental Analysis, calculations and results Calculated C.O.G for whole plate(X): Using the following formula, the “X” coordinate can be determined. x = A1X1 + A2X2 + A3X3 / A1 + A2 + A3 x = 914 647,92 mm2 / 18 111,84 mm2 x = 50,5 *measured value of C.O.G for the whole plate (x) = 50,3 mm Percentage difference = (50,5 – 50,3) ÷ 50,3 × 100 = 0,40%

Calculated C.O.G for whole plate(Y): Using the following formula, the “Y” coordinate can be determined. y = A1Y1 + A2Y2 + A3Y3 / A1 +A2 + A3 y = 905 592 mm2 / 18 111,84 mm2 y = 50 *measured value of C.O.G for the whole plate (y) = 51 mm Percentage difference = (50 – 51) ÷ 51 × 100 = 1,96%

Discussion From the recorded results on the table one can clearly see that the results obtained by experiment and calculation are slightly the same. The aim of the experiment was to determine experimentally the position of the centre of gravity of the shape and verifying the results by calculation, and due to the results, the successful experiment was achieved. The results show that the centre of gravity can either be obtained using calculation or analytically by experiment. These methods were both used on the geometrical shape to obtain the centre of gravity of the shape. The results that were obtained showed that the centre of the three shapes that made the 2D geometric shape were around the same point with the maximum error of 1.96% on this scale. These errors may have arisen simply because of the lack of accuracy when tracing the lines on the board to the paper when trying to the point of the centre of the geometric shape to the paper. Two semi circles and one rectangle shapes were used in determining the Centre of gravity for the shape. The big shape was then split into three shapes as named before. The radii of the two semi centres was then measured and they were used in determining the areas of the two semi circles separately even though they are equal. The dimensions were then done for the rectangle and the area for it was then calculated. Below lays a comparative table of values obtained by geometric calculation and experimental methods: Shape number

Method Chosen

Distance on x axis from Distance on y axis from “Ο" (mm) “Ο" (mm)

1 – Semi-circle

Calculated

50.5mm

121.4mm

Experimental

50.3mm

121.64mm

Calculated

50.5mm

50 mm

Experimental

50.3mm

51 mm

Calculated

50.5mm

21.4mm

Experimental

50.3mm

21.34mm

2 – Rectangle

3 –Semi-circle

To find the coordinates for the entire shape(big) the x-coordinates for the small shapes are added together and divided by the sum of the areas of the shapes to find the x-coordinate for the big shape [X = A1X1 + A2X2 + A3X3 / A1 + A2 + A3]. To find the y-coordinate for the big shape, the y-coordinates are added together and divided by the sum of the tree areas. [ Y = A1Y1 + A2Y2 + A3Y3 / A1 +A2 + A3] The x-coordinate that is calculated for the entire shape is a bit bigger than the one measured [measured value of C.O.G for the whole plate (x) = 50,3 mm < 50,5mm] making a percentage error of (0,40). •

Percentage difference = (50,5 – 50,3) ÷50,3 × 100 = 0,40%

The y-coordinate that is calculated is less than the one which was measured [measured value of C.O.G for the whole plate (y) = 51 mm > 50mm] making a percentage error of 1,96. •

Percentage difference = (50 – 51) ÷51 × 100 = 1,96%

As mentioned before in the experimental analysis, the percentage difference between the theoretical and experimental calculations for the X and Y axis were 0.40% and 1.96% respectively. The result meaning will prove the theory of the centre of gravity and the significant of the accurate result is very important, the calculated value must be close to the measured value by a percentage difference that is less 5%. There is a degree of inaccuracy, however it is still oblivious to see that the measured results versus the calculated results are very similar. These calculations and results are so important to the experiment as without them, the success of the experiment could not be proved.

Conclusion Overall, the experiment was a success. From this investigation, one can learn to appreciate the necessity of knowing the centre of gravity of a body. The measured values and calculated values had inconsistencies, however, if working to a high level of accuracy was required then one would select the calculation method. On the other hand, the experimental method was an exceptional technique for verification of the centre of mass. Human judgement was also required at many times during this investigation, lighting conditions and human error in judgement could have also caused inaccuracies. For example, shadows being cast of the line, increased distance of the line from the page and perplexing are possible areas of awareness. The strengths of this experimental method are that is it fast, fairly accurate and simple to determine a centre of a mass. If this investigation was to be repeated, one could possible adapt the moving parts with friction pads in order to stop/lock the mass from rotating while marking the points. A simple push button on the reverse side of the fixture/mount can be a placed in a way to provide a “locking” mechanism. The vertical plum line could also be temporarily fixed after initial calibration and alignment to the base of the vertical board. This reduces the chances of disturbing the line while marking the points. From this investigation revealed the importance of knowing a centre of mass of a body, it can aid one’s understanding of how everyday objects have been designed with centre of gravity in mind. However, despite all this, fairly accurate results were achieved.

Recommendations Looking at the results achieved in the experiment, it is safe to say that the experiment was a success. The message difference between measured and calculated results were less than 2%. To help improve the accuracy of this experiment, what should be recommended is that multiple drawings of the shape should be taken along with different members of a group drawing lines of centre of gravity, and then taking an average across all the drawings. Other methods to calculate centre of gravity could be used to increase accuracy. The use of better measuring equipment would be a recommendation to improve the experiment, resulting in more accurate results as errors could lie in the equipment used, being worn, internally damaged or not well lubricated, for example the ball bearing mount. The plumb line could have been chafing or rubbing against the plastic shapes while settling vertical and may not have been brought to the investigator’s attention. Other factors such as wind movements and magnetic disturbance due to the ferrite conditions of the plum line weight “plum bob” are inevitable and infinitesimal....